70 - 14,070

MATRE, Marc David, 1937- SOME IMPLICATIONS OF THE DEVELOPMENT OF THE PUERTO RICAN HIGHWAY SYSTEM FOR THE URBAN ECOLOGY OF .

The State University, Ph.D., 1969 Sociology, regional and city planning

University Microfilms, Inc., Ann Arbor, Michigan

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED SOME IMPLICATIONS OF THE DEVELOPMENT OF THE PUERTO RICAN

HIGHWAY SYSTEM FOR THE URBAN ECOLOGY OF PUERTO RICO

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Marc David Matre, B.A., M.A. .

The Ohio State University 1969

Approved By

Adviser Department of Sociology ACKNOWLEDGMENTS

This research was supported with a dissertation fellowship from the Mershon Center for Education in National Security. Dr. James

A. Robinson, director of the Mershon Center, Mrs. Anne F. Trupp, assistant to the director, and other persons at the Mershon Center helped me in many ways during the research process. Mrs. Kay Neves and Mrs. Gloria Werth did the typing of the dissertation. I am grateful to all of these people for creating an atmosphere of freedom, cooperation, and encouragement.

I am grateful to Dr. Kent P. Schwirian for the many ways he has helped me as both adviser and teacher. I also thank the members of my dissertation committee, Dr. Raymond F. Sletto, Dr. Patrick T.

Cleaver, and Dr. Donald L. Noel.

Members of the staff of the Ohio State University computer center gave me timely assistance during several phases of the data processing. I was helped especially by Mrs. Carol Estep and Mr.

Pravin Gandhi.

Mr. Henry J. Harm, of the General Drafting Company, responded generously to a request I made for a series of road maps of Puerto Rico.

These naps were a very useful source of information about the Puerto

Rican road network.

ii VITA

Born: Hamilton, Ohio 1 May 1937

B.A., Miami University, Oxford, Ohio 1960

M.A., The Ohio State University 1966

FIELDS OF STUDY

Major Field: Urban Sociology

Studies in Urban Sociology and Urban Ecology. Professors Kent P. Schwirian and Christen T. Jonassen

Studies in Research Methods. Professors Raymond F. Sletto, Robert P. Bullock, Patrick T. Cleaver, and Kent P. Schwirian

Studies in Race and Ethnic Relations. Professors Donald L. Noel, Brewton Berry, and James W. Vander Zanden

iii TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ...... ii

VITA ...... iii

LIST OF T A B L E S ...... vi

LIST OF ILLUSTRATIONS...... vii

Chapter

I. INTRODUCTION TO THE RESEARCH PROBLEM ...... 1

II. THEORY OF REGIONAL URBAN ECOLOGY ...... 5

Background Literature The Object of Study

III. BASIC CONCEPTS AND OPERATIONAL DEFINITIONS...... 15

The Urban Center Population Size Occupations and Division of Labor Literacy Graph Theoretic Concepts and Definitions Applications of the Graph Concepts and Definitions to the Puerto Rican Road Network Data

IV. PROPOSITIONS RELATING BASIC CONCEPTS ...... 63

V. PRESENTATION OF THE D A T A ...... 68

The Data Tables Descriptive Summary of the Tabled Data

iv VI. TESTS OF THE HYPOTHESES

The Notation for Partial Correlations The Cross-lagged Panel Comparisons The Calculation of T-Values The Tests of Network Specific Hypotheses The Tests

VII. SUMMARY AND CONCLUSIONS......

Interpretation of the Findings Implications of the Findings for the Central-Place Model Limitations of the Study Further Research

SOURCES CONSULTED LIST OF TABLES

Table Page

1. Seventy-five Puerto Rican Urban Centers ...... 17

2. Binary Connectivity Matrix for Simple Road Network . . . 42

3. Binary Connectivity Matrix for Complex Road Network . . . 42

4. Valued Connectivity Matrix for Simple Road Network . . . 44

5. Valued Connectivity Matrix for Complex Road Network . . . 44

6. Shortest Path Matrix and Indices of Vertex Centrality for Simple Road Network...... 49

7. Shortest Path Matrix and Indices of Vertex Centrality for Complex Road Network ...... 50

8. Road Type Codes and Assumed Average Safe Travel Speeds . 56

9. Measures of Central Tendency and Dispersion for Distributions of Vertex Specific Variables ...... 69

10. Correlation Matrix, Zero Order Correlations for Thirty-two Vertex Specific Variables ...... 73

11. Measures of Network Structure, Puerto Rican Road N e t w o r k ...... 80

12. Correlations, Partial Correlations, and T-Values .... 91

vi LIST OF ILLUSTRATIONS

Figure Page

1. Geographical Locations of Seventy-five Urban Centers on the Main Island of Puerto R i c o ...... 18

2. Example of Simple Road Network with Six Vertices ...... 33 ) 3. Example of Complex Road Network with Six Vertices...... 35

4. Possible Correlations Between X and Y at Two Points in T i m e ...... 87

vii CHAPTER I

INTRODUCTION TO THE RESEARCH PROBLEM

This is a study of social and ecological change. It describes cer­

tain alterations in the urban ecology of Puerto Rico, which have occurred

since the island was seized by the United States in 1898. What has

occurred in Puerto Rico is used as data for the testing of general prop­

ositions drawn from the literature of urban sociology and human ecology.

The period in the since its occupation by the

United States is especially interesting from the point of view of urban

ecology for several reasons. First, it is only since the beginning of

the century that many regions of Puerto Rico have been linked with the modern world in other than tenuous ways. For most of its history Puerto

Rico was little more than a military outpost of the Spanish empire and an

isolated location for the cheap production of sugar, coffee, and tobacco

for the world market economy.* This meant that nearly all towns and villages were local service centers for petty trade and administration.

It also means that the social characteristics of towns and villages should have been altered by increasing integration with one another and with

*Sara Jane Deyo, "The Economic Aspects of Cultural Conflict in Porto Rico," (Masters Thesis, Ohio State University, 1933); C. Wright Mills, Clarence Senior, and Rose Kohn Goldsen, The Puerto Rican Journey, (New York: Russell and Russell, 1950), pp. 3-21.

1 o dominating metropolitan centers. That is, the cities, towns, and vill­ ages should have responded in accordance with theories about the expansion of the dominance of modern large-scale, industrial society.^

Another aspect of Puerto Rican urban development which makes it especially interesting to the human ecologist is the system of municipal towns established by the Spanish.^ The territory was divided, for admin­ istrative purposes, into , roughly comparable to counties in the United States. For each of these municipalities a town was desig­ nated as the administrative center. Officials in the municipal towns were directly subordinate to officials in San Juan, the .

This encouraged the establishment of numerous small towns scattered rather evenly over an isolated, rural hinterland.^ The tendency toward localism was reinforced by geographic conditions, especially in the mountainous regions of central Puerto Rico. The extremely rough terrain, combined with a rudimentary transportation network, made commercial

^For a study of social change in a Puerto Rican town see; Joseph W. Scott, "Sources of Social Change in Community, Family and Fertility in a Puerto Rican Town," American Journal of Sociology LXXV, (March, 1967), pp. 520-30.

/ 15 JFor differing perspectives see: Amos H. Hawley, Human Ecology (New York: The Ronald Press, 1950); Scott Greer, The Emerging; City. (New York: The Free Press, 1962); Leo F. Schnore, The Urban Scene, (New York: The Free Press, 1965); W. Fred Cottrell, Energy and,Society. (New York: McGraw Hill Book Company, Inc., 1955); Gerald Breese, Urbanization in New Developing Countries, (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1966); Karl Polanyi, The Great Transformation (Boston: Beacon Press, 1957).

^William F. Willoughby, "The Reorganization of Municipal Government in Porto Rico: Political," Political Science Quarterly, XXIV (September, 1909), pp. 409-443.

-*For a discussion of factors influencing urban development in Spanish colonies see: T. Lynn Smith, "The Changing Functions of Latin American Cities," The Americas. XXV, (July, 1968), pp. 70-83. relations with the outside world costly and one-sided.^ Even in those

areas where large-scale commercial agriculture was practiced, the lives

of the overwhelming majority of the population still reflected the local­

ism and of poverty of rural life at near subsistence levels.^ All of

these factors led to rather meager urban development, except at certain

seacoast locations, such as San Juan, Ponce, and Mayaguez. Hence, the

kinds of economic and technological change since 1900, especially the

development of a highway transportation network, should have considerably altered the ecological basis for the locations and functions of urban centers all over Puerto Rico.

There is another fact about this period in Puerto Rican development which makes it attractive for study by the urban ecologist. This is the fact that extensive data gathering has been regularly undertaken by vari­ ous official agencies since the beginning of control by the United States.

One of the first acts of the military government was to organize a census.8 This first census by the United States government was carried out by the War Department in 1899. Since 1910, Puerto Rico has been in­ cluded in the decennial census of the United States. In addition, from t^,me to time special studies have been undertaken which contribute to the

^Bailey U. Diffie and Justin Whitfield Diffie, Porto Rico: A Broken Pledge, (New York: The Vanguard Press, 1931); pp. 26-28.

^This is not meant to imply that Puerto Rico has been characterized by subsistence farming in a closed economy. For a description of how most Latin American communities are incorporated in the market economy see: Rodolfo Stavenhagen, "Changing Functions of the Community in Under­ developed Countries," Sociologia Ruralis, IV (1964); pp. 315-31.

^Morris Clark Taber, "The Acquisition of Puerto Rico by the United States," (Masters thesis, Ohio State University, 1956). fund of social data concerning the island. For the last several decades the Commonwealth government has carried out policies designed to en­ courage planned economic development. An integral part of these efforts has been the collection and publication of large amounts of social data.

There are probably few areas undergoing rapid change which have such an extensive record of social conditions during the period of development.

The fact that Puerto Rico is a relatively large island makes it ideal for studying the implications of the development of the highway system. The island is large enough to contain an extensive hinterland area, a large part of which is quite isolated in the absence of a good road network. At the same time the rough topography and moderate size of the island have effectively discouraged the development of any signi­ ficant public rail transportation.^ This means that for the great majority of urban centers in Puerto Rico, roads are the first and most important link with distant points. In addition to being the most signi­ ficant transportation network, the road system has many properties of a closed system, since the shores of the island constitute a natural boundary for the network.

t All of the factors just described combine to make Puerto Rico a very advantageous place for the study regional aspects of urban develop­ ment. Therefore, this study takes as its theoretical starting point the literature concerned with the development of metropolitan regions and with regional influences on local communities. Cities, towns, and villages in Puerto Rico will be primarily conceived as part of a system of urban centers linked by a developing transportation network.

^Bailey W. Diffie and Justin Whitfield Diffie, 0£. Cit., pp. 112-16. CHAPTER II

THEORY OF REGIONAL URBAN ECOLOGY

The theoretical basis for this study is drawn from the writings of urban sociologists and urban geographers. It deals with the city, town, or village from a regional perspective. From a regional per­ spective, any urban place is regarded as one center of settlement and activity in a much larger ecological complex. From this per­ spective, an urban center is only one important location in the whole spatial pattern of human settlement; just as a human community is only one important element in the complete structure of ecological relationships.

Background Literature

An important part of the literature concerning the regional

characteristics of urban settlements is that body of writings about the theory of location for cities. The idea is to explain the location

6f urban centers from facts about the natural and socio-cultural

environment in which they have their setting, part of that environment

*®The advantages of the ecological perspective are set forth in: Otis Dudley Duncan and Leo F. Schnore, "Cultural, Behavioral, and Ecological Perspectives in the Study of Social Organization," American Journal of Sociology, LXV, (September, 1959), pp. 132-46.

5 being, of course, other urban centers.This approach to location can be quite abstract, as when little more than certain ideal conditions are postulated, such as assumptions of a uniform land surface, a constant travel time, or a regular transportation cost gradient.

However, the explanatory formulations are not necessarily simple, since in the real world there are many powerful and dynamic factors in the natural and socio-cultural environment of cities.^

Another approach toward explaining the differential growth and development of cities emphasizes the functional specialization of the 13 urban center. Sometimes the most important function of a city is taken as the historical explanation for the founding and perpetuation of that city; just as it is not uncommon for major cities to have their pre-eminence in being the locale for the control of one or more dominant forms of social organization, such as the political, the financial, the military, the informational, or the religious.

Nevertheless, when cast in the regional perspective, all explanations for the development of cities must at some point also consider the

' ^-Edward Ullman, "A Theory of Location for Cities," in Paul K. Hatt and Albert J. Reiss, Jr., eds., Cities and Society, 2nd ed. (New York: The Free Press, 1957), pp. 227-36.

*^For discussion of factors affecting the size and location of cities see: Charles J. Stewart, Jr., "The Size and Spacing of Cities," The Geographical Review, XLVIII, (April, 1958), pp. 222-45. See also: Allan Pred, "Industrialization, Initial Advantage, and American Metropolitan Growth," The Geographical Review. LV, (April, 1965), pp. 158-85.

l%or example, see: Leo F. Schnore and David W. Varley, "Some Concomitants of Metropolitan Size," American Sociological Review, XX (August, 1955), pp. 408-19; Noel P. Gist and Sylvia Fleis Fava, Urban Society, 5th ed., (New York: Thomas Y. Crowell Co., 1964), pp. 79-86. 7

spatial and temporal aspects of relations between a city and its environ­ ment. Regardless of relative size, importance, or functional speciliza-

tion, each of the cities in a region is taken to be influenced by the others. In this way, the characteristics of any urban center are conceived as related to the position it has in relation to other

locations in the region and as related to the niche it holds in the

ecological complex.^

Formulations with regard to the relative size, function, and location of cities have often been made by using the metropolitan community as a point of focus.^ The metropolis is taken as the center of a large productive area and all other places in the region are seen primarily in ways which are related to the dominance of the metropolitan city. The basic idea is that the size of any city is a

1AThe " ecological complex" is Otis Dudley Ducan's term for the full range of relationships postulated as within the conceptual realm of theories of human ecology. It includes the notion of functional interdependence between four basic analytic categories of elements in any ecosystem: population, social organization, technology, and natural environment. For a full description see: Otis Dudley Duncan, "From Social System to Ecosystem," Sociological Inquiry. XXXI (1961), pp. 140-49; and Otis Dudley Duncan, "Human Ecology and Population Studies," in Philip M. Hauser and Otis Dudley Duncan, The Study of Population. (Chicago: University of Chicago Press, 1959), pp. 678-716.

l^See, for example: Otis Dudley Duncan, "Urban Influence on Rural Areas," in Jack P. Gibbs, Urban Research Methods, (Princeton, New Jersey: D. Van Nostrand Co., Inc., 1961), pp. 550-56; Leo F. Schnore, "Metropolitan Growth and Decentralization," American Journal of Sociology. LXIII, (September, 1957), pp. 171-80; Charles M. Grigg, "A Proposed Model for Measuring the Ecological Process of Dominance," Social Forces, XXXVI (December, 1957), pp. 128-31; Lewis W. Jones, "The Hinterland Reconsidered," American Sociological Review, XX (February, 1955), pp. 40-44; Harold F. Goldsmith and James H. Copp, "Metropolitan Dominance and Agriculture," Rural Sociology. XXXIX, (December, 1964), pp. 385-95. function of the size of the tributary area which it dominates in supplying certain services. When trade centers of varying size are

involved, a hierarchy of dominance can be found which is related to

the number of cities of each size in the region and to the average distance separating cities of similar size. Studies have verified the validity of these ideas for rural settlements, as well as for cities. I The studies of metropolitan regions are actually illustrative of a very general explanation of the location of cities. This more general formulation is called the central-place model.^ According 18 to Mark and Schwirian, three basic ideas are contained in the model:

(1) Cities are involved in symbiotic economic relationships with the immediately surrounding agricultural population and are centrally located in their trade area.

(2) Within a region manifesting a central-place locational pattern a hierarchy of central places emerges as a result of ecological position and functional differentiation.

(3) The distance between cities of the same size exhibits regularities which are a function of the mode of transportation at the time of location.

' The specification by Mark and Schwirian that "the mode of trans-' portation at the time of location" is important alerts one to the importance of the transportation network. Since the locations of

•^Edward Ullman, Ojj. Cit.

^Edward Ullman, 0£. Cit.

*-®Harold Mark and Kent P, Schwirian, "Ecological Position, Urban Central Place Function and Community Population Growth," American Journal of Sociology, LXIII (July, 1967), pp. 30-41. cities are usually fixed at a very early point in the social history of a region, ecological adjustments to the development of the trans­ portation network must primarily involve factors other than the actual physical location of cities. In other words, since location is a constant, adjustments to new modes of transportation must be made in terms of relative growth, functional specialization, or other non-

locational characteristics. In fact, Mark and Schwirian present an evolutionary sequence of stages for understanding community growth.

In each of the stages succeeding the original period of agricultural settlement, improvements in the transportation system are an integral part of the factors which lead to the emergence of new forms of 19 regional ecological patterns.

Previous work with the central-place model consistently points to the importance of transportation systems. Thus, it seems likely that the development of central place theory can be furthered by focusing more closely on the transportation system. For example, if the network properties of the system are taken into account, proposi­ tions about the "distances" between places can be sharpened. In particular, the concept of space takes on directional qualities; while the notion of separation often can be specified in terms of more meaningful properties like time, travel distance, quality of 20 route, or rate of traffic flow. By locating urban centers on a transportation network, propositions about particular urban centers

^Ibid.

^®See: Leslie Peter Cummings, "The Structure of Networks and Network Flows, (Ph.D. dissertation, University of Iowa, 1967). 10 and about whole systems of urban centers can be tested by relating the characteristics of urban centers to the results of node specific and network specific analyses of the transportation system. 91

Put in the context of the literature of urban ecology, this study of Puerto Rico is relevant to urban ecological theory in the several ways. First, it tests the relevance of traditional European and

North American formulations in a setting with a strong Spanish-American 22 cultural heritage. Second, it refines and operationalizes certain concepts of the central-place model with respect to space and trans­ portation. Third, it deals with longitudinal data, so as to explore the dynamic properties of systems conceived largely on the basis of cross-sectional studies. Fourth, it extends the work of urban geographers by investigating propositions relating concepts from urban sociological theory and from the study of transportation networks.

The Object of Study

The object of study is a region of urban centers. In general, the following characteristics of the local populations are studied in relation to their locations on the road network: 1) the relative sizes

91For an example of previous work see: Howard L. Gauthier, 'Highway Development and Urban Growth in Sao Paulo, Brazil: A Network Analysis," (Ph.D. dissertation, Northwestern University, 1966).

^Daniel P. Lincoln, "The Development of Porto Rico as a Part of the United States," (Masters thesis, Ohio State University, 1927), p. 4. Lincoln says: "The revolutions of South and Central American caused the Spanish to retreat to Porto Rico and received there by a Spanish governor many of them made the island their home, entered into the agricultural livelihood and prospered. Thus the Spanish ideals, customs, and ten­ dencies spread throughout the island and it became the center of Iberian culture." 11 of local populations, 2) the occupations and division of labor of local populations, and 3) the literacy of local populations. Once cross-sectional relationships are obtained, cross-lagged panel correla­ tions are used to investigate longitudinal relationships between 23 variables.

There is no attempt to take into account the fact that extensive government intervention has occurred in Puerto Rico to encourage industrialization and economic development in rural areas. It is assumed that over the long run these government efforts have not been of the size and nature which would have dramatically altered ecological relationships already observed by social scientists studying other parts of the modern Western system. This assumption is not based on the notion that certain ecological processes are in themselves a

"natural system," independent of organized human intervention.^

Rather, the assumption is based on the conclusion that intervention in Puerto Rico, while novel in some respects, has been rather typical of Western liberal regimes. It is therefore assumed that the same basic conditions of social organization apply to Puerto Rico, just

/ •

^This procedure is thought to be a better one for the study of change than the correlation of gain scores. For a discussion of gain scores versus the cross-lagged panel correlation technique see: George W. Bohrnstedt, "Observations on the Measurement of Change," in Edgar F. Borgatta (ed.), Sociological Methodology 1969, (San Francisco: Jossey-Bass, Inc., 1969).

^For criticism of the position taken here in ignoring socio­ cultural factors see: Sidney M. Willhelm, "The Concept of the 'Ecological Complex:' A Critique," American Journal of Economics and Sociology, XXIII, (July, 1964), pp. 241-48; George A. Theodorson, Studies in Human Ecology, (Evanston, Illinois: Row, Peterson and Company, 1961), pp. 253-321. 12 as they apply to the operation of the ecological complex in similar 25 parts of the modern Western system.

Socio-cultural variables are largely beyond the purview of this study. The rationale for the socio-cultural approach, whether made explicit or not, often seems to include the inclination to gain knowledge by directly studying variables held to be unique to human communities and at the same time causal in ecological processes.

For example, by taking a socio-cultural approach to transportation and regional ecology in Puerto Rico, it would be possible to identify the goals and norms of the persons who were instrumental in the building of the Puerto Rican road network. The interests and power relationships of various groups and elites could be studied, so as to explain the formation of policies and the allocation of resources for the development of transportation on the island. For current times, it might be possible to inquire into the meanings and perceptions

Puerto Ricans have of their road system and its travel characteristics.

Any of these types of studies would proceed from the basic assumption that the ecology of human communities is primarily a product of human social organization. Such studies would usually require interviews, surveys, intensive library study, and examination of codes and docu­ ments. Explanatory hypotheses would be conceived linking socio­ cultural variables to ecological conditions, with the socio-cultural variables taken as the independent vrriables.

25For a brief sympathetic account of government efforts toward the Development of Puerto Rico see: Earl Parker Hanson, Puerto Rico: Ally For Progress, (Princeton, New Jersey: D. Van Nostrand, Inc., 1962). 13

This study ignores to a considerable extent the objects of study

of the socio-cultural ecologists. This should not be taken as a

repudiation of the idea that human ecology reflects the influences

of human social organization. Because of the limited time and

resources and because of the impossibility of obtaining many kinds

of data from the past, it is expedient to work with the object of study

at a rather abstract level of analysis. Since the sources of informa­

tion about the object of study are census reports and maps, it is not

possible to directly incorporate assessments of values, goals, meanings,

social relationships, and other uniquely human phenomena as variables

in hypothesis testing. As a result, opportunities for working with more richly qualitative and more intuitively pleasing information

have been sacrificed for the sake of concentrating on information

gathered regularly by codified methods.

The important advantages in working at the abstract level come

from using relatively complete and reliable data which can be compared

longitudinally. This means that in this study extensive data gathering,

reliability, scientific rigor, quantitative methods, longitudinal

comparisons, and systematic hypothesis testing are gained; while, to

the extent that the socio-cultural ecologists are correct in their

basic assumptions, validity may have been lost. If the results of

this study fail to confirm traditional theories about the relative

size, location, and characteristics of urban centers from the regional

perspective, alternative theoretical and methodological formulations

should be conceived. These formulations may very well, and quite as legitimately, include socio-cultural concepts„ Our present concern is with local populations as they spatially and demographically manifest themselves in a region with a changing road network,, CHAPTER III

BASIC CONCEPTS AND OPERATIONAL DEFINITIONS

This chapter is devoted to explaining the nominal and operational definitions of terms. Problems of gathering, processing, and interpret­ ing the data are considered. An operational definition is specified for each basic concept taken as a variable in the analysis. Problems of measurement are described. After questions of validity, reliability, and level of measurement have been considered, statistics appropriate for each type of data are discussed.

In the case of graph theoretic concepts the discussion is somewhat more detailed and systematic. The discussion begins with the more simple definitions and progresses to the more elaborate definitions, so that all concepts are operationalized only after more primitive concepts have been defined. When appropriate, questions of validity, reliability, and level of measurement are introduced and statistics appropriate to the data are discussed.

The Urban Center

The definition of urban center for this study is an arbitrary one. It is made this way by the necessity of using data which have been collected and classified by official data gathering, agencies.

These agencies have not been consistent in their categorization of data by place. Therefore, in order to attain an acceptable degree

15 16 of continuity in the data from decade to decade, it has been necessary

to make certain allowances and estimates for a few of the places

studied.

The primary criteria for including a location within the category u of urban center for purposes of this study are, in the order of their

importance: 1) the status of the place as a city, town, or village on the main island of Puerto Rico in 1960, the year of the latest census, 2) the availability of data for the city, town, or village and for the corresponding for several decades. The application of these criteria results in the selection of seventy- five places as urban centers, only one of which is not a municipal 26 town. Table 1 lists the seventy-five urban centers. Figure 1 shows the approximate geographical locations of these seventy-five urban 27 centers on the main island of Puerto Rico.

Population Size

Population size is always taken to mean the total population of the urban center at the times of the various censuses. There are a few problems of continuity because of changes in the boundaries of some / places. These problems are noted where appropriate.

26A Puerto Rican municipal town is roughly comparable to a county seat in some mainland states of the United States. The municipality is the basic unit of local government in Puerto Rico.

^Data are available for the municipality of Vieques, consisting of an island east of the main island of Puerto Rico. Vieques town, the municipal town, would have qualified as a urban center, except that its not being on the main island of Puerto Rico would have complicated the analysis of road network data. 17

TABLE 1.--Seventy-five Puerto Rican Urban Centers

Urban Center Urban Center Urban Center

City, Town or City, Town or City, Town or Number Village Name Number Village Name Number Village Name

1 Adjuntas 26 Guanica 51 Naranjito 2 Aguada 27 Guayama 52 Orocovis 3 Aguadilla 28 Guayanilla 53 Patillas 4 Aguas. Buenas 29 Guaynabo 54 Penuelas 5 Aibonito 30 Gurabo 55 Ponce 6 Anasco 31 Hatillo 56 Quebradillas 7 Arecibo 32 Hormigueros 57 Rincon 8 Arroyo 33 Humacao 58 Rio Grande 9 Barceloneta 34 Isabela 59 Sabana Grande 10 Barranquitas 35 Jayuya 60 Salinas 11 Bayamon 36 Juana Diaz 61 San German 12 Cabo Rojo 37 Juncos 62 . San Juan 13 Caguas 38 Lajas 63 San Lorenzo 14 Camuy 39 Lares 64 San Sebastian 15 Carolina 40 Las Marias 65 Santa Isabel 16 Catano 41 Las Piedras 66 Toa Alta • 17 Cayey 42 Loiza 67 Toa Baja 18 Ceiba 43 Luquillo 68 Trujillo Alto 19 Ciales 44 Manati 69 Utuado 20 Cidra 45 Maricao 70 Vega Alta 21 Coamo 46 Maunabo 71 Vega Baja 22 Comerio 47 Mayaguez 72 Villalba 23 Corozal 48 Moca 73 Yabucoa 24 Dorado 49 Morovis 74 Yauco 25 Fajardo 50 Naguabo 75 Loiza Aldea •Figure 1.— Geographical Locations of Seventy-five Urban Centers on the Main Island of Puerto Rico

34 « • • 31 24*, 9 71* ft . *6767 1 6 * ^ ^ 6 2 75* b 3 70 •4 8 6i ^ 2 ♦ . t lV «8- 15 1,1 ^57 64* 23* 51* 6* 39 4# 52* *40 2 2 * 35 50 N p 4 7 10 *45 CM • 72 £ 61 59 36 21 12 • ft 73, •38 53. 4 6 . 74 55 ,65 60 27 # 8,

Scale in Kilometers 0 10 20 30 40 ______1 i______i______i______| 19

The population size variable always refers to the populations of a place in the road network where an urban center is considered to exist. It is important to keep in mind that these places do not always qualify as urban places, as this term is conventionally defined by the

U. S. Bureau of the Census. Some towns and villages in Puerto Rico have never had populations as large as 2,500 inhabitants, even though special population counts have been reported for them in census 28 publications.

In nearly every case, the special status of these places for purposes of census taking depends on their political function as the seat of government for a municipality. This raises the possibility that other places, not accorded political functions, could be over­ looked, even though they might be centers of other important urban activities or urban functions. To some extent this has happened in

Puerto Rico.

From 1899 to 1940 the emergence of new centers seems not to have occurred in ways which cause many problems with population data. There are a few minor problems of data collection during this period, because / early in the 20th Century the number of minor civil divisions was raised from sixty-nine to seventy-six by sub-dividing and reorganizing existing municipalities. Fortunately, the reorganization was done in a way which created new aggregates of to constitute the new municipalities. Since data from previous censuses had been tabulated

^This and the following discussion are based largely on the various census reports for Puerto Rico. These reports are listed as sources consulted under.public documents in the bibliography. 20 for barrios, it was nearly always possible to obtain good population counts for the new aggregates that were given separate political status.

The number of municipalities was raised for seventy-six to seventy-seven between 1920 and 1930 in a similar manner. In any case, reasonably good data can be had for the cities, towns, villages, and municipalities for the whole period up to the 1940 census. Apparently, the reorganizations of municipalities were made in ways which reflected the realities of geography and administration, so that the municipal towns were generally the dominant centers of their local hinterlands.

There was one serious attempt to reduce the number of municipali­ ties in order to simplify and economize local government. This is re­ ported by William F. Willoughby, who supervised the reorganization of 29 government finances in Puerto Rico beginning in 1900.

"... While the Spanish system of having the island divided into districts for the purpose of local government was pre­ served, the number of these districts was materially reduced in the year 1902, 20 of the smaller districts being consolidated with adjoining districts, so as to leave the island divided into only 46 municipalities. This consolidation was effected by a special act passed at the same session of the legislature at which the new municipal law was enacted. The motive dictating this action was the desire to lessen the expense of local government, as a number of the existing districts scarcely ' had the population or resources to warrant the maintenance by them of independent governments."30

Had this economy move been successful, it would have removed the political function of twenty of the smaller municipal towns. Their small populations and loss of urban function could have resulted in no separate population counts being reported for them in subsequent censuses.

^U.S. Bureau of the Census, Insular and Municipal Finances in Porto Rico for the Fiscal Year 1902-3, Census Bulletin Humber 24 (, D.C.: Government Printing Office, 1905). 30 .• Ibid., 17-18. 21

However, only four years after reporting the reduction in the number of

municipalities, Willoughby published an article saying that the number

of municipalities had been returned to sixty-six. The reasons for this

failure of the effort to lessen the expense of local government are an

indication that local social structure was somewhat strongly organized

around the municipal towns as they existed prior to the reorganization.

The hardships of travel in rural areas are also indicated as part of the

reasons for local resistance to the reorganization. Willoughby writes

in 1909:

"... the opinion did strongly prevail that the island had been excessively subdivided, and that a great economy and increased efficiency could be obtained by abolishing the less important districts as entities, and annexing them to adjoining districts. In consequence, at the same session of the legislature, that of 1902, in which the general municipal law was enacted, there was passed another act providing for the consolidation of twenty of the smaller and less populous districts with the remaining forty-six; This act worked badly from the very start. In practical operation the dis­ tricts absorbed, though paying their share of the taxes, were almost entirely neglected in the employment of the funds so received. Their inhabitants resented bitterly having their special status abolished, and found real hardships in the greater distances they had to go in order to attend to any matters requiring action before public authorities. The act was accordingly repealed March 5, 1905, and a return made to the original division of the island into sixty-six , districts."31

This interlude in the reorganization of local government has been

reported here because it illustrates some of the local bases in social

and ecological organization for the continued existence of municipal

' towns. It also suggests that during the early part of the 20th Century

William F. Willoughby, "The Reorganization of Municipal Govern­ ment in Porto Rico: Political," Political Science Quarterly, XXIV (September, 1909), 424. 22 political functions were a very important basis for the continued existence of the smaller urban centers.

Three striking cases of change at the end of the period up to the

1940 census highlight the importance of the political seat as the recognized center in the municipality. One change is the shift in the seat of municipal government in Loiza Municipality from Loiza Aldea to

Loiza. The data suggest that the shift of the seat of government came only after the town of Loiza had already become a powerful rival center in Loiza Municipality. The other two cases of change are the emergence of rival population centers in Salinas and Guanica municipalities. In the case of Guanica, the village of Ensenada emerges as the near equal of Guanica Town. In Salinas, Central Aguirre Village appears as a

Vival of Salinas Town. The difficulty with respect to data collection arises because it is not possible to obtain longitudinal data for the newly recognized places.

In the period since the 1940 census six more rival centers have been given village status in Puerto Rico. Two of these are in Salinas

Municipality near Central Aguirre, so that the three villages rivaling

Salinas Town are all in one , Aguirre Barrio. Hence, longitudinal data for Aguirre Barrio are of no use in studying the population of these villages, especially since Aguirre Barrio is very large and con­ tains a relatively large rural part.

A similar problem arises in Guayama Municipality, where Jobos

Village and Puerto Jobos village have recently been recognized. However

Guayama Town is very large in comparison to the two new villages, which are themselves quite small. There can be no doubt that Guayama Town, 23 for which there are acceptable data, is the only important urban center in the municipality.

The remaining two of the six newly recognized villages are in

Barceloneta Municipality. Unlike the situation in Guayama Municipality,

Barceloneta Town is very much overshadowed by the two villages. They are

Florida Adentro and Florida Afuera. Florida Afuera is contiguous to

Barceloneta Town and seems to be growing to a greater size than Barce­ loneta Town. The barrios in which the two villages are located have very large rural areas, so that longitudinal data for the barrios are far from valid for the villages.

In view of all the foregoing one can conclude that' there are a few problems with the population size variable as it is operationalized in this study. Problems of validity chiefly arise from the question of the historical role of the municipal town vis a vis other centers of dense, settlement in the municipality. In nearly every case, the municipal town does seem to have been the only considerable urban center in the area. In some cases, the validity and reliability of the population size figures are impaired, especially when new municipalities were organized, by the pecessity of depending on data for barrios, rather than on data for the towns themselves. On the other hand, many of the barrios involved were termed "Pueblo" and were virtually conterminous with the limits of the newly recognized town.

Subject to the qualifications already given, it seems reasonable to regard the population size data as interval scale measures of the sizes of urban centers in Puerto Rico for the period from 1899 to 1960.

Achievement of this level of measurement places few restrictions on the 24 statistics chosen for assessing the degree of association with other variables..

Occupations and Division of Labor

Occupation is defined in terms of categories of employment. These categories have been extensively revised from decade to decade and have been considerably refined and elaborated. Because of these extensive changes in categories, it would be impossible to accurately trace trends with respect to particular occupations or even for some gross categories of occupations. However, an attempt is made to assess relative degrees of division of labor by comparing the amount of dispersion across occupa-

OO tional categories at each census.

The concept of occupation is operationalized in terms of the cate­ gories of occupations used by census takers to describe the social and economic characteristics of the population. These categories have been redefined several times in the last 70 years. Furthermore, for some of the earlier censuses in Puerto Rico, the published data are not complete enough for the purpose of this study. To further complicate the matter, when it is possible to get data for the whole island, these data are / tabulated by municipalities, rather than by cities and towns.

The kinds of occupation data available reflect four different data collection procedures which have been used by census takers in Puerto

Rico. The first data collection procedure was used only one time,

^This approach to assessing division of labor is suggested by: Omer R. Galle, "Occupational Composition and the Metropolitan Hierarchy: The Inter and Intra-Metropolitan Division of Labor," American Journal of Sociology, LXIX, (November, 1963), pp. 260-69. For criticism of the use of conventional categories of occupations see: A.J. Jaffe, People, Jobs and Economic Development, (Glenco, Illinois: The Free Press, 1959). 25 which was for the 1899 census. This gave complete data for all of the municipalities in existence at that time. Five categories of occupations were used to classify persons by employment. These were: 1) Agriculture,

Fisheries, and Mining, 2) Trade and Transportation, 3) Manufacturing and

Mechanical Industries, 4) Professional Service, and 5) Domestic and

Personal Service.

A different procedure with respect to occupation data was used in

1910 and in 1920. The number of categories was boosted to eight in 1910 and then to nine in 1920. Nevertheless, the basic breakdown was very similar to that used in 1899. The additional categories were obtained in 1910 by distinguishing between Trade and Transportation and by adding

Public Service (not elsewhere classified); while in 1920 the one addition was a new distinction between agricultural pursuits and mining occupa­ tions. Unfortunately, the only data published in 1910 and 1920 are for a very few cities. For 1910 only Caguas, Mayaguez, Ponce, and San Juan data are available. For 1920 data are available for the same four cities and for Arecibo and Bayamon as well.

In 1930 data are available for all municipalities, but the catego­ ries of occupations have been greatly elaborated and regrouped. The number of occupations totals more than thirty. These specific occupa­

tions are listed by six industry groups, rather than by the convention­ ally accepted occupation types. These groupings by industry seem

especially ill-conceived. Some of the groupings are mostly aggregates of dissimilar jobs. This system was replaced in 1940 with a new system

incorporating two kinds of classification. One classification lists the

number of employed persons by industry group. It-is rather like the 26 systems used in 1899 and 1930. The other classification is more directly concerned with types of occupations; so that categories are based on dis­ tinctions between the kinds of work individuals do, such as professional, clerical, sales, service, labor, and so on.

The three censuses taken in 1940, 1950, and 1960 are very much alike in their presentations of occupation data. There is complete cov­ erage of all municipalities. The categories of occupations vary slightly, but these variations involve details of distinction. For purposes of assessing the degree of division of labor, the minor differences are not very important.

The degree of division of labor is operationalized by calculating the index of dispersion for occupation data as they are grouped by category.

The formula is

where

D = Index of Dispersion H = The Number of Occupation Categories n£ = The Frequence Count of the ith Category H N = The Sum of all Category Frequencies, or £_K\\ i-1

In terms of the data, this involves the calculation of an index of

dispersion for every municipality for each census year. The number of

occupation categories will vary depending on the year of the census;

while N and n^ are the frequency counts for types of occupations.

The Index of Dispersion has a lower limit of 0.0 and an upper limit

of 1.0. Minimum possible dispersion across occupational categories 27

would be indicated by 0.0. Maximum possible dispersion across occupa­

tional categories would be indicated by 1.0. This index is being used

because it is sensitive to concentration of workers in one particular

type of work. The relevance to Puerto Rico becomes obvious when the cb- u servation is made that traditionally the overwhelming majority of the

Puerto Rican work force has been employed in agriculture. Most munici­

palities in Puerto Rico will have relatively low index of dispersion

scores. Higher scores will indicate for the most part employment in

other than the farm labor category.

The most consistent difficulty with the occupation data, is the

fact that the municipality is the only unit of analysis which can be

used to compare any given part of Puerto Rico with another. It is cer­

tainly invalid to argue that these data represent the state of affairs

in cities, towns, and villages. The unit of analysis becomes the muni­

cipality.

On the other hand there is in Puerto Rico one special factor which

partially justifies the use of data for the municipality. This is the

fact that nearly every one of the cities and towns apparently has been

the most important center of town life in the municipality. For this

reason it is reasonable to think of the municipal town as an integral

part of the surrounding municipality. In many respects, the municipal

town and its corresponding municipality share the same location in the

. road network and have a similar niche in regional ecology. Every urban

center except Loiza Aldea is, in fact, the municipal town for its muni­

cipality. 28

The changes in the number of categories of occupations are some­ what less serious impairments of reliability and validity than might be imagined, because a farm labor category has been included in every class­ ification scheme. More serious than the changes in the number of cate- gorier are their redefinitions from census to census. The changes made between 1930 and 1940, are so sweeping that comparisons spanning this period for individual municipalities would be extremely questionable.

The only period during which reliable cross-census comparisons for individual municipalities might be in order is that one from 1940 through 1960.

The occupation data from the 1899 census have to be interpreted with special caution. Puerto Rico was raked from east to west by an extremely destructive hurricane just three months previous to the taking of the census. This hurricane brought the economy to a state of prostra­ tion, with especially disastrous effects on the coffee industry. Re­ covery was slowed by the chaotic financial situation following the change from Spanish to United States sovereignty. J The census returns for occupations reflect this in that very large numbers of persons reported themselves as unemployed.

Subject to the qualifications already given, it seems reasonable to regard the index of dispersion scores for the occupation data as interval scale measures of the degree of division of labor. However, because of inter-censal changes in the occupational categories,

longitudinal analyses should be limited to the period from 1940 through

1960.

3 3 Daniel P. Lincoln, Op. Cit., 10. Literacy

Literacy is defined as the reported ability to read and write. It is meant to be an index of the degree of "modernization" in a given local­ ity. This interpretation of the significance of literacy is made on the basis of two facts from Puerto Rican history. First, there is the fact that under Spanish rule the so-called "old Castillian System" of education was the general pattern.^ This meant that only the small Spanish elite was afforded schooling. Second, there is the fact that a vigorous edu­ cational program at the elementary level was put into operation soon after occupation by the United States. This program was beset by many problems, including a lack of competent teachers, limited funds, ethno­ centric North American goals and restrictions, and administrative 35 crises. Some of these problems continue to impair the Puerto Rican system of education. Over the long run, however, the educational effort in Puerto Rico has been an important part of the spread of new cultural elements. Literacy, rather than school attendance or ability to speak

English, is selected, because the fact of establishment of schools or the teaching of English represented more the effort toward change than the actual impact of "progressive" social change.

For every census, starting with 1910, the operational definition of literacy has been the per cent of persons 10 years and over who report that they are able to read and write. The data on literacy in

^Earl Parker Hansen, 0j>. Cit., pp. 115-17.

^^For a description of some difficulties and failures in Puerto Rican education see: Morris Clark Taber, Op. Cit., pp. 76-78. 30

reports of the census of 1899 are not presented in a usable form. The

only censuses for which complete listings for cities and towns are

available are those for 1930 and 1935.^ Consequently, the municipality

is the only unit for which complete coverage of the island can be had

for eveiy census year since 1910.

The major question about the reliability and validity of literacy

data concerns the truthfulness with which reports are made. For example,

in their study of Philippine barrios, De Young and Hunt found that a

sizable minority of persons who claimed the ability to read and write were actually "non-functional" (claimed to read but had no comprehension

OO of a written passage subsequently given them). They report that 63.87.

of the sample claimed literacy in response to the usual census question;

whereas it was then discovered that 19.27. of the group claiming to be

literate could not perform at the lowest level of functional literacy in

any language.

Even though there is reason to question the accuracy of reported

literacy, there is no alternative open in this study but to draw on the

census data as given. There is no way to know if false reports in Puerto

Rico systematically bias these data. It is assumed that any misre-

porting was distributed rather evenly throughout the island, so that

differences in the levels of literacy from one area to another are not

spurious.

37 A special census was taken in 1935 under the auspices of the Puerto Rican Reconstruction Administration. The data available from the 1935 census have not been used in this study. 38 John E. DeYoung and Chester L. Hunt, "Communication Channels and Functional Literacy in the Philippine Barrio," The Journal of Asian Studies, XXII (November, 1962), 67-77. 31

Subject to the qualifications already given, it seems reasonable to regard the literacy data as interval scale measures of the ability of local populations to read and write at the times of the various cen­ suses. Achievement of this level of measurement places few restrictions on the statistics chosen for assessing the degree of association with other variables.

Graph Theoretic Concepts and Definitions

In this section the graph theoretic concepts are explained and illustrated.39 Simple examples are used to show how index scores are calculated.Problems in the application of these formulae to the

Puerto Rican data will be noted in a following section.

The graph theoretic concepts are of two types: vertex specific concepts and network specific concepts. In the case of vertex specific concepts one is concerned with the relative location and linkage of a particular place in the network. Indicators can .be used to express the number of places to which the place is adjacent, the number of routes intersecting the place, the accessibility or centrality of the place,

, J7This discussion of graph theoretic concepts is drawn from several sources. It is based most directly on the following sources: Charles A. Dailey, "Graph Theory in the Analysis of Personal Documents," Human Relations, XII (1959), pp. 65-74; John D. Nystuen and Michael F. Dacey, "A Graph Theory Interpretation of Nodal Regions," Papers and Proceedings of the Regional Science Association. VII (1961), pp. 29-42; Oystein Ore, Graphs and Their Uses, (New York: Random House, 1963); William L. Garrison, "Connectivity of the Interstate Highway System," Papers and Proceedings of the Regional Science Association, VI (1950), pp. 121-37.

^ F o r systematic mathematical derivations of the graph theoretic concepts see: Robert G. Busacker and Thomas L. Saaty, Finite Graphs and Networks (New York: McGraw-Hill Book Co., 1965); and Karl J. Kansky, Structure of Transportation Networks (Chicago: University of Chicago Press, 1963). . 3 2 and the number of parallel routes to another place. In using network specific concepts one is concerned with the properties of the whole sys­ tem of places and routes, so that indicators can be used to express summary information about the size, areal expansion or contraction, de­ gree of connectivity, degree of redundancy, shape, and diameter of the network.

A graph is a figure consisting of points (often called vertices or nodes) and line segments (often called edges or links) connecting some or all of these points. A highway network connecting a number of places can be represented as a graph, so that highway routes are signified by line segments and places are signified by points. Figure 2 shows a simple hypothetical road network connecting six points.

An edge progression, or route, is a finite sequence of (not necess­ arily distinct) edges, such that one end point of the first edge is also an end point of the second, the remaining end point of the second is also an end point of the third, and so on to the last edge of the se­ quence. The edge progression is closed if the "free" end point of the first edge in the sequence is the same vertex as the "free" end point pf the last edge in the sequence, and open otherwise. In Figure 2 the- edge progression, AB, BD, DF, would be open; while the edge progression,

DF, FE, EC, CD, would be closed.

A path is a route in a graph that goes through no edge more than once. An arc is a route in a graph that goes through no vertex more than once. A circuit is an arc that returns to its starting point.

When two vertices are connected by two different routes these two routes are said to be parallel routes. Figure 2.— Example of Simple Road Network with Six Vertices

u> 34

A connected graph is one in which every vertex is connected to every other vertex by some arc. This means that there are no isolated vertices and no isolated subgraphs in a connected graph. A subgraph is a graph x*hich is contained in another graph, such that every element of the subgraph is an element of the larger graph and only some of the elements of the larger graph are elements of the subgraph. The graph in Figure 2 is a connected graph.

The area of a graph is the sum of all its edges, or the sum of all edge values for a valued graph. The size of a graph is the number of vertices in it. Areal expansion of a graph is said to occur when new edges are added which connect to the graph vertices or subgraphs pre­ viously isolated from it. Areal intensification of a graph is said to occur when new edges are added which do not connect to the graph vertices or subgraphs previously isolated from it. Figure 3 shows a graph which results when edges are added to the graph shown in Figure 2.

The small circles in Figure 2 are the nodes or vertices of the graph. They correspond to the cities, towns, or villages on a map.

Each of these vertices is identified with a letter. The lines connecting fhe vertices are the edges or links of the graph. These edges or links correspond to roads connecting the cities, towns, or villages on a map.

The edges of a graph may be directed, indicating that only one way

passage is possible on certain edges. This is called a directed graph.

A directed graph might be used to represent a road system having one way

streets or divided highways. In this study all roads in the network

are considered to be two-way, or undirected, roads. Figure 3.— Example of Complex Road Network with Six Vertices 36

Two edges are said to be adjacent if they have at least one common end point. In Figure 2 edge AB is adjacent to edge BD, because both of these edges have an end point at vertex B. Two vertices are sail to be adjacent if they are joined by an edge. In Figure 2 vertex A is adjacent to vertex B and Vertex B is adjacent to vertex D, but vertex A is not adjacent to vertex D.

The local degree of incidence of a vertex is the number of edges with end points at the vertex. A vertex is said to be isolated if there is no edge with an end point at the vertex. None of the vertices in

Figure 2 is isolated. The local degree of adjacency of a vertex is the number of vertices to which it is adjacent. In Figure 2 the local degree of adjacency of vertex C is two, while the local degree of adja­ cency of vertex D is three.

Arithmetic values may be assigned to the edges of a graph, indicat­ ing some property of the edges, such as distance, flow, cost, or quality.

Such a graph is called a valued graph. In this study all graphs are considered valued graphs.

Since the size of a graph refers to the number of its vertices and the area of a graph refers to the sum of its edge values, the size of a graph and the area of a graph may vary independently of one another.

For example, an increase in size does not necessarily mean an areal ex­ pansion, as when only isolated vertices are added to the graph. Areal expansion does not necessarily mean in increase in size, as when pre-

«. .'ously isolated vertices or subgraphs are connected by new edges. The removal of edges from a graph can always be accomplished without a reduction in size. However, whenever a reduction in size would remove 37 the end point for an edge, some reduction in area must occur, because the edge whose end point was removed must also be removed. The area of a graph depends on two quantities. It depends on the number of edges and it depends on the values assigned to those edges. When edge values of a valued graph signify travel time, travel cost, or mileage, it is quite possible that some changes in edge values will be toward lower edge values. This means, even though areal intensification may be occurring, some edge values could at the same time undergo reductions which in the aggregate contribute to a reduction in the area of the graph.

A completely connected graph (often called a complete graph, a universal graph, or a maximally connected graph) is one in which every two distinct vertices are adjacent. A completely connected graph has n(n-l)/2 edges, where n is the number of vertices in the graph.

A tree is any connected graph without circuits. A minimally connected graph is one that is connected by a tree. For such a graph the number of edges will be n-1, where n is the number of vertices in the graph. This number of edges is also called the number of edges in

£he complete tree of the graph.

The circuit rank, or cyclomatic number, of a graph is the number of edges that would have to be removed to reduce the graph to a tree.

The cyclomatic number can be computed directly from the number of edges, the number of vertices, and the number of subgraphs. If m is the number of edges, n is the number of vertices, and p is the number of subgraphs; the cyclomatic number equals m-n+p, where p is taken to be equal to one ' for a connected graph. 38

The ratio of redundancy, or alpha index, is an index of connective­ ness. It is designed to indicate the change from a simple network struc­ ture to a more elaborate network structure. It is based on the compari­ son of the number of circuits actually present with the number of circuits which would be present if the graph were maximally connected

The alpha index is calculated from the cyclomatic number of a graph and the number of circuits which would be present if the graph were a completely connected graph. The alpha index equals the ratio of the cyclomatic number to the difference between the number of edges in a com­ pletely connected graph of the same number of vertices and the number of edges in the complete tree of the graph.

The formula for alpha is: m-n+p c x . =

where;

m is the number of edges in the graph,

n is the number of vertices in the graph,

p is the number of subgraphs.

In the calculation of the alpha index parallel edges are counted as a single edge. Alpha has lower and upper limits of 0.00 and 1.00 respec­ tively. Its calculation results in a value of 0.00 for a tree and for any less connected graph. As edges creating circuits increase the alpha index increases, until the condition is reached where the graph is maximally connected. For a complete graph alpha reaches its upper limit at 1.00. 39

The alpha index for the graph in Figure 2 is easy to calculate.

The graph has six edges and six vertices; it has no isolated subgraphs, so the number of subgraphs is set equal to one. The alpha index for the graph in Figure 2 is equal to 0.10.

Calculating the alpha index for the graph in Figure 3 is also easy, once some conventions have been established for counting the number of edges. Notice that vertices A, B, and C in Figure 3 are connected by a Y-shape road. It will be necessary to treat this Y-shape road as though it were instead three edges as follows: AB, BC, and CA. Since there was already a direct link between vertex A and vertex B, the

Y-shape road is merely a parallel route as far as the link between vertex A and vertex B ic concerned.

Another convention in counting edges must be applied when by-pass roads exist. Figure 3 shows that a by-pass system has been constructed around vertex D. Thus, vertex C is directly connected with vertices B and F through the by-pass. Vertex B and vertex F are also directly connected through the by-pass. The by-pass route connecting vertex C to vertex B is a parallel route. However, the by-pass route connecting vertex C to vertex F and the by-pass route connecting vertex B to vertex F are to be counted as direct links between these two pairs of vertices.

Applying these conventions for counting the number of edges gives a total of ten edges for the graph ir. Figure 3. As with Figure 2, there are six vertices and no isolated subgraphs. The alpha index for the graph in Figure 3 is equal to 0.50. 40

A different index of the connectivity of a graph is the gamma index. This index of the degree of connectivity refers to the ratio between the actual number of links present in a graph and the number of links that would be present if the graph were completely connected.

Since the number of edges in a completely connected graph is n(n-l)/2, the degree of connectivity of a graph can be calculated by dividing the actual number of edges present by the value n(n-l)/2. This division will yield a number with lower and upper limits of 0.00 and 1.00 respectively. In the calculation of the gamma index parallel edges are counted as a single edge. The gamma index for the graph in Figure 2 is equal to 0.40. The gamma index for the graph in Figure 3 is equal to

0.67.

The important difference between the alpha index and the gamma index comes from the fact that they are sensitive to changes in the connectivity of networks at different stages in the development of networks. The alpha index is sensitive to changes in connectiveness which occur after the graph has become minimally connected, but it is insensitive to changes in less connected graphs. The gamma index is sensitive to any change in the number of links in a graph. Hence, the gamma index is a simple index of connectivity, while the alpha index is designed to reflect redundancy in the connectivity of the network.

Much of the literature in graph theory is written on the assumption that the links between vertices are not weighted. That is, only the existence or non-existence of a route between vertices is indicated by the presence of a zero or non-zero element in a connectivity matrix.

A connectivity Matrix with binary operators can be used to express this kind of information, such as.the one presented by Table 2. Table 2 41 is the binary connectivity matrix for the graph in Figure 2. Table 3 is the binary connectivity matrix for the graph in Figure 3. The diagonals of these matrices are set to zero, since no vertex has a link with itself. The matrices are symmetrical with respect to their diagonals. If the entries are summed, the sum for each matrix is equal to twice the number edges in the corresponding graph. Thus, summing a connectivity matrix is one way to obtain the number of edges for computing alpha and gamma.

Gauthier has shown that in dealing with transportation networks 41 it is possible to weight the values of routes between vertices. In the case of roads, a valued graph would represent differences in roads due to varying length, quality, and other cost or time gradients.

The use of valued graphs alters slightly the definitions of some graph theoretic concepts, but it mainly involves the selection of ways to standardize values assigned to links in the network, so that matrix manipulations will yield meaningful indices.

The alpha index and the gamma index, as has been shown, can be calculated without paying any attention to edge values. If changes in the quality and length of roads were important aspects of the / development of a highway network, the numerator used in calculating gamma could be the sum of edge values. However, it would be necessary to standardize edge values for all graphs before any calculations were made. That is, every link in every network would have to be assigned its edge values on the basis of a scale of values used for all other links in every other network to which comparisons were to be made.

/^Gauthier, 0£. Cit., 42-79. 42

TABLE 2.--Binary Connectivity Matrix for Simple Road Network

A B DDE F

A 0 1 0 0 0 0

B 1 0 0 1 0 0

C 0 0 0 1 1 0

D 0 1 1 0 0 1

E 0 0 1 0 0 1

F 0 0 0 1 1 0

TABLE 3.--Binary Connectivity Matrix for Complex Road Network

A B C DE F

, A 0 1 1 0 0 0

B 1 0 1 1 0 1

C 1 1 0 1 1 1

D 0 1 1 0 0 1

E 0 0 1 0 0 1

F 0 1 1 1 1 0 43

\ If such an effort were made, the gamma index could be made to reflect a reduction in travel times due to highway improvements.

One possible scale for assigning edge values would be to con­ sistently use the reciprocal of the travel time between places in minutes, with the provision that no travel time be taken as less than one minute. Using this system, the gamma index would be sensitive to both new roads and to improvements in old roads. This gamma index for using standardized edge values will be called the valued gamma index. It will be written like this: X s where,

m is the number of edges

t^ is the travel time for the ith edge, and

n is the number of vertices.

Note that for a universal graph with vertices one minute apart, the valued gamma index would reach its maximum at 1.00. For a set of isolated vertices it would reach its minimum at 0.00. In the case of parallel links, it is understood that the edge value will be taken as equal to the edge value of that parallel link with the lowest edge value.

In order to illustrate the calculation of the valued gamma index, let us assume that the entries in Table 4 and Table 5 represent the travel times in minutes for the links in the graphs in Figure 2 and

Figure 3. By referring to Table 4 it can be seen that the travel time from vertex A to vertex B in Figure 2 is assumed to be 15.5 minutes. 44

TABLE 4.--Valued Connectivity Matrix for Simple Road Network

ABC D E F

A 0.0 15.5 0.0 0.0 0.0 0.0

B 15.5 0.0 0.0 35.5 0.0 0.0

C 0.0 0.0 0.0 20.0 21.5 0.0

D 0.0 35.5 20.0 0.0 0.0 45.5

E 0.0 0.0 21.5 0.0 0.0 47.0

F 0.0 0.0 0.0 45.5 47.0 0.0

TABLE 5.--Valued Connectivity Matrix for Complex Road Network

A BC D EF

A 0.0 14.0 30:0 0.0 0.0 0.0

B 14.0 0.0 23.5 28.5 0.0 74.0

C 30.0 23.5 0.0 17.0 21.0 62.5

D 0.0 28.5 17.0 0.0 0.0 44.0

E 0.0 0.0 • 21.0 0.0 0.0 46.5

F 0.0 74.0 62.5 44.0 46.5 0.0 45

By referring to Table 5 it can be seen that the travel time from vertex

A to vertex B in Figure 3 is assumed to be 14.0 minutes. Similar comparisons can be made for other corresponding links in Figure 2 and Figure 3. All zero entries in the connectivity matrices mean that no direct link exists between the two vertices for which the matrix cell contains information.

Once the reciprocals of the connectivity matrix cell entries have been obtained, the calculation of the valued gamma index is easy. The valued gamma index for the graph in Figure 2 is equal to

0.015. The valued gamma index for the graph in Figure 3 is equal to

0.024. Just as with the regular gamma index, the more complex graph has a higher index score, but with the valued gamma index there is some basis for thinking that the difference between index scores reflects more than just a difference in the number of links. A valued alpha index seems more difficult to construct and interpret; it will not be attempted in this study.

On a connected graph more than one path can exist between any two given vertices. The length of a path is the sum of the links in it. The shortest distance between two vertices on a connected graph is the length of any shortest path joining them. For valued graphs, the shortest distance between two vertices will be that path which gives the lowest sum of edge values for edges traversed connect­ ing them.

The associated number of a vertex on a connected graph is the maximum of the shortest distances between the vertex and each of the other vertices. The largest associated number for the vertices of a graph, or the maximum of the shortest distances between any pair 46 of vertices in the graph, is called the diameter of that graph. The central point, or central vertex, of a connected graph is the point with the smallest associated number. This central point is considered the point most accessible to all others.

The accessibility of a vertex is measured by its degree of centrality. That is, the most accessible vertex is the one with the highest degree of centrality. Hence, relative accessibility of vertices can be based on the comparison of the associated numbers of places.

A more comprehensive index of accessibility for any given vertex is the point index of centrality. This index is based on the ratio of the sum of the shortest distances from the vertex to all other vertices to the sum of the shortest distances from all vertices to all other vertices. In other words, if d^j represents the shortest distance between vertex i and vertex j , then for every vertex i one can sum every d ^ . This is the sum of the shortest distances from the vertex i to all other vertices. The result of this summation can be represented by s^, so that the formula for s^ is:

If the values of s^ for all vertices are summed the result is a grand sum for all vertices of the graph. This is the sum of all shortest distances from all vertices to all other vertices. The result of this socond summation can be represented by S, so that the formula for Sis: Then, a point index of centrality, if represented by C., is: C:= s/s,

42 This is the so-called Bavelas point index of centrality.

Beauchamp suggests that a relative point index of centrality be calculated, in order to standardize the values of the index of centrals 43 ity. This standardization is done by using a different numerator than that suggested by Bavelas. Beauchamp’s relative point index of centrality is calculated by the following formula:

where, n is the number of vertices.

The advantage gained in using Beauchamp's relative point index of centrality is that RC^ has an upper limit of 1.00 for any vertex which is a unit distance from every other vertex. The reasoning on which Beauchamp's index is based is simply that the value of n-1 is the smallest possible value that the sum of d ^ can have for any vertex, if the edges of the graph are all assigned a unit value. For valued graphs, the upper limit of RC^ could be different from 1.00, unless the values of edges were standardized so that the minimum possible value of an edge is 1.00. When using travel time in minutes as values for graph edges, it is sufficient to be sure that no travel time is

^Murray A. Beauchamp, "An Improved Index of Centrality," Behavioral Science. X (April, 1965), 161-163. less than one minute in order to have an upper limit of 1.00 for the relative point index of centrality.

In order to obtain the point index of centrality for each vertex of a graph it is convenient to use the matrix form of data presentation.

Since the calculation of the point index of centrality calls for the summation of shortest paths from each vertex to every other vertex, the cells of the matrix contain the value of d^^ for each pair of ver­ tices. Table 6 presents the data for the calculation of the point index of centrality for each vertex of the graph in Figure 2. Table

7 presents the data for the calculation of the point index of centrality for each vertex of the graph in Figure 3. Table 6 and Table 7 also list the indices of centrality for each of the vertices of the graphs in Figure 2 and Figure 3.

There are a few measures of network structure which are based on simple ratios between terms which have already been defined as graph indicators. The ratio of the area of the graph to the number of edges in the graph is called the eta index. The formula for eta can be written as follows:

where,

A is the area of the graph,

m is the number of edges in the graph.

The division defined by this ratio yields a number which is the length of an average edge in whatever units have been used to assign values to the edges of a valued graph. TABLE 6.— Shortest Path Matrix and Indices of Vertex Centrality for Simple Road Network

0.0 15.5 71.0 51.0 92.5 96.5

15.5 0.0 55.535.5 77.0 81.0

71.0 20.0 21.5 65.555.5

35.5 20.0 0.0 41.5 45.551.0

E 92.5 77.0 21.5 41.5 0.0 47.0

F 96.5 81.0 65.5 45.5 47.0 0.0

326.5 264.5 233.5 193.5 279.5 335.5

Assoc'd. No. 96.5 81.0 71.0 51.0 92.5 96.5

RC4 0.015 0.019 0.021 0.026 0.018 0.015

■p* VO TABLE 7.— Shortest Path Matrix and Indices of Vertex Centrality for Complex Road Network.

A B C D E F

J A 0.0 14.0 30.0 42.5 51.0 86.5

B 14.0 0.0 23.5 28.5 44.5 72.5

C 30.0 23.5 0.0 17.0 21.0 61.0

D 42.5 28.5 17.0 0.0 38.0 44.0

E 51.0 44.5 21.0 38.0 0.0 46.5

F 86.5 72.5 61.0 44.0 46.5 0.0

224.0 183.0 152.5 : 170.0 201.0 - 310.5 di Assoc'd. No. 86.5 72.5 61.0 44.0 51.0 86.5

R ^ 0.022 0.027 0.033 0.029 0.025 0.016 The theta index is computed by forming a ratio between the area of the graph and the number of vertices in the graph. The formula for theta can be written as follows: 6 -A/ft where,

A is the area of the graph,

n is the number., of vertices in the graph. \ The division defined by this ratio yields a number which is the average length of edge per vertex, another way of expressing average edge length. Theta has limited usefulness in this study because the size of the graphs does not vary and areal expansion does not occur.

The pi index is computed by forming a ratio between the area of the graph and the diameter of the graph. It is termed the pi index because it is analogous to the relationship between the circumference of a circle and its diameter. The formula for the pi index can be written as follows:

where,

A is the area of the graph,

D is the diameter of the graph.

The magnitude of the pi index is a reflection of the "shape" of the graph, since the index has a lower limit of one when the diameter is 52 equal to the area of the graph and assumes greater magnitudes as area increases per unit of diameter.

A very simple index of network structure is the beta index.

This index if formed by dividing the number of edges in the graph by the number of vertices in the graph. The formula for the beta index can be written as follows:

where

m is the number of edges in the graph,

n is the number of vertices in the graph.

The beta index has a value of zero when the graph has no edges. It has a value of one for a graph with one circuit and takes on higher values as the number of redundant edges increases.

Application of the Graph Concepts and Definitions to the Puerto Rican Road Network Data

This section is devoted to explaining how the graph indicators are applied to the Puerto Rican data. The road network data for

?uerto Rico were compiled from a number of maps and tables. Frequently, it was necessary to consult several sources of information to construct a complete valued connectivity matrix for each.of the stages in the development of the network. Because of the multiple data sources and because of the large amounts of data involved in the analyses of valued graphs with seventy-five vertices, it is not feasible to give a detailed accounting of how the Puerto Rican road network data were generated and processed. Consequently, this discussion is organized 53 and presented so as to highlight the special conventions necessary for operationalizing each of the graph concepts for the Puerto Rican data.

The number of vertices in the graphs of the Puerto Rican road 0 network was always arbitrarily taken to be seventy-five. This means that many actual intersections of roads were ignored, even though these intersections did join links in the networks. There are, how­ ever, some good reasons for ignoring many intersections. First, there is no source of demographic information specific to every road inter­ section. Hence, there is no fund of data to which vertex specific indices for all of the intersections could be related. Second, as the Puerto Rican road network increases in complexity, the number of intersections becomes very large. The size of the graphs for such complex networks would be enormous. Third, there do exist longitudinal demographic data for the seventy-five urban centers chosen as vertices.

Vertex specific measures for these urban centers can be related to other locally specific data.

The restriction of the number of vertices to seventy-five created the need for several conventions to be applied to counting the number of edges in the graphs of the road network. First, a direct link was considered to exist between two vertices if it was possible to find a link between them which did not intersect another recognized vertex.

In rare cases long, concatenated links through back roads, which tech­ nically joined two vertices, were ruled out as direct links when the distance traveled on them reached ridiculous lengths in comparison with alternative routes. Second, Y-shape roads and by-pass roads 54

were taken to constitute multiple direct links, in the way this con­

vention was illustrated with the examples in the previous section.

Third, foot paths and trails, not indicated to be horse trails or

country roads, were not considered links between vertices. Fourth,

roads which crossed urbanized areas around metropolitan centers were

not considered direct links between vertices when neither of the

vertices to be linked was the metropolitan center itself.

In this study all graph indicators based on the values of

edges are expressed in terms of estimated travel times. The travel

time estimates for links and routes between vertices were made by means of a rather complicated set of calculations. These calculations

involve numerous assumptions about the effects of distance, relief,

population density, road type, and mode of transport on the travel

times over roads in Puerto Rico. These assumptions and the reasons

for making them will now be explained.

Travel time in minutes was selected as the most useful unit for

assigning values to edges, because it seemed that many other character­

istics of road networks could be translated into units of time.

Furthermore, units of time seem better than units of distance, since

the meaning of distance is profoundly altered by changes in the

technology of transportation. Other possible units for assigning

values to edges, like quality of route, are not as useful, because

they do not permit the full use of arithmetic. Travel time in minutes

seems nicely suited for longitudinal comparisons, since the unit has

had a consistently precise meaning over the decades. This observation,

of course, leaves aside questions of the various cultural definitions

of time, which may differ over the decades and between peoples. 55

In order to obtain reasonable estimates of travel times, it seemed essential to take into account the effects of the following factors:

1) road distances, 2) quality of road surfaces, 3) speed of available vehicles, 4) curvilinearity of roads, and 5) traffic congestion and speed limits in densely populated areas.

Maps and tables were assembled which would give road distance data and road type data for the Puerto Rican road network at the times of the seven censuses for which demographic data had been gathered.

Road distance connectivity matrices were compiled, with distances expressed in kilometers. Road type connectivity matrices were compiled using a four category system of classification as follows: 1) Horse

Trail, Unimproved Road, or Country Road; 2) Third Class Road, or

Improved, Unsurfaced Road; 3) Second Class Road, or Surfaced Highway;

4) First Class Road, or Modern Highway. It was assumed that a given type road implied a given average safe travel speed under ideal road conditions. Table 8 shows these assumed average travel speeds for each of the road types.

During the process of coding the road types it was convenient to take into account the maximum possible speed of vehicles available • for travel over the network. This was done in anticipation of using road types to estimate travel speeds. For example, even though there were a few surfaced roads in Puerto Rico in 1900, all roads were coded as either type 1 or type 2. The purpose in restricting the codes for

1900 roads to type 1 and type 2 was to reflect differences in the quality of road surfaces without at the same time implying an un- realistically high travel speed on the better roads. The point to be TABLE 8.— Road Type Codes and Assumed Average Safe Travel Speeds

Type of Road Road Type Assumed Average Safe Travel Speed Code Number Miles per Hour Kilometers per Hour Kilometers per Minute

Modern Highway 4 50.00 80.00 1.333

Surfaced Highway 3 37.50 60.00 1.000

Improved, Unsurfaced 2 .25.00. 40.00 0.667 Road

Country Road 1 6.25 10.00 0.167

Kjy o n 57 made here is that in coding data from maps and tables an attempt was made to systematically take into account distance, quality of road surface, and speed of available vehicles for traveling over the road networks.

Nearly all inland roads in Puerto Rico are very curvilinear.

This is because the island is covered from end to end with ranges of low mountains and hills. The peaks of these land forms seldom exceed three or four thousand feet, but the relief is extremely rough due to the weathering effects of heavy rainfall. Consequently, many roads are so treacherous that travel speeds must be reduced considerably below what the road surface would permit for straight-away driving.

In order to obtain some indication of the degree of curvilinearity of roads, the scaled or tabled road distances between vertices were compared to the airline, or desire line, distances between vertices.

The idea here is simply that the greater the ratio of the road distance to the straight line distance the more curvilinear the road. The airline distances for this ratio were computed from the scaled map coordinates of vertices.

, Once all of these data about the distances and conditions of travel over the road network had been assembled, it became possible to make rough estimates of the travel times in minutes for links of the network. This was done by combining the information about the links into the following computing formula:

s y 58 where,

T^j is the estimated travel time in minutes from vertex i to vertex j,

j is the road distance in kilometers from vertex i to vertex j,

A^j is the airline distance" in kilometers from vertex i to vertex j,

S^j is the speed constant in kilometers per minute for the type of road linking vertex i to vertex j.

This computation, when carried out for each link, yielded an estimated travel time connectivity matrix for each of the road networks. No more than face validity is claimed for this method of obtaining travel time connectivity matrices.

One factor influencing travel times through the networks has not been taken into account up to this point. This is the travel time re­ quired to pass through densely populated areas. This could be called vertex time, since it is time used to pass through what are represented as vertices on the graphs. Two possible travel situations could make accounting for vertex time important. One of these situations is the existence of many by-pass routes, especially if these by-pass routes are located in sequence along a given highway. The other situation could

/ arise from the existence of large metropolitan centers which add con­ siderable travel time to any route passing through them. Both of these situations arise numerous times in the analyses of the Puerto Rican road network data.

In order to estimate the vertex times for the seventy-five Puerto

Rican urban centers, a computing formula was devised which yields a 59 travel time estimate proportionate to the population size of the urban center. The formula is as follows: •ti = (i/s) J P/tt D where,

ti is the estimated travel time in minutes through vertex i

s is a speed constant in kilometers per minute

P^ is the population at vertex i

D is the population density constant in persons per square kilometer for Puerto Rican urban centers

7t is the constant, 3.1416.

The speed constant, s, was assigned the following values for the various graphs:

0.167 for the graphs of the 1900 network and the 1910 network,

0.333 for the graph of the 1920 network,

0.500 for the graph of the 1930 network,

0.667 for the graphs of the 1940 network, the 1950 network, and the 1960 network.

The density constant, D, was in every case given the value of 13,812 persons per square kilometer. This was considered a reasonable average density figure after careful study of topographic maps with a 1:20,000 scale.^ The estimate of average density was computed by dividing the census population by the scaled area of dense settlement in square

^ T h e maps consulted are those in the series: U. S. Geological Survey, Topographic Maps of Puerto Rico and the , Scale: 1:20,000, Washington, D. C.: U. S. Geological Survey, 1922-1964. 60 45 kilometers. No more than face validity is claimed for this method of obtaining estimates of travel times through centers of population.

The assumptions and computations which have just been described may seem highly questionable. Nevertheless, in the absence of actual clocked travel times for the Puerto Rican road network since the year

1900, they have been undertaken in the hope that they somewhat accurate­ ly reflect the relative time distances between urban centers, That is, the actual travel time estimates may be quite inaccurate; but it is hoped that they do reflect relative time distances between urban centers, as these relative time distances have been influenced by differences in distances, quality of road surfaces, speed of available vehicles, curvilinearity of roads, and travel times through centers of dense settlement.

The edge values and vertex times for the various graphs of the

Puerto Rican road network were punched into IBM cards as travel time connectivity matrices and arrays of vertex travel times. These data could then be processed by computer to yield network specific and vertex specific indicators. The computer program was written so that it would pot only perform the usual analyses of network data, but also would do analyses utilizing edge values and vertex times. In particular, these analyses involved the calculation of the valued gamma index and the

^ F o r a descriptive discussion of this procedure see: Jack P. Gibbs, "Methods and Problems in the Delimitation of Urban Units," in Jack P. Gibbs, Urban Research Methods. (Princeton, New Jersey: D. Van Nostrand Co., Inc., 1961), pp. 57-77. 61 generation of shortest travel time matrices from matrices of valued edges and arrays of vertex times.^

The valued gamma index does not incorporate the vertex times, since it is defined in terms of the reciprocal of edge values. However, the vertex specific indicators do reflect vertex times, since the short­ est travel time matrices are generated using both link times and vertex times. The area of the graph and all indices derived using the area as an element incorporate the sum of vertex times.

The point index of centrality, as formally defined, is a measure of the accessibility of a given vertex to all other vertices in the network. However, it is not really necessary to define accessibility in terms of all other vertices. It is just as possible to identify cer­ tain vertices as important centers and compute accessibility to only these selected vertices. In the analysis of the Puerto Rican road net­ work data this kind of special measure of accessibility was undertaken.

The three metropolitan cities in Puerto Rico, San Juan, Ponce, and Mayaguez, were selected as metropolitan centers. The point index of centrality was computed for every vertex, using only the shortest

travel times to the metropolitan centers in the summations to get the .

^ T h e generation of shortest travel time matrices is quite simple for non-valued graphs. It is sufficient to power binary connectivity matrices to obtain the number of links between vertices. A network search method was used to generate shortest travel time matrices for this study. Consequently, it was possible to incorporate vertex times as data. It should be obvious that a computer is essen­ tial for this analysis of valued networks, because of the many, many summations and comparisons required for generating the shortest travel time matrices. 62

terms in the formula. The index of centrality scores obtained using this technique are henceforth called accessibility to metropolitan centers.

Since the sum of vertex times can easily be obtained, a new index is possible to formulate. This will be called the mu index. It ex­ presses the ratio between the sum of vertex times and the sum of edge or link times. It may be written as follows:

where,

t£ is the vertex time for the ith vertex,

n is the number of vertices in the graph,

Tj is the travel time for the jth edge,

m is the number of edges in the graph.

This index expresses the relative magnitude between travel time spent in vertices and travel time spent on edges, for the matrix as a whole.

When applied to the graph of a road network, it would reflect increasing congestion and/or speed regulation in urban centers as a region becomes

increasingly urbanized and urban centers cover more area. CHAPTER IV

PROPOSITIONS RELATING BASIC CONCEPTS

The concepts introduced in the foregoing chapter concern either social characteristics of local populations or graph theoretic terms.

In this chapter the concepts are related to one another through sets of propositions bearing on ecological theory. The propositions are generally grouped according to the concepts involving the social characteristics of local populations. The propositions are usually stated so as to emphasize the relationships of three concepts to all others. These three concepts are: 1) population size; 2) accessi­ bility to all urban centers, or centrality; and 3) accessibility to metropolitan centers. Special attention is paid to these three concepts because it is assumed that they have high explanatory potential.

The propositions are labeled PI through P19, so that they can be easily found in the text when references are made to them. The following propositions concern the size of the populations of urban centers:

PI: The greater the population of the urban center the greater the centrality of the urban center.

P2: The greater the area of the road network the lower the dispersion of the relative population sizes of urban centers.

P3: The greater the average edge length of the road network the lower the dispersion -of the relative population sizes of urban centers.

63 64

P4: The greater the connectivity of the-road network the greater the dispersion of the relative sizes of urban centers.

P5: The greater the redundancy of the road network the greater the dispersion of the relative sizes of urban centers.

These propositions are all based on the idea that a hierarchy of dominance exists in a metropolitan region and that city size reflects the niche of the urban center in this hierarchy. The propositions conform to the central-place model for explaining the size and location of cities because they relate relative city size to distance between potentially competing service centers. That is, the propositions are consistent with the idea that as distances in the network are diminished the hierarchy of dominance becomes more pronounced.

A more complicated proposition involving three concepts can be ad­ vanced using the central-place model.

P6: For urban centers of equal centrality, the greater the accessibility to metropolitan centers the lower the population.

This proposition expresses the idea that metropolitan centers dominate satellite cities, reducing their rates of growth. The relationship may be quite specific with respect to location, since metropolitan dominance seems to affect suburbs, satellite cities, and competitive hinterland centers differently according to location.^

The following propositions concern the division of labor of popula­ tions in and around urban centers:

P7: The greater the centrality of the location the greater of degree of division of labor of the local population.

^See: Leo F. Schnore, The Urban Scene. (New York: The Free Press, 1965), pp. 77-200. 65

P8: The greater the average distance between urban centers the greater the dispersion of the relative degrees of division of labor of local populations.

P9: The greater the areal intensification of the road network the lower the dispersion of the relative degrees of division of labor of local populations.

P10: The greater the connectivity of the road network the lower the dispersion of the relative degrees of division of labor of local populations.

Pll: The greater the redundancy of the road network the lower the dispersion of the relative degrees of division of labor of local populations.

These propositions are based on the ideas of Durkheim about the effects of changes in "dynamic density" on the division of labor. ^ In this instance, it is proposed that any change in transportation and communication leading to an increase in the frequency of interaction will also lead to an increase in the degree of division of labor.

Since Durkheim also posited density of population as a cause of

"dynamic density," it should be interesting to explore the relationships between population size, centrality, accessibility to metropolitan centers, and division of labor.

P12: The greater the population size of the urban center the greater the degree of division of labor of the , local population.

P13: For urban centers of equal size, the greater the centrality the greater the degree of division of labor of the local population.

P14: For urban centers of equal size, the greater the accessibility to metropolitan centers the greater the division of labor of local populations.

Literacy is being taken as an index of "modernization," because in

most areas it seems to be greatly influenced by educational policies-and

48 Emile Durkheim, The Division of Labor in Society, (New York: The Free Press, 1964), pp. 257ff. 66 programs fostered and controlled by officials in modern, metropolitan centers. It is assumed that the ability to read and write is not only an indication of exposure to cultural innovation in the school, but also constitutes a condition of greater receptivity to ideas spread through modern mass communication systems. It is proposed that any change in transportation and communication leading to greater contact with metro­ politan centers will be associated with an increase in the rate of literacy. The following propositions are ba:;ed on these ideas about the diffusion of literacy and on the work of DeYoung and Hunt in their study of communication channels and functional literacy in Philippine barios:^

P15: The greater the population size of the urban center the greater the rate of literacy of the local population.

P16: The greater the connectivity of the road network the smaller the dispersion of rates of literacy between all urban centers.

P17: The greater the accessibility of the urban center to metropolitan centers the higher the rate of literacy of the local population.

P18: The smaller the average distance between urban centers the smaller the dispersion of rates of literacy between all urban centers.

P19: For urban centers of equal centrality, the greater the accessibility to metropolitan centers the higher the rate of literacy of the local population.

The nineteen propositions which have been set forth in this chapter can be used to derive numerous hypotheses for test using the Puerto Rican data. Chapter V will be devoted to presenting those data which arc rele­ vant to the propositions. The data will be organized so as to facilitate the testing of hypotheses drawn from the nineteen propositions. In

49 DeYoung and Hunt, Op. Cit. Chapter VI hypotheses will be tested using the data to be presented in

Chapter V, so that conclusions can be drawn about the validity of the propositions for Puerto Rican data. CHAPTER V

PRESENTATION OF THE DATA

The purpose of this chapter is to present data in an organized way, so that hypothesis testing will be facilitated. The statistics essential for hypothesis testing are listed in three tables. Two of these tables present summary statistics for thirty-two vertex specific variables. The remaining table presents the statistics for fourteen network specific measures.

After the contents of the tables have been outlined and explained, a brief summary of continuities and trends in the data is given. This summary is designed to familiarize the reader with general character­ istics of the data, before the data are used to test detailed hypoth­ eses.

The Data Tables

Table 9 contains summary statistics for the distributions of vertex specific variables. Notice that the variables listed in Table 9 are identified with short descriptive names and with mnemonic labels.

To conserve space, the mnemonic labels are employed in place of the descriptive names in following tables.

The organization of Table 9 contributes to familiarization with the mnemonic labels. The variables are listed by groups, so that there are five groups of variables in ordered sequence. The first group of variables is comprised of only centrality measures for

68 TABLE 9.— Measure? of Central Tendency and Dispersion for Distributions of Vertex Specific Variables

Variable Mnemonic Standard Coeff. of Descriptive Name Label Mean Deviation Variation

Centrality, 1900 CO 0.0020 0.0004 0.200

Centrality, 1910 cl 0.0041 0.0008 0.195 Centrality, 1920 C2 0.0058 0.0008 0.138

Centrality, 1930 C3 0.0079 0.0010 0.127

Centrality, 1940 C4 0.0100 0.0013 0.130

Centrality, 1950 C5 0.0107 0.0014 0.131

Centrality, 1960 C6 0.0111 0.0014 0.126

Accessibility to Metropolitan Centers, 1900 M0 0.0620 0.0196 0.316

Accessibility to Metropolitan Centers, 1910 Ml 0.1127 0.0239 0.212

Accessibility to Metropolitan Centers, 1920 M2 0.1575 0.0308 0.196

Accessibility to Metropolitan Centers, 1930 M3 0.2019 0.0319 0.158

Variable Mnemonic Standard Coeff. of Descriptive Name Label Mean Deviation Variation

Accessibility to Metropolitan Centers, 1940 M4 0.2541 0.0428 0.168

Accessibility to Metropolitan Centers, 1950 M5 0.2707 0.0422 0.156

Accessibility to Metropolitan Centers, 1960 M6 0.2807 0.0436 0.155

Population Size, 1900 PO 2882.1 5201.0 1.805

Population Size, 1910 PI 3740.3 7248.9 1.938

Population Size, 1920 P2 4583.6 10105.8 2.205

Population Size, 1930 P3 6401.7 16164.8 2.525

Population Size, 1940 P4 8245.1 23322.4 2.829

Population Size, 1950 P5 12419.8 42641.1 3.433

Population Size, 1960 P6 13120.4 51233.1 3.905

Degree of Division of Labor, 1900 DO 0.59 0.14 0.230 TABLE 9.— Continued

Variable Mnemonic Standard Coeff. of Descriptive Name Label Mean Deviation Variation

Degree of Division of Labor, 1930 D3 0.65 0.15 0.236

Degree of Division of Labor, 1940 D4 0.83 0.08 0.095

Degree of Division of Labor, 1950 D5 0.84 0.08 0.096

Degree of Division of Labor, 1960 D6 0.90 0.06 0.071

Rate of Literacy, 1910 LI 30.4 7.0 0.236

Rate of Literacy, 1920 L2 41.9 7.0 0.166

Rate of Literacy, 1930 L3 55.8 6.1 0.110

Rate of Literacy, 1940 L4 65.9 5.8 0.088

Rate of Literacy, 1950 L5 72.6 4.2 0.057

Rate of Literacy, 1960 L6 80.5 3.8 0.047 72 the decades from 1900 through 1960. The mnemonic labels for these variables are CO through C6. The "C" in each label stands for centrality and the diget following the "C" corresponds with the decennial to which the data apply. A similar procedure is followed in labeling the remaining twenty-five variables listed in Table 9.

In addition to introducing the system of mnemonic labels, Table 9 presents measures of central tendency and dispersion for the dis­ tributions of the thirty-two vertex specific variables. The coeffi­ cients of variation for the distributions are included, because they, rather than the standard deviations, will be used to make compari­ sons of degrees of dispersion in the distributions of vertex specific 50 variables. In particular, since the coefficients of variation are standardized relative to the magnitudes of the means of the distributions, the coefficients of variation can be used to assess differences between the degrees of dispersion from decennial to decennial for the distributions of the five groups of time sequenced variables.

Table 10 is composed entirely of a thirty-two by thirty-two ipatrix of correlations. These are the zero order correlations between, the thirty-two vertex specific variables. Rows and columns of the matrix are identified with the mnemonic labels.introduced in Table 9.

Only a few of the zero order correlations in Table 10 are used directly for the testing of hypotheses. Nevertheless, the whole

^The Coefficient of Variation is computed by dividing the Standard Deviation of the distribution by the Arithmetic Mean of the distribution. For a full explanation see: Samuel B. Richmond, Statistical Analysis, (2d. ed., New York: The Ronald Press, 1964), pp. 89-90. TABLE 10.— Correlation Matrix, Zero Order Correlations for Thirty-Two Vertex Specific Variables

CO Cl C2 C3 C4 C5 C6 MO Ml M2 M3

CO 1.000

Cl .743 1.000

C2 .748 .861 1.000

C3 .558 .692 .909 1.000

C4 .609 .736 .912 .972 1.000

C5 .563 .655 .869 .957 .979 1.000

C6 .538 .680 .881 .957 .972 ' .980 1.000

MO .809 .529 .474 .186 .228 .128 .127 1.000

Ml .613 .783 .615 .348 .385 .248 .291 .784 1.000

M2 .434 .486 .527 .308 .304 .173 .214 .747 .853 1.000

M3 .154 .215 .345 .309 .268 .152 .184 .473 .603 .868 1.000 TABLE 10. — Continued

CO Cl C2 C3 C4 C5 C6 MO Ml M2 M3

M4 .206 .249 .339 .255 .255 .130 .162 .549 .662 .900 .975

M5 .128 .130 .268 .212 .201 .109 .135 .478 .557 .837 .961

M6 .082 .073 .254 .234 .217 .132 .179 .404 .478 .784 .939

P0 .239 .186 .201 .126 .116 .109 .104 .325 .295 .306 .246

PI .270 .216 .228 .154 .147 .146 .137 .316 .287 .269 .195

P2 .264 .207 .229 .164 .156 .161 .154 .287 .254 .229 .155

P3 .242 .182 .203 .139 .134 .144 .135 .263 .224 .198 .123

P4 .227 .169 .188 .129 .125 .136 .130 .245 .208 .182 .111

P5 .225 .175 .189 .132 .131 .143 .139 .227 .193 .155 .079

P6 .221 .174 .187 .133 .131 .143 .142 .221 .186 .148 .072

DO .361 .197 .200 .173 .169 .184 .134 .307 .206 .154 .161

4> TABLE 10. — Continued

CO Cl C2 C3 C4 C5 C6 HO Ml M2 M3

D3 .158 -.035 .073 .055 .028 .017 -.025 .295 .179 .306 .396

D4 .260 .219 .338 .278 .287 .259 .253 .267 .254 .298 .277

D5 .339 .281 .412 .370 .362 .362 .354 .250 .230 .230 .183

D6 .427 .394 .442 .368 .356 .360 .362 .277 .273 .168 .016

LI .302 .177 .150 .054 .024 .016 .011 .384 .323 .278 .214

L2 .148 .109 .061 -.001 -.036 -.047 -.024 .269 .299 .250 .258

L3 .171 .038 .012 -.083 -.126 -.142 -.119 .327 .255 .240 .192

L4 .229 .149 .133 .044 .023 -.016 .027 .354 .337 .306 .241

L5 .276 .139 .135 .034 -.001 -.001 .005 .349 .265 .216 .125

L6 .348 .103 .168 .091 .043 .049 .031 .368 .183 .178 .084

<*n TABLE 1CL --Continued

M4 M5 M6 PO PI P2 P3 P4 P5 P6

M4 1.000

M5 .975 1.000

M6 .944 .980 1.000 e

PO .260 .283 .251 1.000

PI .211 .228 .195 .987 1.000

P2 .169 .188 .161 .966 .991 1.000

P3 .140 .162 .137 .946 • .975 ’ .992 1.000

P4 .129 .152 .130 .926 .960 .982 .997 1.000

P5 .099 .117 .097 .890 .934 .968 .985 .992 1.000

P6 .092 .108 .091 .874 .920 .958 .975 .985 .998 1.000

DO .143 .170 .118 .534 .517 .483 .456 .435 .392 .366 TABLE 10.— Continued

M4 M5 M6 PO PI P2 P3 P4 P5 P6 DO

D3 .377 .414 .369 .517 .520 .487 .456 .430 .384 .353 .553

D4 .273 .281 .261 .362 .372 .360 .337 .319 .300 .284 .210

D5 .171 .181 .166 .402 .408 .396 .370 .351 .327 .305 .360

D6 .023 ■^.001 -.026 .312 .316 .305 .287 .267 .245 .226 .439

LI .222 .232 .197 .657 .670 .659 .652 .641 .610 .589 .670

L2 .248 .275 .252 .614 .629 .622 .613 .601 .567 .545 .537

L3 .192 .207 .180 .506 .519 .509 .502 .490 .464 .446 .456

L4 .256 .257 .247 .394 .396 .392 .391 .388 .365 .354 .358

L5 .131 .143 .109 .444 .453 .449 .449 .441 .419 .403 .504

L6 .085 .089 .056 .395 .411 .407 .403 .392 .365 .349 .455 TABLE Id - -Continued

D3 1)4 D5 D6 LI L2 L3 L4 L5 L6

D3 1.000

D4 .519 1.000

D5 .569 .778 1.000 c

D6 .372 .348 .711 1.000

LI .608 .240 .421 .436 1.000

L2 .671 .230 .345 .385 .784 1.000

L3 .665 .237 .372 .446 • .757 ' .896 1.000

L4 .467 .285 .426 .422 .696 .712 .829 1.000

L5 .499 .239 .467 .608 .778 .749 .840 .837 1.000

L6 .489 .218 .446 .626 .683 .632 .730 .720 .881 1.000

00 79 matrix is tabled. Most of the zero order correlations contained in

Table 10 are needed for the calculation of partial correlations for

hypothesis testing. These partial correlations are used in two ways:

1) those used for specifying relationships between cross-sectional

data, and 2) those used for cross-lagged panel correlations to investi­

gate longitudinal relationships between variables. In both cases,

the additional zero order correlations serve to introduce control

variables in the calculations of partial correlations. This phase

of the hypothesis testing is described more fully in Chapter VI.

Table 11 presents the values of the network specific indicators

for the seven points in time for which Puerto Rican road network

data were analyzed. The fourteen network specific indicators listed

in Table 11 are those described in detail in Chapter III. All entries

in Table 11 for Sum of Vertex Values, Diameter, and Area are in minutes of travel time. Units for other entries are given in the

table as needed. Many of the values tabled are ratio numbers, to which units of measure are inapplicable.

Descriptive Summary of the Tabled Data

This section is intended to give the reader a general perspective

on the data, before the data are used to test specific hypotheses.

It includes a brief review of striking continuities and gross trends

revealed by inspection of the tables.

Some continuities and trends in the Puerto Rican Data for the

period 1900 to 1960 are:

1. The road network develops from one with relatively simple structure to one with relatively complex structure, Table 11. 80

TABLE 11.--Measures of Network Structure, Puerto Rican Road Network

Date

Network Characteristic 1900 1910 1920 1930

Number of Vertices 75 75 75 75

Number of Edges 115 152 157 184

Sum of Vertex Values 197.8 221.8 120.6 92.3

Diameter 1514.7 567.5 398.3 305.3

Area 12343.5 18880.6 14608.1 12929.9

Cyclomatic Number 41 78 83 110

Alpha 0.02 0.03 0.03 0.04

Beta 1.5333 2.0267 2.0933 2.4533

Gamma 0.0414 0.0548 0.0566 0.0663

Valued Gamma Index • 0.0008 0.0018 0.0021 0.0031

Eta 107.3 124.2 93.1 70.3

Pi 8.15 33.27 36.68 42.35

Theta 164.6 251.7 194.8 172.4

Mu 0.016 0.012 ' 0.008 0.007 TABLE 11.--Continued

Date Network Characteristic 1940 1950 1960

Number of Vertices 75 75 75

Number of Edges 186 236 269

Sum of Vertex Values 76.0 87.6 86.6

Diameter 244.0 223.0 207.3

Area 5758.2 7030.5 8516.9

Cyclomatic Number 112 162 195

Alpha 0.04 0.06 0.07

Beta 2.4800 3.1467 3.5867

Gamma 0.0670 0.0850 0.0969

Valued Gamma Index 0.0040 0.0051 0.0055

Eta 31.0 29.8 31.7

Pi 23.6 31.5 41.1

Theta 76.8 93.7 113.6

Mu 0.013 0.013 0.010 82

2. Travel times in the network are reduced, Table 11.

3. Urban centers become more accessible to one another and the differences between urban centers in accessi­ bility are reduced, Table 9.

4. Urban centers become more accessible to metropolitan centers and the differences between urban centers in accessibility to metropolitan centers are reduced, Table 9.

5. The correlation between centrality (or accessibility ; to all urban centers) and accessibility to metropolitan centers decreases from high positive to low positive, suggesting that these two indicators of accessibility become less similar and more complementary, Table 10.

6. The populations of urban centers increase and the differences between the population sizes of urban centers increase, Table 9.

7. The degree of division of labor increases and the differences between local populations in degree of division of labor decrease, Table 9.

8. The rate of literacy increases and the differences between local populations in rates of literacy de­ crease, Table 9.

9. The correlations between time sequenced vertex specific variables are moderately high to high, suggesting gradual change in the period from 1900 through 1960, Table 10.

10. The correlations between the time sequenced population size variables are very high, suggesting little change , in the relative population sizes of Puerto Rican urban centers, Table 10.

11. The correlations between non-time sequenced vertex specific variables are consistently low positive to moderate positive, suggesting the existence of many weak to moderately strong relationships between the three different characteristics of the population and the two different indicators of accessibility, Table 10.

In the most general terms, these continuities and trends suggest that Puerto Rico has experienced urbanization, increasing division of labor, increasing literacy, and considerable development of its road network during the sixty years from 1900 through 1960. The tests of hypotheses will reveal more precisely the nature and strength of the relationships between particular variables. CHAPTER VI

TESTS OF TlflS HYPOTHESES

In this chapter twenty-nine hypotheses are tested. Each of these hypotheses is derived from one of the nineteen general propositions set forth in Chapter IV. If the results of the tests of the hypotheses do not contradict expectations about the data based on the general propo­ sitions, then there will be reason to have greater confidence in the validity of the propositions for explaining the regional urban ecology of Puerto Rico. If the results of the tests of the hypotheses contra­ dict these expectations, then there will be reason to question the gen­ eral propositions.

The data being used here to test hypotheses were not gathered by a random process sampling procedure. The data were gathered with the intention of describing conditions in Puerto Rico as accurately and completely as possible, within the limitations of the scope and methods of the study.

The main purpose in testing the hypotheses is not to generalize to some clearly defined larger population of urban centers or regions of urban centers. Nevertheless, conventional statistical tests of sig­ nificance are applied to establish cut-off values for accepting or rejecting hypotheses. The rationale for this use of tests of signif- cance is different from that on which predictive studies rest. The

84 85 rationale here is to use statistical tests as a criterion for deciding whether or not the observed relationships depart significantly from a condition of entirely random variation. That is, a null hypothesis of no relationship between variables will be accepted if the probability of getting an association of the observed strength and direction under conditions of random variation is greater than 0.05. Since the purpose in following this hypothesis testing procedure is not to generalize to a sampled population, or to predict subsequent events, the tests of significance should be regarded as convenient, but arbitrary, standards for making decisions about the data from an elaborate case study.

The Notation for Partial Correlations

Some of the tests of hypotheses are made with first order partial correlations.^^ These partial correlations are calculated from the zero order correlations presented in Table 10. The formula for the first order partial correlation is rather simple, but it will be pre­ sented here to illustrate the notation for indicating which are the related variables and which is the control variable.

If r^j stands for the zero order, or total, correlation between

( variable i and variable j, then the first order partial correlation, rij.k* is:

y _ Ej -(riQOfr) j,k jt-u

■^See: Hubert B. Blalock, Jr., Social Statistics, (New York: McGraw-Hill Book Company, Inc., 1960), pp. 326-336. 86

Notice that the subscript notation indicates two related variables and one control variable. Thus, r^j ^ means the correlation between variable i and variable j, controlling for variable k.

The mnemonic labels introduced in Chapter V will be used to indi­ cate related and control variables in this study. For example, rPl D1 Cl means t^ie correlation between population size in 1910 and degree of division of labor in 1910, controlling for centrality in 1910.

The Cross-lagged Panel Comparisons

Cross-lagged panel comparisons are used to test hypotheses con- 52 cerning the longitudinal relationships between variables. The logic of this procedure can be illustrated with a simple line diagram. See

Figure 4.

Assuming some kind of causal relationship between variable X and variable Y, the issue becomes whether or not X causes Y or vice versa.

The simplest approach to settling this issue would be to compare rX]Y2 with rY^X2’ 0n t*ie assumPt^on that the stronger of the two correlations would indicate the more likely causal relationship. However, these time- lagged zero order correlations fail to take into account the effect of / * Y^ on X2 through x^ and the effect of Xi on Y2 through Yi. That is, the time-lagged zero order correlations are likely to overestimate the strength of the putative causal relationship. This is because they do not correct for the original cross-sectional correlation, rx^Y^’ just as they do not correct for the correlation between the scores of a var­ iable at successive points in time, rXjX2 or rY],Y2'

■*^This discussion of cross-lagged panel comparisons is adapted from: George W. Bohrnstedt, 0j>. Cit., pp. 114-120. Figure 4.— Possible Correlations Between X and Y at Two Points in Time

X, X A A

fx.Y, rx;

fy, x V Y i These considerations lead to the conclusion that the time-lagged

panel correlations used to investigate longitudinal relationships

should be the first order partial correlations. For example, one might

inquire about the longitudinal relationships between centrality and

population size by comparing the magnitudes of the two time-lagged

partials, r ^ p£ ^ and rpj^ Q2 ^ q^. Such comparisons are undertaken

whenever appropriate in the course of hypothesis testing.

The Calculation of T-Values CO The test of significance which is employed is the t-test. For

partial correlations the formula for t is:

where,

r is the partial correlation,

N is the number of cases,

k is the number of control variables used in calculating the partial correlation.

Since N always equals seventy-five for the correlations between vertex specific measures, it is possible to find a cut-off value for the

0.05 level of significance. This value of t is 1.67. Therefore, any

correlation corresponding to a t-value of 1.67 or higher will be con­

sidered grounds for rejecting a null hypothesis.

CO JJFor a discussion of tests of significance for total and partial correlations see: Helen M. Walker and Joseph Lev, Statistical Inference, (New York: Holt, Rinehart and Winston, 1953), pp. 231-344. 89

The Tests of Network Specific Hypotheses

Many of the hypotheses tested refer to relationships between net­ work specific measures and measures of central tendency and dispersion for the distributions of vertex specific variables at the seven decenn­ ials. These hypotheses concern whether or not trends are related.

Comparisons of the values in Table 9 and Table 11 show that trends in the variables to be related are usually very clear and simple in nature. In many instances when an applicable measure of association might be calculated, such as the rank-order correlation of mean scores, it would be equal to plus or minus one, indicating "perfect" association.

Hence, for purposes of hypothesis testing it will sometimes be suffi­ cient to note that the trends are regular and concomitant in some specified direction to reject a null hypothesis. Since very few cal­ culated measures of association will be needed for this part of the hypothesis testing and since the number of cases for these measures is always less than or equal to seven, no correlation matrix is being pre­ sented for them.

When one or the other of the trends to be related is not completely regular, a rank-order correlation between values will be the measure of association. Its calculated value will simply be reported at the time of hypothesis testing. A rank order correlation of 0.71 or higher is 54 sufficient to reject a null hypothesis with N equal to seven.

The Tests

The hypotheses will now be listed and tested one by one. They will be labeled in a way which identifies the general proposition from

54Ibid., p. 478. 90 which each is derived. In particular, each hypothesis will be identi­ fied with a whole number and a decimal number. The whole number will refer to the proposition from which the hypothesis is derived, while the decimal number will distinguish between multiple hypotheses derived from the same proposition. For example, H7.1 and H7.2 are the labels for the first two hypotheses derived from proposition seven. Proposition numbers were assigned during the discussion of the propositions in

Chapter IV.

Hl.l: The size of the population of the urban center is directly related to the centrality of the urban center.

Entries 1 through 7 in Table 12 show the results of tests for cross-sectional relationship between population size and centrality.

The null hypothesis can be rejected only on the basis of the data for

1900, 1910, and 1920.

A cross-lagged panel analysis of longitudinal relationships betweai population size and centrality yields uniformly low, non-significant correlations, with only one substantively unimportant exception. These uniformly low partials and t-values are not tabled. Since the vertex specific data fail to.support any substantive hypothesis about longitu­ dinal relationships between centrality and population size, the null hypothesis can not be rejected.

111.2: The average population size of urban centers is directly related to the average centrality of urban centers.

Table 9 shows that in the aggregate there is a strong positive association between the trend in mean population size and the trend in TABLE 12.--Correlations, Partial Correlations, and T-Values

T-Value T-Value Total Partial Related Control of Partial- of Total Number Correlation Correlation Variables Variable Correlation Correlation

1 0.239 CO P0 2.103

2 0.216 Cl PI 1.890

3 0.229 C2 P2 2.010

4 0.139 C3 P3 1.199

5 0.125 C4 P4 1.076

6 0.143 C5 P5 1.234

7 0.142 C6 P6 1.226

8 0.325 0.231 PO M0 CO 2.011 2.936

9 0.287 . 0.194 PI Ml Cl 1.679 2.560

10 0.229 0.131 P2 M2 C2 1.121 2.010

11 0.123 0.085 P3 M3 C3 0.724 1.059

12 0.129 0.101 P4 M4 C4 0.863 1.111 TABLE 12.— Continued

T-Value T-Value Total Partial Related Control of Partial of Total Number Correlation Correlation Variables Variable Correlation Correlation

13 0.117 0.103 P5 M5 C5 0.879 1.007

14 0.091 0.067 P6 M6 C6 0.573 0.781

15 0.361 CO DO 3.307

' 16 0.055 C3 D3 0.471

17 0.287 C4 D4 2.560

18 0.362 C5 D5 3.318

19 0.362 C6 D6 3.318

.20 0.362 0.230 C4 D5 D4 2.010 3.318

21 0.356 . 0.285 C4 D6 D4 2.525 3.255

22 0.360 0.157 C5 D6 D5 1.345 3.297

23 0.259 -0.113 D4 C5 C4 -0.961 ■ 2.291

24 0.253 -0.115 D4 C6 C4 -0.985 2.234

VON> TABLE 12.--Continued

T-Value T-Value Total Partial Related Control of Partial- of Total Number Correlation Correlation Variables Variable Correlation Correlation

25 0.354 -0.004 D5 C6 C5 -0.035 3.234

26 0.534 P0 DO 5.396

27 0.456 P3 D3 4.378

28 0.319 P4 D4 2.876

29 0.327 P5 D5 2.956

30 0.229 P6 06 1.982

31 0.351 0.173 P4 D5 D4 1.488 3.203

32 0.267 0.176 P4 D6 D4 1.513 2.367

33 0.245 . 0.019 ?5 D6 D5 0.160 2.159

34 0.300 -0.137 D4 P5 P4 -1.178 2.687

35 0.284 -0.185 D4 P6 P4 -1.595 2.531

36 0.305 -0.357 D5 P6 P5 -3.246 2.736

VO TABLE 12.— Continued

T-Value T-Value Total Partial Related Control of Partial of Total Number Correlation Correlation Variables Variable Correlation Correlation

37 0.361 0.284 CO DO , PO 2.516 3.307

38 0.055 -0.010 C3 D3 P3 -0.081 0.471

39 0.287 0.263 C4 D4 P4 2.311 2.560

■ 40 0.362 0.337 C5 D5 P5 3.038 3.318

41 0.362 0.342 C6 D6 P6 3.090 3.318

42 0.307 0.167 M0 DO PO 1.436 2.756

43 0.396 0.385 M3 D3 P3 3.538 3.685

44 0.273 0.247 M4 D4 P4 2.160 2.425

45 0.181 0.152 M5 D5 P5 1.306 1.572

46 ‘ -0.026 -0.048 M6 D6 P6 -0.408 -0.222

47 0.670 PI LI 7.711

48 0.622 P2 L2 6.787

to TABLE 12.— Continued

T-Value T-Value Total Partial Related Control of Partial of Total Number Correlation Correlation Variables Variable Correlation Correlation

49 0.502 P3 L3 4.959

50 0.388 P4 L4 3.597

51 0.419 P5 L5 3.943

52 0.349 P6 L6 3.182

53 0.323 ' 0.301 Ml LI Cl 2.680 2.916

54 0.250 0.257 M2 L2 C2 2.255 2.206

55 0.192 0.230 M3 L3 C3 2.002 1.672

56 0.256 0.259 M4 L4 C4 2.273 2.263

‘ 57 0.143 0.144 M5 L5 C5 1.234 1.234

58 0.056 0.051 M6 L6 C6 0.436 0.479

VOUl 96 mean centrality score. This relationship between mean scores would pro­ duce a rank-order correlation of 1.00. This is sufficient to reject the null hypothesis.

H2.1: The magnitude of the coefficient of variation for popula­ tion size is inversely related to the magnitude of the area of the road network.

Inspection of Table 9 and Table 11 reveals that the association between these variables is considerably less than "perfect." Since the rank-order correlation is only equal to -0.68, the null hypothesis can not be rejected.

H3.1: The magnitude of the eta index is inversely related to the magnitude of the coefficient of variation for population size.

Inspection of Table 9 and Table 11 reveals that the association between these variables is less than "perfect." However, since, the rank-order correlation is equal to -0.86, the null hypothesis is rejec­ ted.

H4.1: The magnitude of the gamma index is directly related to , the magnitude of the coefficient of variation for popula-- tion size.

Inspection of Table 9 and Table 11 shows that there is a "perfect" direct rank-order relationship between gamma and the coefficient of variation for population size. The null hypothesis is rejected.

H4.2: The magnitude of the valued gamma index is directly related to the magnitude of the coefficient of variation for population size. 97

Inspection of Table 9 and Table 11 shows that there is a "perfect"

direct rank-order relationship between the valued gamma index and the

coefficient of variation for population size. The null hypothesis is

rejected.

H5.1: The magnitude of the alpha index is directly related to the magnitude of the coefficient of variation for popula­ tion size.

Inspection of Table 9 and Table 11 shows that there is a "perfect" direct rank-order relationship between the alpha index and the coeffi­ cient of variation for population size. The null hypothesis is rejected.

H6.1: The size of the population of the urban center is inversely related to its accessibility to metropolitan centers, when the centrality of the urban center is controlled.

Entries 8 through 14 in Table 12 show the results of tests for

this specifying hypothesis. Notice that all of the correlations for

these data are positive. Consequently, all of the relationships are

opposite to the direction stated in the hypothesis. Furthermore, the

data for 1900 and for 1910 yield correlations which are positive and

Jiigh enough to be statistically significant. Since no correlation is

in the direction stated by the substantive hypothesis, the null

hypothesis can not be rejected.

H7.1: The degree of division of labor at a location is directly related to the centrality of the location.

55 Since the gamma index and the valued gamma index have a "perfect" direct rank-order correlation, they will always produce the same rank order correlations with any third variable. Therefore, no more hypotheses will be tested using the valued gamma index. 98

Entries 15 through 19 in Table 12 show the results of tests for this hypothesis. The null hypothesis can be rejected with these cross- sectional data from every decennial except 1930,

H7.2: Assuming causality, high centrality at time^ produces high degree of division of labor at time2*

Entries 20 through 22 in Table 12 show that analyses of cross - lagged panel correlations yield significant correlations in two out of three instances. Furthermore, the partial correlation which is not statistically significant is the one for the latest and shortest time span, the time span for which a relatively low correlation might be expected. These three time-lagged partial correlations support rejection of the null hypothesis.

H7.3: Assuming causality, high degree of division of labor at time^ produces high centrality at time2>

Entries 23 through 25 in Table 12 show that analyses of cross - lagged panel correlations fail to yield significant correlations for any of the partials. The null hypothesis can not be rejected.

/ H8.1: The magnitude of the eta index is directly related to the magnitude of the coefficient of variation for the degree of division of labor.

Inspection of Table 9 and Table 11 reveals that the association between these variables for the 1940, 1950, and 1960 decennials, during which trend data for degree of division of labor are considered 99 reliable, is completely opposite to that stated in the hypothesis. The null hypothesis is not rejected. °

H9.1: The magnitude of the area of the road network is inversely related to the magnitude of the coefficient of variation for.degree of division of labor.

Inspection of Table 9 and Table 11 reveals that the association between these variables for the 1940, 1950, and 1960 decennials is less than perfect. The null hypothesis is not rejected.

H10.1: The magnitude of the gamma index is inversely related to the magnitude of the coefficient of variation for degree of division of labor.

Inspection of Table 9 and Table 11 reveals that the association between these variables for the 1940, 1950, and 1960 decennials is almost opposite to that stated in the hypothesis. The null hypothesis is not rejected.

Hll.l: The magnitude of the alpha index is inversely related to the magnitude of the coefficient of variation for degree of division of labor.

/ Inspection of Table 9 and Table 11 reveals that the association between these variables for the 1940, 1950, and 1960 decennials is almost opposite to that stated in the substantive hypothesis. The null hypothesis is not rejected.

H12.1: The size of the population of the urban center is directly related to the degree of division of labor of the local population.

An N of only three is too small to reject the null hypothesis even with a rank-order correlation of plus or minus one. 100

Entries 26 through 30 in Table 12 show the results of tests for cross-sectional relationship between population size and degree of division of labor. The null hypothesis is rejected in every test.

H12.2: Assuming causality, large population at time^ produces high degree of division of labor at time2-

Entries 31 through 33 in Table 12 reveal that analyses of cross­ lagged panel correlations fail to yield any significant partials. Con­ sequently, these cross-lagged panel correlations do not support rejection of the null hypothesis.

H12.3: Assuming causality, high degree of division of labor at time^ produces large population at time2.

Entries 34 through 36 in Table 12 show that analyses of cross­ lagged panel correlations fail to yield any significant partials. These cross-lagged panel correlations do not support rejection of the null hypothesis.

H13.1: The degree of division of labor is directly related to centrality, when the population size of the urban center is controlled.

Entries 37 through 41 in Table 12 show the results of tests for this specifying hypothesis. The null hypothesis can be rejected with these cross-sectional data for every decennial except 1930.

H14.1: The degree of division of labor is directly related to accessibility to metropolitan centers, when the population size of the urban center is controlled. 101

Entries 42 through 46 in Table 12 show the results of tests for this specifying hypothesis. The null hypothesis can be rejected only with cross-sectional data from 1930 and 1940. The null hypothesis can not be rejected with cross-sectional data from 1900, 1950, and 1960.

H15.1; The size of the population of the urban center is directly related to the rate of literacy of the local population.

Entries 47 through 52 in Table 12 show the results of the tests for cross-sectional relationship between population size and rate of literacy. The null hypothesis is rejected in every test.

H15.2: Assuming causality, large population at time^ produces high rate of literacy at time2 .

The results of the tests of the cross-lagged panel correlations for this hypothesis are not tabled. Of the fifteen tests of partial correlations, only two t-values were above the cut-off point for signi­ ficance. The partial correlations themselves are nearly all clustered near zero. Thus, with only two exceptions, the null hypothesis can not be rejected.

H15.3: Assuming causality, high rate of literacy at time^ pro­ duces large population at time2.

The results of the tests of cross-lagged panel correlations for

this hypothesis are not tabled. Of the fifteen tests of partial correlations, not one t-value was above the cut-off point for signifi­ cance. All of the partial correlations were negative and near zero.

The null hypothesis can not be rejected. 102

H16.1: The magnitude of the gamma index is inversely related to the magnitude of the coefficient of variation for the rate of literacy.

Inspection of Table 9 and Table 11 shows that there is a "perfect" inverse rank-order relationship between gamma and the coefficient of variation for rate of literacy. ^ The null hypothesis is rejected.

H17.1: Accessibility to metropolitan centers is directly related to the rate of literacy.

Entries 53 through 58 in Table 11 show the results of tests for cross-sectional relationship between accessibility to metropolitan centers and literacy. The null hypothesis can be rejected only with the data for 1910, 1920, 1930, and 1940.

H17.2: Assuming causality, high accessibility to metropolitan centers at timei produces high rate, of literacy at time2.

The results of the tests of cross-lagged panel correlations for this hypothesis are not tabled. Of the fifteen tests of partial correlations, not one t-value was above the cut-off point for signifi­ cance. The null hypothesis can not be rejected.

/

H17.3: Assuming causality, high rate of literacy at timei pro­ duces high accessibility to metropolitan centers at time2*

The results of the tests of cross-lagged panel correlations for this hypothesis are not tabled. Of the fifteen tests of partial

■^The trend data for literacy are from six decennials. The cut­ off value for significance at the 0.05 level for rank-order correla­ tion is 0.83. 103

correlations, not one t-value was above the cut-off point for signifi­

cance. The null hypothesis can not be rejected.

H18.1: The magnitude of the eta index is directly related to the magnitude of the coefficient of variation for the rate of literacy.

Inspection of Table 9 and Table 11 shows that there is a somewhat

less than "perfect" direct relationship between eta and the coefficient

of variation for rate of literacy. However, since the value of the

rank-order correlation between these two variables, with N equal to

six, is 0.83, the null hypothesis is rejected.

H19.1: Accessibility to metropolitan centers is directly related to rate of literacy, when centrality is controlled.

Entries 53 through 58 in Table 12 show the results of tests for

this specifying hypothesis. The null hypothesis can be rejected only

with cross-sectional data from 1910, 1920, 1930, and 1940. However,

there may be additional justification for rejecting the null hypothesis

entirely. All of the correlations are in the expected direction.

Those two correlations which are not significant are for decennials

'when correlations might be expected to be relatively low because of

low variances in the distributions. CHAPTER VII

SUMMARY AND CONCLUSIONS

This chapter is composed of four parts. The first part gives an interpretation of the findings. The second part summarizes the impli­ cations of the findings for the central-place model. The third part identifies the most important limitations of the study. The fourth part suggests possible lines for further research.

Interpretation of the Findings

The first six propositions set forth in Chapter IV are presumed to be consistent with a central-place model of a region of urban centers. Hypotheses derived from these propostions have been tested using Puerto Rican data. Now that the hypothesis testing is complete, it is appropriate to pose two basic theoretical questions. The first question concerns the applicability and universality of the central- place model for explaining the size and location of cities. That is, to what degree do the Puerto Rican data support or contradict the idea that the central-place model explains the size and location of cities?

The second question concerns the influence of the development of the highway system on the region of urban centers. That is, how is highway development related to the size and location of cities?

The Puerto Rican data seem only partially consistent with the central-place model. The cross-sectional data are inconclusive on this

104 105 point. A strong positive correlation between the size of urban centers and centrality would have been consistent with the idea that highway network structure reflects a dominance hierarchy. In other words, a set of strong positive correlations would have been consistent with the idea that large urban centers dominate other centers by being more centrally located in the network. The correlations are consistently positive, but they are not strong enough to be conclusive.

Significant cross-lagged panel correlations for centrality and population size would have been consistent with the existence of a dominance hierarchy. Such correlations might have indicated that large centers were being given preference in the design of the network, or conversely that changes in the network were changing dominance patterns, as reflected by differential population growth.

A clear-cut hierarchy of dominance may exist between urban centers in Puerto Rico. If it does, the correlations for vertex specific measures of size and centrality do not reflect it very clearly.

The trend data are very consistent, suggesting that highway develop­ ment may contribute to higher degrees of dominance. A plausible cycle jor causal sequence might run as follows:

1) As network structure becomes more complex,travel times in the network are diminished.

2) As travel times in the network are diminished, competition between urban centers is increased.

3) As competition between urban centers is increased, some urban centers grow relatively more rapidly than others. 106

Tests of the hypotheses derived from the first six propositions give some support to the idea that a central-place model is applicable to the urban centers of Puerto Rico. Evidence supporting the central place pattern is provided by aggregate trend data and network specific data. These data give results consistent with the idea that a hierarchy of dominance is becoming more pronounced as the road network develops.

However, if there is a central place pattern, one would also expect vertex specific data to demonstrate relationships consistent with a central-place model. Unfortunately, the vertex specific data fail to conclusively demonstrate such relationships.

The tests of hypotheses derived from propositions seven through fourteen all directly concern the division of labor. Three basic theoretical questions will be posed in the light of the hypothesis tests for these propositions.

The first question concerns the components of "dynamic density" posited by Durkheim as leading independently or together to increased division of labor. That is, do large numbers of people contribute to a high degree of division of labor? Does high accessibility contri­ bute to a high degree of division of labor?

The second question concerns the influence of the development of the highway system on the division of labor. -That is, can specific statements be made about network structure and relative degrees of division of labor?

The third question concerns the relationship between the central- place model and information about degrees of division of labor. That is, can information about the division of labor be used to support or contradict the central-place model? 107

The Puerto Rican data support Durkheim's conception of the com­

ponents of "dynamic density" leading to division of labor. High

centrality (which is taken here as equivalent to high exposure to trans­

portation and communication for Durkheim) is directly related to degree

of division of labor cross-sectionally. Furthermore, cross-lagged

panel correlations support the hypothesis that high centrality leads to,

or produces, high degree of division of labor.

The cross-sectional data also support the idea that large popula­

tion size is directly related to high degree of division of labor.

However, cross-lagged panel correlations are not significant for popu­

lation size and degree of division of labor. Thus, in Puerto Rico

degree of division of labor may be more dependent on centrality than

on size of population.

Very little can be said about changes in network structure and

changes in degrees of division of labor. The small number of cases

for hypothesis testing and the subsequent failures to reject null

hypotheses generate very little substantive knowledge. However, if

all of the data are considered from 1900 through 1960, there is an

/ apparent long range trend for the aggregate degree of division of

labor to increase as the degree of complexity of the road network

increases.

There seem to be no crucial ways in which information about the

relative degree of division of labor contradicts the central-place

model. There is a disturbing degree of variety in the magnitudes

and directions of correlations and partial correlations, when all of

the relationships are considered together. Division of labor seems 108 most consistently related to centrality. It is also directly related to population size. Mutual direct association between population size, centrality, and degree of division of labor is entirely consistent with the central-place model. That is, dominant cities have large popula­ tion, central location, and complex division of labor; while other cities in the region are less developed in these respects.

The tests of hypotheses derived from propositions fifteen through nineteen all directly concern literacy. Two theoretically relevant questions will be posed in the light of the findings. The first question concerns the relationship between the central-place model and information about rates of literacy. That is, can information about rates of literacy be used to support or contradict the central- place model? The second question concerns the influence of the development of the highway system on rates of literacy. That is, can changes in the road network be related to changes in literacy?

The results of the tests of the hypotheses involving literacy can be interpreted rather simply as follows: The larger population centers have consistently had relatively high rates of literacy for the last sixty years. Literacy rates generally decline as one travels away from metropolitan centers, these patterns of rates and distribu­ tions have been rather stable for the last sixty years, but the relative differences in rates, or the degrees of dispersion of rates, have been declining during the whole period, so that in the last few decades the dispersion is getting quite small.

The development of the highway system seems directly associated with the diffusion of higher rates of literacy. However, the process 109

is apparently so regular and gradual, that cross-lagged panel compar­

isons fail to detect significant change.

Implications of the Findings for the Central-Place Model

According to the central-place model, urban centers in any region

are different from one another in size, level of specialization, and

centrality. Furthermore, the region exhibits a hierarchy of dominance

among service centers; so that larger, more highly specialised centers

dominate surrounding smaller, less highly specialized towns and vil­

lages. Finally, size and spacing of urban centers reflect the hierarchy

of dominance.

The Puerto Rican data give clear indications of differences

between urban centers. The size of population and the degree of

division of labor differ from location to location. Furthermore,

these two variables are directly associated with one another and with

rate of literacy. Such gradients and relationships are typical of

rural-urban, or village-metropolitan, differences.

If it is assumed that typical village-metropolitan gradients

'are evidence of a hierarchy of dominance, the crucial question

about the applicability of the central-place model becomes one

concerning relative location of differing urban centers. That is,

are the urban centers in Puerto Rico located relative to one another

in ways consistent with the central-place model?

For purposes of this study, relative location means relative

location on the road network. The relative location and spacing of

cities are expressed in terms of centrality and accessibility to 110

metropolitan centers,, Therefore, one way to demonstrate the plausibil­

ity of the central-place model is to demonstrate relationships between

relative location on the road network and niche in the hierarchy of

dominance. This implies significant positive correlations between

indicators of dominance and indicators of centrality.

In general, the Puerto Rican data do not consistently yield

significant positive correlations between indicators of dominance

and indicators of centrality. This is especially true of correlations of variables for decennials later than 1940. On the other hand, many correlations are in the direction expected, even though their strength

is not sufficient to give a t-value above the cut-off for significance.

These results, being somewhat inconclusive, permit at least four different judgments about the implications of this study for the central-place model. 1) The central-place model has rather limited applicability to the regional urban ecology of Puerto Rico and its explanatory power may be decreasing with each passing decade. 2) As

the road network of Puerto Rico becomes more complex in structure, relative location on the network becomes less highly associated with niche in the hierarchy of dominance and with location in a central- place pattern. 3) The road network data are not sufficiently reliable indicators of relative location and are not suitable for a conclusive

test of the central-place model. 4) The weak positive correlations obtained between indicators of dorainace and indicators of centrality are, in the aggregate, support for the applicability of the central- place model to the regional urban ecology of Puerto Rico. There is also the possibility that all of these judgments are partially true. Ill

Limitations of the Study

The study ignores socio-cultural factors which might have more satisfactorily explained the relationships and trends found in the data.

There is no way to assess, within the context of this study, the possi­ bilities for explanation ignored by restricting the scope of inquiry.

That assessment must be made in the light of comparable studies by those working in the socio-cultural tradition of urban ecology. The socio­ cultural tradition has already been discussed in Chapter II.

The most serious limitations of the study were products of the questionable reliability of the travel time estimates. This applies to both edge values and vertex times.

The techniques and formulae employed in this study are capable of yielding quite reliable indicators of properties of valued graphs. It is unfortunate that these techniques could not have been used with highly accurate data. The vertex specific measures seem to be the indicators which suffered most from unreliability. In addition, the network specific measures depending on sums of travel times for their calculation yield somewhat erratic trend data. All of these problems with the data could come from crude estimates of travel times.

Another shortcoming was the lack of some kinds of demographic data

for 1900, 1910, and 1920. The division of labor data suffered most in

this respect. Furthermore, changes in operational definitions of categories cut down the period of time for which time-lagged panel comparisons could be confidently applied to the division of labor data.

This was especially unfortunate, since the division of labor data yielded some of the most substantively interesting results. Further Research

This section contains suggestions for future research. The sug­ gestions are based on the experience gained in doing this study, as well as on the results of the study.

There should be advantages in focusing more intensively on smaller regions. Not only could more accurate data be gathered, but the range of information could be extended. The summary statistics appearing in census reports are very limited in the amount of information they convey about local places. This suggestion applies especially to local towns and service centers where small scale activities and fine degrees of CO functional specialization prevail.

Another potentially fruitful line of research would involve the use of typological classifications of cities, towns, and villages. For example, urban centers might be classified according to their socio­ economic functions or according to their sub-regional locations. After meaningful classifications had been made, the central-place model and road network data could be applied.

CO For example, the central-place model and road network data could be formally applied in conjunction with the kinds of anthropolo­ gical data Marvin Harris gathered for towns and villages in rural Brazil. Harris often mentions factors like enter-town rivalries, functional specialization, urban-rural differences, and transportation costs in his descriptions. See: Marvin Harris, Town and Country in Brazil, (New York: Columbia University Press, 1956). SOURCES CONSULTED

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