35 Kinship Network Analysis

Klaus Hamberger, Michael Houseman, and Douglas R. White

KINSHIP IN A NETWORK PERSPECTIVE kinds of kinship ties between spouses may be overrepresented. Marriage rules and prohibitions, Kinship, like language, is a structure, not a sub- but also residential organization, social morphol- stance.1 The distinctive features of kinship networks ogy, and so forth, affect the relative frequencies of reside less in how their constitutive ties – be they different types of cycles in a kinship network. biological, jural, ritual, symbolic, or whatever – are Analyzing the distribution of cycles therefore defined and established than in the way these is the key to kinship network classification and ties are organized. Kinship network theory interpretation. is thus not just another “application” of general network theoretic methods to a particular social domain but a specific branch of social network theory in itself, defined by its own axioms and Paths and cycles in kinship networks described by its own theorems. Kinship networks are characterized by the Kinship network theory thus rests on a theory of interplay of three fundamental principles: filia- cyclic configurations. We call a path an alternat- tion, marriage, and gender. We ordinarily repre- ing sequence of vertices and lines (edges or arcs sent filiation by a set of arcs (descent arcs) that are of whatever direction), where every vertex is inci- directed from parents to children, and marriage dent with the lines that precede and follow it in the by a set of undirected edges (marriage edges) sequence, and all vertices are distinct. If, by con- between spouses (for alternative representations trast, the first and the last vertex are identical (all of kinship networks without edges, see 2.2 below). others being distinct), we obtain a cycle. A path is Kinship networks thus are mixed graphs, contain- said to be closed by a line connecting its first and ing both arcs and edges. Gender is usually taken its last vertex, so that adding that line turns the into account by a partitioning of the vertex set path into a cycle. A path or cycle is called oriented (the gender partition), usually into two or three if all lines are arcs oriented in the same direction.3 disjoint classes (male, female, and possibly A weakly acyclic network is one that contains no unknown sex). oriented cycles and an acyclic network contains The characteristic features of kinship networks no cycles whatsoever. can be described in terms of cyclicity. While kin- Alternatively to their definition as (open or ship networks do not contain oriented cycles closed) sequences of vertices and lines, paths and (nobody can be his or her own descendant2), they cycles are also often defined as the graphs made may contain cycles (where arc direction does not up of these vertices and lines (in this perspective, matter): people may marry persons with whom a cycle is a connected graph where every vertex they are already linked by kinship or affinity. has degree 2, a path a connected graph where two Now, such cyclic configurations occur not just vertices have degree 1, and all others degree 2). randomly but in ways that are informative about There is, however, a crucial difference between the self-organizing behavior of the network. Some the two concepts. If we define a path as a kinds of relatives hardly ever marry, while other sequence, the starting point matters. A path ABC 534 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS is distinguished from its inverse CBA, or to take A weakly acyclic graph is one that contains no ori- a kinship example, the path “father’s wife’s ented cycles and an acyclic graph contains no daughter” (FWD) is distinguished from its inverse: cycles whatsoever. “mother’s husband’s son” (MHS). If, by contrast, A chain is a graph whose vertices and arcs form a we define a path as a graph, ABC and CBA are single path (alternatively, it may be defined as a only two different notations for one and the same connected graph where two vertices have degree mathematical object – for the kinship case, this 1 and all others have degree 2). means that FWD and MHS are two different ways A circuit is a graph whose vertices and arcs form to express the same kinship chain. a single cycle (alternatively, it may be defined The ambiguity is less severe in the case of as a connected graph where every vertex has cycles, where, by convention, graph theorists treat degree 2). starting and ending points as irrelevant: the The first and the last vertices of a path are called sequences ABCA and BACB are considered to be endpoints. Any line connecting the endpoints of identical. Nevertheless, in kinship network analy- a path is said to close it; the line’s addition to the sis we often need to distinguish them. In an ego- path transforms the latter into a cycle. centric perspective, marrying one’s “father’s wife’s daughter” (FWD) is different from marry- Armed with these basic concepts, we can now ing one’s “son’s wife’s mother” (SWM). By con- describe kinship networks entirely in structural trast, in a socio-centric perspective – as it is terms, without making statements regarding the required if, for example, we want to count all nature of the relations involved. Characterizing cycles of a given type in a kinship network – we kinship networks by weakly acyclic filiation, acy- have to treat FWD and SWM marriages as two clic gendered descent, nonoccurrence of certain different aspects of one and the same configura- types of circuits, and so on, is not equivalent to tion. In other words, we should define paths and stating that no one can be engendered by his or her cycles as sequences when adopting an ego-centric own offspring, that it takes a man and a woman to view and as graphs when adopting a socio-centric produce children, or that self-organization is view. In order to avoid any ambiguity, we shall brought about by incest prohibitions that prevent reserve the terms “path” and “cycle” to sequences people in certain kinship relations from having and use the terms “chain” and “circuit” when we sex with each other. Filiation does not necessarily speak of the corresponding graphs.4 A chain is involve a biological tie, procreation is not the only thus a graph made up by the vertices and lines of basis for parenthood, and marriage prohibitions a single path, and a circuit is a graph made up by are not defined everywhere in terms of sexual the vertices and lines of a single cycle (alternative relations. Kinship network structures may have definitions in terms of degrees and connectedness biological explanations, but these are culturally are given below). determined, variable from one society to another, and far from universal. A path is an alternating sequence of vertices and Biology is the privileged model for kinship lines (arcs or edges), where each vertex is inci- inasmuch as it affords a universally intelligible dent to the preceding and the succeeding line, code for expressing the latter’s fundamental rela- the vertices preceding and succeeding a line are tions: procreation as a model for filiation, sex as a its endpoints, and all vertices are distinct. model for marriage, and so forth. However, being A cycle is a sequence of vertices and lines sharing a model for a relationship does not mean being the the properties of a path, except that the first and essence of this relationship. There are many social the last vertex are identical (all other vertices networks that are clearly kinship networks, both being distinct). according to structural criteria and in the minds of A path or cycle is called oriented if all its lines are those concerned, but where filiation, marriage, and arcs oriented in the same direction. gender cannot be defined exclusively in reference

acacacac

b bbb Path Cycle Oriented path Oriented cycle

Figure 35.1 KINSHIP NETWORK ANALYSIS 535 to biological givens. Kinship is a type of social to the same child vertex necessarily implies the structure, and if people all over the world have existence of a marriage edge between the parent chosen to express its basic relations in a biological vertices. idiom, there are numerous cases where these relations are defined independently of, and some- A kinship network is a mixed graph G(V, E, times even in opposition to, biological relation- A, ~), where V is a set of vertices, called individual ships. Defining kinship in network terms recognizes vertices, E is a set of edges, called marriage this fact. edges, A is a set of arcs (directed from parents to children), called descent arcs, and ~ is an equivalence relation on V partitioning it into n disjoint classes Vi (i = 1, …, n), called genders (usually n = 2 for {male,female}), with the NETWORK REPRESENTATIONS following property: OF KINSHIP 1 the network is weakly acyclic Ore-graph representation The descent graph for the ith gender is the subgraph of a kinship network produced by all descent arcs What we have described in our introductory state- springing from individuals of the gender i. They ment is the most conventional representation of are called agnatic for male gender and uterine for a kinship network, where vertices represent female gender. individuals, arcs represent filial ties, and edges represent marriages. Such networks are called A kinship network is regular if Ore-graphs.5 If not specified otherwise, kinship networks are treated in this article as Ore-graphs. 2 no vertex in a subgraph G(V, Ai) (the descent Their characteristic feature is that they are weakly graph for the ith gender) has an indegree 6 acyclic. As a consequence, kinship networks con- higher than 1 (which rules out cycles in descent tain a directed generational hierarchy. graphs), where A is the subset of arcs with Gender in Ore-graphs is represented by a parti- i origin in Vi [unique descent]. Descent graphs tion of the vertex set. This gender partition gives of regular kinship networks are acyclic. rise to an analogous partition of arcs, according to the gender of the vertex from which they origi- A regular kinship network is standard if nate. The subgraph produced by arcs originating from parental vertices of the same gender can be 3 the subgraphs G(Vi, E ) are empty [hetero- termed a descent graph. Cycles in descent graphs sexual marriage]. are ruled out as soon as we impose the condition of unique descent, that is, filial arcs for only one A standard kinship network is canonical if parent of each gender: one father and one mother. 4 any two vertices that are arc-adjacent to the Under this condition, descent graphs are acyclic, same vertex8 are edge-adjacent to each other and their connected components are trees (the [married co-parents]. well-known unilinear descent trees of “lineages”). This condition is closely related to the condition Every kinship network G(V, E, A, ~) is par- of heterosexual marriage, according to which a tially ordered on V, as is any weakly acyclic marriage edge can only link vertices from a differ- graph. This is the generational partial order rela- ent gender. We speak of a standard kinship tion. In addition, a kinship network may be network if these two conditions – unique descent ordered on V by assigning an arbitrary unique and heterosexual marriage – are satisfied. This identity number to each individual (this is impor- allows for the possibility of nonstandard kinship tant for computation issues, but also for data networks, which, because they admit multiple storage in general, see below). descent7 and homosexual marriage, require a more Two individuals linked by a descent arc are complex analysis. In this chapter, we will restrict called parent and child with respect to each other. ourselves to discussing standard kinship networks. Two individuals linked by an oriented path of In many cases, marriage is the correlate, some- descent arcs are called ascendant and descendant times even the condition or equivalent of having with respect to each other. Two individuals linked children in common. It is therefore useful to dis- by a marriage edge are called the spouses of each tinguish, as a particular category of standard other. Two individuals are called co-parents kinship networks, those networks that meet if they are parents of the same child (arc-adjacent the condition of married co-parents: a standard to the same vertex), siblings if they are children kinship network will be called canonical if the of the same parent (arc-adjacent from the same presence of descent arcs from two parent vertices vertex), and co-spouses if they are spouses 536 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS

Kinship network Regular kinship Standard kinship Canonical kinship network network network

Figure 35.2 of the same spouse (edge-adjacent to the same P-graphs vertex). 10 We can reformulate conditions (1)–(4) as P-graphs represent couples as vertices and indi- follows: viduals as individually and gender-labeled lines. As these individuals are at once born of one 1 In a kinship network, no individual can be his or couple and may become partners in another, the her own ascendant or descendant. arcs that represent them run from the couple 2 In a regular kinship network, no individual has formed by an individual and his or her spouse to more than one parent of each gender. the couple formed by his or her parents. Unmarried 3 In a standard kinship network, parents are individuals are treated like couples. regular and spouses are always of different P-graphs have the advantages of incorporating gender. fewer lines and vertices, allowing marriage cycles 4 In a canonical kinship network, spouses are to be more easily detected. An individual who standard and all co-parents are spouses. marries several times is represented by several lines that are numbered with individual IDs or can be given names. The way to distinguish two lines representing the same individual from those Simple digraph representations representing two same-gender full siblings is by either the line IDs or vectors for individual male There are a variety of ways of representing stand- or female IDs for each vertex. Apart from these ard kinship networks as digraphs. In addition to an two differences, P-graphs share the structural Ore-graph, two of these, P-graphs (White and properties of Ore graphs, such as weak acyclicity Jorion, 1992, Harary and White 2001, also see and generational partial ordering. White, this volume) and bipartite P-graphs (proposed by White, implemented by Batagelj and Mrvar, 2004), are represented in Figure 35.3 where letters indicate individuals (a married Bipartite P-graphs couple, wife A and husband B, who have a daugh- ter C and a son D).9 These are all isomorphic once Bipartite P-graphs are two-mode networks arc-labels for P-graphs are included. where individuals and couples are represented by

CD

AB CD

AB

C D AB

Ore-graph P-graph Bipartite P-graph

Figure 35.3 KINSHIP NETWORK ANALYSIS 537 vertices. There are therefore no marriage edges, and direction enter into its definition; for example, but there are arcs that run from individuals to the definition of “widow” requires a supplemen- couples and from couples to individuals. tary partition of vertices (alive vs. dead), the defi- Bipartite P-graphs are weakly acyclic and, nition of “elder brother” makes reference of a because marriages are represented by vertices, it is partial order relation defined on filial arcs, and possible to represent sibling relations even if the so forth. siblings’ parents are unknown. In addition, mar- riages can be easily partitioned, for example, A kinship path is a path in a kinship network. The according to marriage dates. first vertex of a kinship path is called Ego; the last one is Alter. The direction of a line in a kinship path is 0 (“horizontal”) if it is a marriage edge, –1 KINSHIP PATHS, KINSHIP RELATIONS, (“descending”) if it is an arc directed to the successor, and +1 (“ascending”) if it is inversely AND MATRIMONIAL CIRCUITS directed. Two kinship paths are isomorphic if there is a Kinship paths and relations bijection between them that preserves the gender of vertices and the sequence and direc- The most general definition of a kinship relation tion of lines. relies on the existence of a path linking one indi- An elementary kinship relation corresponds to a vidual to another in a kinship network. If we maximal set of isomorphic paths in a kinship abstract from the individual identity of vertices, network. Any of these paths represents the kin- kinship paths can be characterized by generic ship relation. Any invariant property of these properties such as the gender of vertices and the paths – beginning with the gender and direction direction of lines. This abstract form of a kinship sequence – can be considered as a property of path is an elementary kinship relation. Elementary the elementary relation. kinship relations thus can be considered as abstract A complex kinship relation is any relation that can be kinship paths (where vertex identity does not obtained by logical junction of several elemen- matter as long as vertex gender, line direction and tary kinship relations. vertex-line incidence are preserved). Whatever property we may state of a path without reference to individual vertices, we may consider as a The notation of kinship paths and relations property of the corresponding elementary kinship The conventional notation of kinship relations relation. uses capital letters for indicating direction and the We shall call a simple kinship relation one that gender of Alter (the gender of Ego must be indi- connects Ego and Alter by a single line. By con- ↑ cated by additional signs such as [male Ego] trast, we shall call a relation compound if it con- or + [female Ego] placed before the initial letter): nects Ego and Alter by several consecutive lines. ascending arcs are F(ather) and M(other), descend- Compound relations can be defined by the compo- ing arcs are S(on) and D(aughter), marriage edges sition of simple relations. For instance, “father” is are H(usband) and W(ife), plus supplementary a simple kinship relation, while “mother’s father” letters for B(rother) and Z(Sister) relations. is a compound one. Examples of this are MBD (mother’s brother’s Elementary kinship relations are defined by a daughter, a matrilateral cross-cousin), ZH (sister’s complete specification of the gender and the husband, a brother in-law), and FWS (father’s direction pattern of a single kinship path connect- wife’s son, a stepbrother). ing Ego and Alter. Complex kinship relations are This conventional notation, a simple abbrevia- obtained by combining elementary relations by tion of English kinship terminology, has the means of one or more logical operations (“and,” advantage of being easy to learn and to apply, but “or,” “not,” etc.). If the logical connective is “or,” it is hardly the best tool for analysis. It may even we obtain a disjunctive (or “classificatory”) rela- obscure the structural similarities and the distinc- tion (e.g., “uterine sibling” is defined as “mother’s tive properties of kinship relations. son or mother’s daughter”). If the connective is In this respect it can be contrasted with the “and,” we obtain a conjunctive (or “multiple”) alternative, positional notation developed by relation (e.g., “full brother” is defined as L. Barry (2004). Here, a kinship relation is repre- “mother’s son and father’s son”). If the connective sented by a sequence of letters specifying vertex is “not,” we obtain a residual relation (e.g., labels (gender) and diacritical signs, which indi- “nonagnatic kin”). cate the presence of a marriage edge (the point Finally, we call a kinship relation mixed if or full stop “.”) and a change of arc direction properties of vertices and lines other than gender (the parentheses “()”). All letters not separated by 538 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS a point represent vertices connected by arcs, verti- homogeneous representation of kinship paths ces in “apical” position (that is, vertices that are (with individual numbers), of kinship relations not a neighbor’s children) are put into parentheses (with gender letters), and of kinship relation and, by convention, all arcs to the left of an apical classes (with gender variables), positional nota- vertex have ascending direction, all arcs to its tion may be used not only as a means of notation right have descending direction, and the marriage but also as a classificatory or programming tool. point implies change of direction. Consider, for example, the kinship path linking a male Ego (1) to a female Alter (5) who is Ego’s The classification of kinship relations paternal sister’s husband’s daughter (Figure 35.4). In order to compare kinship relations, and to In positional notation, this relation is written as 1 analyze the ways they combine so as to give rise (2) 3 . (4) 5 where 2 is Ego’s and 3’s parents, 4 to particular network structures, these relations is Ego’s sister’s husband and 5 is the latter’s may be classified according to different criteria. daughter. We restrict ourselves here to some basic defini- By abstracting the concrete identity of the indi- tions (for a more extensive treatment, see viduals concerned (represented by their number) Hamberger and Daillant 2008 and Hamberger and retaining their gender only (H for male and F forthcoming). for female11), one obtains the kinship relation in question: H(H)F.(H)F.12 See Table 35.1 for an A kinship relation is linear if it is oriented (we speak example of how standard and positional notations of “oriented” paths but of “linear” relations). (for paths and relations) are used. Any linear kinship relation can be represented by a The principle of positional notation also applies characteristic number λ to P-graphs. In P-graph notation, with labels for κ ∑ ()σ ).2i arcs rather than for vertices, the ZHD relation of i =0 i Figure 35.4 is written HfH.hf, where H and F give the parents of a male and a female, respectively,13 where κ is the degree of the relation (see note -1 -1 relation inverses h=H , f=F give sons and daugh- 15) and σi is the gender number (0 = male, ters, and marriages are thus written fH and hF, 1 = female) of the ith individual in ascending respectively. The full stop “.” in H.h identifies the direction (starting with Ego i=0), for example, same individual in a different couple – a distinc- λ = 1 for male Ego, λ = 3 for F/S, λ = 5 for M/S, tion that is necessary to clarify that 5 is the child λ = 7 for FF/SS, λ = 13 for MM/DS, and so on.14 of 4 but not of 3. Characteristic numbers impose an order on all Positional notation has a number of important linear kinship relations. advantages. It incorporates the gender of Ego as A kinship relation is canonical if it contains no link- well as that of Alter, and it provides a clear repre- ing children positions (that is, if the kinship path sentation of the structural properties of kinship does not pass through parental triads as defined relations, which is not radically changed by sym- below). metry transformations. Thus, for example, a man A canonical kinship relation is consanguineous if it who marries his HF()HF is his wife’s FH()FH contains no marriage edge. (in p-graph version: HFhf and FHfh), whereas A consanguineous component of a kinship network in conventional notation, a man who marries is a maximal set of individuals linked to each his MBD is his wife’s FZS. Allowing for a other by consanguineous paths (an individual

2

2

1344 1 3 4

5 5

Ore-graph P-graph

Figure 35.4 KINSHIP NETWORK ANALYSIS 539

Table 35.1 Matrimonial census

111 marriages (2.06%) involving 211 (1.41%) individuals (105 men, 106 women) in 1,114 circuits of 37 different types (average frequency 3.08) of width 1 and height 3 ID Standard Positional P-Graph Marriages Circuits % Circuits 1 FBD HH()HF HHhf 3 3 2.63 2 FZD HH()FF HHff 5 5 4.39 3 FFSD HH(H)HF HHH.hhf 4 4 3.51 4 FFDD HH(H)FF HHH.hff 10 10 8.77 5 MBD HF()HF HFhf 9 9 7.89 6 MZD HF()FF HFff 2 2 1.75 7 MFSD HF(H)HF HFH.hhf 13 13 11.4 8 MFDD HF(H)FF HFH.hff 2 2 1.75 … 372 marriages (6.89%) involving 667 (4.47%) individuals (339 men, 328 women) in 267 circuits of 152 different types (average frequency 1.76) of width 2 and height 2 46 FWFSD H(H).F(H)HF HH.hfH.hhf 6 3 1.12 47 FWFDD H(H).F(H)FF HH.hFH.hff 2 1 0.37 48 FWFFSD H(H).FH(H)HF HH.hFHH.hhf 2 1 0.37 49 MHBD H(F).H()HF HF.fHhf 2 1 0.37 50 BW H()H.(F) HhF 20 10 3.75 51 BWMD H()H.F(F)F HhFF.ff 2 1 0.37 … 62 marriages (1.15%) involving 112 (0.75%) individuals (53 men, 59 women) in 23 circuits of 16 different types (average frequency 1.44) of width 3 and height 1 190 FWBWZ H(H).F()H.F()F HH.hFhFf 6 2 8.7 191 FWZHD H(H).F()F.(H)F HH.hFfH.hf 6 2 8.7 192 FWFSWFD H(H).F(H)H.F(H)F HH.hFH.hhFH.hf 3 1 4.35 193 FWFDHD H(H).F(H)F.(H)F HH.hFH.hfH.hf 6 2 8.7 194 BWBWFD H()H.F()H.F(H)F HhFhFH.hf 3 1 4.35 ...

without consanguineous kin constitutes a consan- together with two arcs linking it to its parents, guineous component in itself). who are, at the moment of birth, the individual’s The length of a kinship relation is the number only direct neighbors in the network. If we assume of arcs and edges it contains. The height of that the parents are already linked by a marriage a kinship relation is the length of the longest edge (i.e., we are dealing with a canonical kinship linear kinship relation it contains. The width of network), the emergence of a new child vertex a kinship relation is the number of consanguine- does not create a chain between individuals who ous components it contains.15 The kinship rela- are not already linked by a shorter chain. Its tion represented by the path in Figure 35.4, for impact on global structure may thus be said to be example, has a length of 4, a height of 2, and a marginal: it enlarges the network but does not width of 2. alter its connectivity. On the other hand, kinship networks also grow by marriage, that is, by the creation of new edges between two vertices, each of which can already Matrimonial circuits be linked to a neighborhood of other vertices by several different lines going in all directions. The The concept of a matrimonial circuit new marriage edge creates new connections Real-world kinship networks grow in two ways. between all these neighbors. In this way, marriage On the one hand, new individuals are born, gener- changes social structure. But at the same time, the ally being assigned a father and a mother from way in which social structure would be changed birth. In other words, a new vertex emerges by a potential marriage influences the marriage 540 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS choice itself, be it explicitly (through marriage well as through all of its lines except the closing rules and incest prohibitions), implicitly (by virtue marriage edge, which links the first and the last of preferences or strategies), directly (by taking vertex of the path. For a matrimonial circuit con- into account kinship ties between potential taining n marriage edges, there are 2n different spouses), or indirectly (by taking into account matrimonial paths. other factors that are in turn correlated with If the kinship network is ordered (for instance, by kinship). The probability of two potential partners arbitrary identity numbers), there is a unique rule becoming a couple thus depends upon the nature of selecting, for any matrimonial circuit, a char- of the kinship chains between them. Now, the acteristic path: it is the matrimonial path that most direct way to study this dependency is to has the lowest possible Ego and (if there are two look for those kinship chains that actually connect such paths) the lowest possible Alter. marriage partners – in other words, to look for A matrimonial circuit type is a class of isomorphic circuits. matrimonial circuits. Any matrimonial circuit type It should be clear from the preceding remarks can be represented as a complex kinship rela- that we are not interested in just any type of cir- tion formed by a marriage relation and another cuit, but in those circuits that contain at least one elementary kinship relation. We can define a marriage edge. However, this restriction is not unique rule for selecting, among these relations, enough. Every triangle formed by a couple and the characteristic relation of a matrimonial circuit their child – a parental triangle – is a circuit con- type: for example, the relation that begins with taining a marriage edge, and yet the couple’s rela- the longest sequence of ascending arcs and (if tion to their common child is hardly a condition of there are several sequences of equal length) with marriage that we wish to consider. More gener- the lowest characteristic number. ally, we want to exclude all circuits containing parents together with their children – parental triads. We can therefore define the types of cir- Circuit inclusion and rings cuits of interest to us as those that contain at least Matrimonial circuits have been defined in a most one marriage edge and no parental triads. But in general manner as circuits that do not pass through fact, the second condition implies the first: as parental triads. However, there may be situations descent is weakly acyclic, the only way to form a where we might want to reduce our analysis to circuit in a kinship network without passing only some of these circuits – namely, those that through a parental triad is to pass through a mar- have no “shortcut” linking two of its vertices. riage edge. We thus arrive at a simple definition of Consider, for example, a marriage with the daugh- a matrimonial circuit: ter of the maternal uncle’s wife (MBWD). Now, if this woman is at the same time the maternal A parental triad is a graph formed by three vertices uncle’s daughter (MBD), many anthropologists and arcs pointing from two of them to the third would consider it improper to count her as an (that is, by parents and their child). It the parents MBWD and would want to distinguish such are joined by a marriage edge, the resulting cir- marriages form marriages with “true” MBWDs cuit constitutes a parental triangle. (stepdaughters rather than daughters of maternal A matrimonial circuit is a circuit that does not uncles). There is no a priori answer to the question contain a parental triad. Alternatively, it can be of whether or not to count circuits that contain defined as a connected subgraph where every “shortcuts” of this kind. The choice depends both vertex has degree 2 but no vertex has arc- on the ethnographical context16 and on the type of indegree 2. Because of the acycliclity of descent circuit in question.17 In either case, it is useful to (condition 1 in the definition of kinship net- distinguish circuits that contain shortcuts from works), this definition implies that a matrimonial circuits without shortcuts that link their vertices. circuit necessarily contains at least one marriage The latter are called rings (White, 2004).18 edge. In P-graph representation (where mar- We say that a circuit A includes another circuit riages are vertices and not edges), every circuit is B if all vertices of the circuit B form part of a matrimonial circuit. the circuit A. This may also be stated by saying `Any vertex of a matrimonial circuit that is adjacent that B forms part of the subgraph induced by to a marriage edge (that is, an individual whose the vertices of A, that is, the graph constituted marriage forms part of the circuit) is called a by these vertices and all of the lines that pivot of the circuit. connect them in the global network (if a circuit Any consanguineous chain within a circuit A contains all the vertices of circuit B, the sub- connecting two pivots is called an arch of the graph induced by these vertices also includes the circuit. lines of B). An induced circuit or ring can thus A matrimonial path is a kinship path that passes be defined as a circuit that is its own induced through all vertices of a matrimonial circuit as subgraph.19 KINSHIP NETWORK ANALYSIS 541

The subgraph of a graph G induced by a vertex set manner can be entirely decomposed into circuits V is the maximal subgraph of G having V as its shorter than itself is called reducible; if it cannot, vertex set (see Harary, 1969: 11). The subgraph it is called irreducible. By definition, every irre- induced by a circuit in the kinship network G is ducible circuit is also a ring. the subgraph of G induced by the vertices of the It should be stressed that reducible circuits are circuit. We also call it briefly the induced sub- not necessarily sociologically less relevant than graph of the circuit. irreducible circuits. If a Fulani man, for instance, A circuit A includes a circuit B if every vertex of B is marries a FFBSD who is at the same time an also a vertex of A (that is, if B lies in the induced MBD (due to an FBD marriage between the hus- subgraph of A). band’s parents), the apparent cross-cousin mar- An induced circuit or ring is a circuit that does not riage MBD may simply be a by-product of include any other circuit (that is, a circuit that is successive parallel-cousin marriages (FBD and its own induced subgraph). Alternatively, it can FFBSD): it is then the longer and not the shorter be defined as a circuit such that no two vertices ring that matters for marriage decisions. Nor does of the circuit are connected by a line that is not the formation of a reducible circuit presuppose the itself part of the circuit. previous formation of some irreducible circuit (see note 17). The study of irreducible matrimonial circuits is Circuit intersection and composition closely related to the idea of a cycle basis in gen- The definition of an induced circuit (or ring) eral graph theory. A cycle basis for a graph is a rules out circuits whose vertices are connected by minimal set of circuits from which all circuits of extra lines. It does not, however, exclude circuits a graph can be composed by a circuit union. whose vertices are connected by extra chains Different sets of circuits can constitute a cycle (consisting of more than one line). Consider, for basis. However, the number of circuits in the example, Figure 35.5, in which a man marries his basis (also called the circuit rank or cyclomatic MMBDD, while his own mother is his father’s number of the graph) is invariant: we can compute FFZD.20 it from the numbers of its arcs, edges, and compo- The chains 1-8-9-5 and 2-8-9-5 “shorten” the nents by means of a simple formula (see below). outer circuit 1-2-3-4-5-6-7-1 (of type MMBDD), In order to find a cycle basis for a kinship forming two inner rings of type FFZD: 8-9-5-4-3- network, it is reasonable to concentrate on irre- 2-8 and 1-8-9-5-6-7-1. But note that the outer ducible circuits. Note, however, that the cycle circuit also constitutes a ring: as the vertices 8 and basis may well be smaller than the set of irreduc- 9 do not belong to it, neither of the inner rings is ible circuits: if a man marries three sisters, we included in it.21 However, it intersects with the have three irreducible circuits, but the circuit two inner rings in the sense that it has one or more rank is only two (as two circuits are sufficient to lines in common with them (for a further discus- compose the third). sion of circuit intersection, see section 4.2). Moreover, the entire outer ring can be composed Two circuits intersect if they have lines in common. from the two inner rings and the parental triangle They intersect matrimonially if they have mar- 1-2-8-1 by taking their union and deleting all lines riage edges in common. that form part of more than one circuit (an opera- The union of two circuits is the graph that has the tion called circuit union). A circuit that in this union of their line (vertex) set as its line (vertex) set. The circuit union of two intersecting circuits consists in taking their union and deleting the 4 lines they have in common. A circuit is irreducible if it cannot be composed by a circuit union from a set of circuits that are all 3 5 shorter than itself. A cycle basis of a graph is a minimal set of circuits whose union contains all the circuits of the graph. 2 9 6 The cyclomatic number or circuit rank of a graph is the number of circuits constituting its cycle basis 8 (which is equivalent to the number of lines one has to remove from a graph to make it acyclic). 1 7 For a graph with e lines, v vertices, and c compo- nents, it is calculated as

Figure 35.5 γ = e – v + c 542 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS

Counting circuits: The matrimonial 1 and height 3, circuits of width 2 and height 2, census and circuits of width 3 and height 1. A matrimonial census provides an exhaustive list of all matrimonial circuits of specified properties in a given kinship network, and it counts the occurrences of every distinct circuit type (which NETWORK REPRESENTATIONS OF may or may not be aggregated into broader CIRCUIT STRUCTURES classes). A circuit search usually has to be restricted to a The set of matrimonial circuits thus obtained can limited set of circuit types. Even if the total in turn be studied with genuine network-analytic number of circuits in a finite network is not infi- tools. nite, it is usually so high that an unbounded search One of these tools is to construct the network exceeds the capacities of the most advanced per- that the circuits compose; this gives us a subnet- sonal computers. For exploratory analysis, it is work of the original kinship network. A second recommended to restrict matrimonial circuit tool consists of constructing the network of the search by criteria that are as neutral as possible, structural relations between the circuits them- such as maximal width and height of circuits. Do selves; this gives us a second order network in not just count first-cousin marriages in order to which the circuits represent the vertices and their decide whether you are dealing with “generalized structural interrelations are represented by lines. exchange,” “arab marriage,” and so on – a look at In the following section, we shall discuss one higher degree consanguineous marriages, and at network of the first type, the matrimonial net- marriages between affines, may change the entire work, and one of the second type, the circuit picture! Kinship structures constitute a whole and intersection network. can only be understood if one considers them as such; they cannot be characterized by the fre- quency of this or that circuit type but only by the proportions and the interdependency of these Networks derived from circuit sets: The frequencies. A matrimonial census must therefore be comprehensive in order to provide the basis for matrimonial network further analysis and interpretation, even if subse- A matrimonial network is a subgraph of a kinship quent circuit searches may be more restricted and network resulting from the union of a set of mat- refined. It is clear that, even for small networks, rimonial circuits, as, for instance, the circuits such a task cannot be accomplished without found by the matrimonial census in Table 35.1. computer support. Note that this is not equivalent to the subgraph Circuit searches can be undertaken on the induced by this set: for an arc or edge to be in the entire network or restricted to certain subsets of subgraph, it is not enough that each of its end- vertices. This restriction does not imply that all points is in some circuit – the arc or edge must vertices of the matrimonial circuit have to belong itself be part of a circuit. The matrimonial network to the subset – but their pivots have to lie within it derived from a set of matrimonial circuits found in (if, for example, we are interested in consanguine- a kinship network is thus simply the network com- ous marriages concluded between people born posed of these circuits. It consists, in other words, after 1800, we, of course, allow consanguineous of the matrimonially “interesting” regions of the chains to pass through ancestors born before original kinship network. The components of the 1800). Subsets may be defined according to matrimonial network (which we call matrimonial “exogenous” criteria recorded for the individuals components) are connected subnetworks of matri- in the network (dates of birth, death or marriage, monial circuits, which may be studied from vari- residence, occupation, etc.) but also according to ous perspectives. On the one hand, we may “endogenous” criteria deriving from the kinship suppose that the frequent occurrence of particular network itself (e.g., sibling group size, number of matrimonial patterns is correlated with other prop- known ascendants, number of spouses, etc.). Such erties of the network region concerned (for restrictions are not only convenient for compara- instance, social class, geographical region, or his- tive analysis and tests of representativity, but they torical period); we may then apply several parti- may also be helpful in determining the optimal tions to the network in order to evaluate the degree network to work with (see 6.2.1). to which partition clusters correspond to matrimo- Table 35.1 presents excerpts from a matrimo- nial components. On the other hand, we may nial circuit census produced by the software Puck interpret the density of circuits as an effect of self- (infra) of a kinship network collected among the reinforcing social mechanisms (behavior trans- Watchi of Togo,22 restricted to circuits of width mission, imitation, or the presence of rules) or as KINSHIP NETWORK ANALYSIS 543 a simple network effect (circuits combining to As a result, matrimonial components (consist- compose other circuits) that we did not consider ing only of circuits) are closely related to matri- when defining the criteria for our initial circuit monial bicomponents: both are line-biconnected search. (two distinct line series link each vertex to every The concept of a matrimonial network is also other), but matrimonial bicomponents have the meaningful in and of itself, independent of any additional feature of being vertex-biconnected as particular circuit set. Even in cases where it is not well (the two interconnecting line series never run possible to precisely identify all matrimonial cir- through the same vertex). cuits (without limits of size) that may exist in a In the kinship network represented in kinship network, it is possible to determine which Figure 35.6, for example, the shaded individuals part of the network is composed of matrimonial and their interconnections within the bold bounda- circuits (of whatever length). The result – the ries constitute the nucleus, which is composed of largest possible matrimonial network – is the two matrimonial components (A and B) and three nucleus of the kinship network, the union of all matrimonial bicomponents (1, 2, and 3), two of existing circuits in the network (see Grange and which overlap (one individual is included in both Houseman, 2008). 1 and 2). The concept of the nucleus can be more strictly delineated by introducing the concept of a matri- Given a set of circuits R in a kinship network K, monial bicomponent, that is, a maximal subgraph the matrimonial network derived from R is the in which every two vertices form part of a matri- subgraph of K resulting from the union of the monial circuit (note that this is a stricter condition circuits of R. In other words, it is a subgraph in than that of simply forming part of a circuit, as is which every line belongs to some matrimonial required for the more general notion of a bicom- circuit of R. The components of a matrimonial ponent). The nucleus is simply the union of all network are called the matrimonial components matrimonial bicomponents.23 of K with respect to R. Matrimonial components

Matrimonial bicomponent 2 Matrimonial bicomponent 3 Matrimonial bicomponent 1

Matrimonial component B

Matrimonial component A

Figure 35.6 544 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS

are line-biconnected but not necessarily vertex- that are simultaneously part of a circuit of the two biconnected. corresponding types. A matrimonial bicomponent is a subgraph of K in which every pair of vertices (no matter how The matrimonial intersection of two circuit types distant) belongs to a matrimonial circuit (hence A and B is the set of all marriage edges belong- no pair of vertices can be separated by removal ing simultaneously to a circuit of type A and to a of a single intermediary vertex). In P-graph rep- circuit of type B. resentation, every bicomponent is a matrimonial The circuit intersection network corresponding to bicomponent. a set of circuit types T is a valued graph G, The nucleus of a kinship network is the network such that resulting from the union of all matrimonial circuits in K. The nucleus is equivalent to the 1 each vertex of G corresponds to a circuit type union of all matrimonial bicomponents of the of T, and the value (size) of the vertex is pro- kinship network. The largest of these matrimo- portional to the frequency of the corresponding nial bicomponents constitutes the kernel of the circuit type kinship network. 2 each edge between two vertices corresponds to a nonempty matrimonial intersection of the cor- responding circuit types and the weights of the edges correspond to the size of the intersections Networks of circuits: The circuit intersection network Circuit graphs are tools for analyzing the interde- pendence of matrimonial circuits. A simple exam- ALLIANCE NETWORKS ple is the circuit intersection network, where circuit interdependence is measured by the fre- All matrimonial structures hitherto discussed have quency of shared marriage edges. been defined in terms of filial and marriage rela- It is often the case that a given marriage forms tions between individuals. But kinship also has to part of two or several matrimonial circuits. Such do with relations between groups derived from overdetermination poses problems of sociological relations between individuals. One convenient interpretation: if a man has married a woman who tool of analyzing these relations is the alliance is at the same time his father’s sister’s daughter, network. his maternal aunt, and his sister-in-law, should we The alliance network corresponding to a given say that he married his cousin, his aunt, or his kinship network (real or simulated) is a network sister-in-law? And how shall we interpret such a composed of vertices representing groups of indi- situation if, for instance, it is considered to be very viduals and arcs representing marriage frequen- good to marry one’s cousin but bad to marry one’s cies between the groups, where the value of an arc aunt? Under such circumstances, it is clearly from A to B indicates the number of marriages of insufficient to simply count the frequency of a woman of A with a man of B. One can think of maternal aunts that are spouses and affirm the alliance networks as resulting from “shrinking” presence of a high number of “bad” marriages, the kinship network with respect to some parti- without considering the fraction of such aunts tion, after having transformed marriage edges into who are at the same time cousins and thus repre- arcs that point in the husband’s direction, that is, sent “good” spouses. But there is a further reason from the group of “wife-givers” to the group of for wanting to determine the frequencies of circuit “wife-takers.” The clusters of the partition are intersection: if a father’s sister’s daughter is at those collectivities related by the marriages the same time a maternal aunt, such a configura- between their respective members. The partition tion mathematically implies that the husband’s used may be exogenous to kinship (for instance, if father has married his sister’s daughter. We shall we are dealing with residential units, professional therefore necessarily find a high number of niece categories, or social classes), but it may also be marriages in our network, and if we did not yet derived from the kinship network itself. Thus, for search for them, such a finding will prompt us to example, clusters may represent the components do so. of the agnatic or uterine subnetwork, that is, Circuit intersection networks are an easy and “lineages” within the kinship network. intuitive way to approach these questions. In these Alliance networks no longer have the particu- networks, vertices represent circuit types, the size larities of kinship networks. They are simple (or vector value) of vertices represent circuit fre- valued digraphs. Circuits in alliance networks quencies, and the values of the lines connecting may be called connubial circuits in order to distin- two vertices represent the number of marriages guish them from matrimonial circuits in kinship KINSHIP NETWORK ANALYSIS 545 networks, and the connubial circuit structure of an checking possible errors and biases but may be alliance network can be represented by its matrix an interesting datum in itself, for instance, to (the alliance matrix). In exogamous systems, the appraise genealogical memory. diagonal will have only zero values; in a perfect • Document missing data; indicate if the spouses system of balanced bilateral marriage alliances and children you have noted for a given individ- (designated by Lévi-Strauss [1949 (1967)] as ual are, according to your knowledge, complete. “restricted exchange”) it will be symmetric; in a • In case of contradictory data, keep records system having a single directed Hamiltonian cycle regarding alternative items of information (and (“generalized exchange”), it will be asymmetric their origin) and on the reasons for your choices. and contain only one nonzero value in each line • Try to avoid biases from the start by always fol- and column. Note, however, that real-world alli- lowing both male and female lines; do not record ance structures are not as clear-cut. On the one only those kinship ties that are easily given but hand, several circuit patterns may be superim- make an effort to search those that are missing, posed, such that, for example, the resulting cumu- even if this may be costly in the case of an exten- lative pattern may appear symmetrical even if sive matrimonial area. partial patterns are asymmetric. On the other • If you are interested in the kinship relations hand, the structure of alliance networks is highly between two individuals, do not be content to sensitive to the definition of wife-giving and wife- ask how they are related but try to establish their taking groups, connubial circuit structures being entire pedigree within given bounds – you will in general not continuous with respect to changes surely find quite a number of additional ties your in the level of group aggregation. Thus, what may informants did not mention spontaneously! appear to be endogamous unions on one level may • Try to determine the relative order of births and be exogamous on another, with combinations of marriages (in particular in the case of polygamy), endogamy and balanced exchange on one level even if absolute dates are not available. In inter- giving rise to asymmetrical patterns on another preting marriages with affines or overlapping (for an example, see Gabail and Kyburz, 2008). matrimonial circuits, this may turn out to be very important. The alliance network corresponding to a partitioned • Keep an account of the research method used kinship network K is a digraph G where and specify the purpose for which the corpus (or part of the corpus) is established. 1 veach vertex of G corresponds to a partition cluster of K 2 each arc between two vertices of G correspond- Storing data without a computer ing to two clusters A and B corresponds to the Data are not only a result but also a means of data existence of a marriage of a woman of A with a collection. They should be easily accessible in man of B, and the weight of the arcs correspond order to guide your research and to cross-check to the number of such marriages. informant answers. When dealing with archives, this is often fairly simple: you can take a computer A connubial circuit is a circuit in an alliance with you. But in many fieldwork situations this is network. not possible. However, noting kinship “by hand” can be extremely fast and efficient, if some basic principles are observed.

• Always use a compact medium, such as a note- APPLICATION ISSUES book. Do not use filesheets or loose papers. You cannot easily use them during interviews, and Data collection and saving there is a high risk of losing some of them. • Separate graphics and text. A good method is Orientations of data collection to use a notebook with the left page for draw- Of course, the way kinship data collection is ori- ing genealogies, the right page for listing the ented depends largely on what one wants to find. individuals and their properties, and numbers However, in order to minimize biases and to allow for identifying these individuals (if numbers get the data to be used by others, there are some large, it is recommended to use, in addition, criteria that every corpus of kinship data should initial letters to prevent identification problems fulfill, regardless of the specific purpose of the in case of numbering errors). collection: • Attribute an identity number to each individual and never attribute that number to another • Document the dates and informant(s) for every individual. If you have “duplicates,” make a link part of the data set; this is not only useful for to the original number but do not re-assign it. 546 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS

Gaps in the series of numbers do not cause any A related problem concerns the delimitation of damage, but ambiguities in identity numbers the network to be used for analytic purposes. To cause much damage and are extremely difficult begin with, it is important that results relating to to detect. unconnected components of the matrimonial net- • Do not use identity numbers as codes. Identity work not be combined indiscriminately, as each numbers serve to identify individuals and noth- component represents an autonomous matrimo- ing else (except, perhaps, to recall the order in nial universe whose patterning may not obey the which you have entered them and to document same rules. This also applies, to a lesser degree, to the history of your corpus). If you want to convey matrimonial bicomponents. However, even when information on individuals’ gender, clan affilia- analysis is limited to the largest matrimonial tion, residence, and so on, do not use identity bicomponent (kernel) of the kinship network, numbers for that purpose. additional restrictions are often necessary. Because • Never forget to make copies and store them bicomponents contain matrimonial circuits of any in different places. This holds for all data, but length, height, and width, including those that especially for kinship data, due to the network incorporate very distant ties whose sociological properties of kinship: one lost notebook may relevance is questionable, it is helpful to confine render all other notebooks useless. analysis to subnetworks formed by matrimonial circuits of a certain maximal height or width. It is not always easy to determine the optimal criteria for such restricted matrimonial networks, choices Data interpretation being largely guided by two contrary principles. On the one hand, the resulting subnetwork should It is necessary to know the biases, gaps, and limits be as large as possible so as to be sufficiently of a genealogical corpus. In some cases, this may representative of the wider kinship network from lead one to conclude that certain analyses simply which it is drawn. On the other hand, in order to cannot reasonably be made. For instance, it makes avoid redundancies and to keep the number of little sense to search for consanguineous marriage circuits and circuit types at manageable propor- circuits in a corpus with very shallow genealogies. tions, it is essential that the latter be kept to a Here, as in all cases, knowledge of the basic minimum. qualities of the corpus is essential for interpreting the findings. In particular it is important to keep in mind that Determining representativity and significance kinship networks are virtually infinite. Every Once a delimited data set has been chosen and corpus is necessarily only a part of a larger whole, first results obtained, another difficulty arises: to and incomplete both with regard to individuals what extent do the regularities observed in the and with regard to the links that connect them. corpus provide information on the organization of Choices pertaining to the delimitation and compo- the real social network? This, of course, is a gen- sition of genealogical corpuses arise not only eral problem in network analysis; however, during fieldwork but also prior to analysis as because issues of incompleteness are so omnipres- a means of reducing the collected data to a ent in genealogical research, questions regarding meaningful core. the representativity of kinship networks are particularly pressing. One central question concerns the extent to Choosing bases and boundaries which a corpus’s boundaries correspond to endog- The usefulness of exogenous reductions bringing enously definable subnetworks of the real kinship sociological, geographical, and demographical network. Another, related question concerns the criteria into play is fairly self-evident. However, over- and underrepresentation of persons belong- the value of endogenous reductions based on ing to certain kinship categories. These issues structural features of the network itself is perhaps apply not only to the individuals (vertices) that less so, and depends on what one is trying to show. make up the corpus, but to the relations (lines) For example, one cannot first eliminate all unmar- between them. Kinship networks are often biased ried individuals if one later wants to compare in that they favor some types of relations over matrimonial circuit frequencies with those of kin- others (agnatic over uterine relations, for example, ship relations (e.g., White 1999). This applies not for a population with patrilocal residence). It only to endogenous reductions but also to endog- might be possible to adjust one’s findings so as to enous augmentations of the network, such as the eliminate such biases. However, these biases may creation of fictive individuals in order to preserve themselves be a function of matrimonial behavior: information on full siblingship in those cases kinship ties that form part of matrimonial circuits where one or both parents are unknown. may be more easily remembered than others. KINSHIP NETWORK ANALYSIS 547

The stronger this correlation between network used by many French historians; Alliance Project, structure and collective memory is, the more dif- developed by S. Sugito and S. Kubota in Japan ficult it is to do away with the biases concerned. (Sugito, 2004), is widely used by Australian The problem of network biases is directly anthropologists; Kinship Editor, written by related to issues of significance: are the observed M. Fischer, is a key element of the European regularities indicative of a behavioral pattern, or Kinship and Social Security (KASS) project, and are they simply due to chance? In order to treat it allows both for systematic data collection and this question by means of a comparison with the modeling of kinship phenomena such as termi- random kinship networks, it is necessary to simu- nological usage. There are also additional soft- late a network that not only reproduces the demo- ware tools developed by demographers or graphic features of the concerned population but historians for managing records and calculating also the biases of the corpus itself. Because of demographic and other variables on the basis of this, random permutation (see White, 1999; White, empirical genealogical data (e.g., CASOAR this volume) or explicit simulation of the data [Hainsworth and Bardet, 1981]). None of these collection process (virtual fieldwork) may prove programs, however, allow for an in-depth analysis more useful than conventional demographic of kinship networks as such. simulation. Such analyses may be undertaken using general purpose social network analysis tools. Perhaps the software most used in this way is the network Interpretation of results analysis and visualization program Pajek, devel- If “interesting” regularities appear in analysis for oped by V. Batagelj and A. Mrvar (de Nooy et al., which the known norms and institutions of the 2005; Batagelj and Mrvar, 2004, 2008 : Mrvar and society do not provide a straightforward account, Batagelj 2004). A number of kinship-centered one is immediately tempted to take them as indi- macros have been developed for Pajek by the cators of some hidden norms or institution. The authors and by others (e.g., Tip4Pajek pack by most important thing in kinship data analysis is K. Hamberger). to resist this temptation. Before formulating a Finally, certain programs have been specifi- sociological hypothesis, one must always check cally developed for the analysis of kinship net- the following: works (including matrimonial circuit censuses). First attempts, such as Gen-Par by M. Selz (1987; • First, does the “interesting” structural feature see also Héritier, 1974) and Pgraph by D. White simply reflect a bias of the corpus? (for instance, (1997, White et al. 1999), have since given way to high incidences of patrilateral marriages in a more flexible and easier-to-use software such as corpus where uterine genealogies are shallow). Genos by L. Barry (2004) and, more recently, • Second, is the structural feature, if it does not Puck by K. Hamberger (Hamberger et al., 2009). correspond to any known rule, the result of a An open depository of kinship networks from combination of known rules? (for example, in historical and anthropological sources, controlled a society where uterine nieces are preferred by a scientific board, is hosted at the site of the spouses and maternal aunts are avoided, pat- kinsources project (http://kinsource.net). rilateral cross-cousins have a greater chance of being at the same time maternal aunts and therefore avoided, even if no rule states that they should be avoided as such). NOTES • Finally, remember that a sociological hypothesis is not validated by the simple fact that no other 1 We are grateful to Isabelle Daillant and interpretation can account for the structural fea- Vladimir Batagelj as well as to the editors Peter ture observed. Sociological hypotheses often can Carrington and John Scott for helpful comments and be validated only in the field. discussions. 2 We are talking of individuals and not of classes. An individual may well belong to one’s parent’s Resources parent’s marriage class, as in Australian alternating generation models. A wide variety of commercial software (Brother’s 3 In digraphs, the notions “path” and “cycle” are Keeper, Family Tree Maker, Legacy, Kith and often restricted to oriented paths and cycles, whereas Kin, etc.) and noncommercial programs (Personal the terms “semipath” and “semicycle” are used if Ancestral File, Gramps, etc.) exist for entering, arcs are not consistently oriented. Because we are storing, and outputting genealogical data. Some talking of mixed graphs (containing edges as well as of them are particularly flexible and are used arcs), we use “path” and “cycle” as the general extensively by social scientists: Généatique is terms (as in the case of undirected graphs) and 548 THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS specify them as “oriented” if all of their lines are arcs marriages may be considered as redoublings of alli- pointing in the same direction. ance relations rather than as consanguineous mar- 4 The terms “chain” and “circuit” are equivalent riages (Dumont, [1953] 1975), counting all MBWDs, to “path graph” and “cycle graph” in graph- whether they are or are not also MBDs, would make theoretic literature. Note, however, that these terms good sense. are also sometimes used in a different meaning, 17 For instance, if two brothers marry two sisters, namely, as synonyms for “walk” (open and closed, one brother dies, and the remaining brother marries respectively). the widow, the fact that the longer circuit (of type 5 Named after the Scandinavian mathematician BWZ) includes two shorter ones (of types BW and Oystein Ore (1960). The representation of marriages WZ) does not in the least make it sociologically less by edges has been introduced by the computer relevant: the BWZ marriage clearly precedes the BW program Pajek (see Batagelj and Mrvar, 2008). (and WZ) marriage (we are grateful to Isabelle 6 Note that the weakly acyclic property of kinship Daillant for this example). networks holds for remarriage cycles because a cycle 18 The distinction between circuits and rings consisting entirely of edges is not an oriented cycle. (induced circuits) is a rather recent one. Much of 7 Kinship networks that incorporate adoptive, what has been said regarding matrimonial rings in godparent, or co-genitor relations often have to Hamberger et al. (2004) refers to matrimonial circuits allow for the possibility of more than one parent of a in general, while rings in White (2004) refers to given gender (multiple descent). induced circuits. 8 A vertex x is arc-adjacent to another vertex y if 19 Note, however, that this restriction may not the arc goes from x to y. It is arc-adjacent from y if mean the same thing in P-graph and in Ore-graph the arc goes from y to x. representation. The subgraph induced by the vertices 9 The upper and lower vertices of the P-graph of a circuit changes meaning according to whether and bipartite P-graph are reversed in Figure 35.3 vertices are defined as individuals (Ore-graph) or as compared to the Ore-graph. This is because their marriages (P-graph). For instance, a marriage with an arrows run in reverse, from children to parents, usu- MMBDD who is at the same time an MBD constitutes ally as a unique function mapping a child to a unique a ring in Ore-graph but not in P-graph representa- parental couple. The inverse of this function gives the tion. If the context is not clear, one should therefore relation of parents to children. speak of Ore-rings and P-rings to avoid ambiguities. 10 The “p” stands for parenté (French for 20 This example holds in a P-graph as well as in “kinship”). Ore-graph representation: in both cases, the outer 11 From the French homme (man) and femme circuit is a ring. In general, Ore-rings are not always (woman). P-rings (see note 19). 12 This positional notation may also be used, 21 This is equally true in P-graph representation, within limits, to represent complex kinship relations, where the two individuals 8 and 9 are represented by by leaving the parentheses empty (or by putting two two lines forming an extra chain between the verti- letters in it) if the relation entails kinship paths that ces (representing marriages) of the outer circuit. pass through apical ancestors of both genders. For 22 See http://kinsource.net/kinsrc/bin/view/ instance, the MBD (mother’s brother’s daughter) KinSources/Watchi. relation is represented as HF()HF, the ZH (sister’s 23 Every line of a matrimonial bicomponent is by husband) relation as H()F.H, and the FWS (father’s definition in some matrimonial circuit. On the other wife’s son) relation as H(H).(F)H. hand, every matrimonial circuit is either a matrimo- 13 Original P-graph notation (White and Jorion, nial bicomponent in itself or forms part of some 1992) used letters G and F (from the French garçon larger matrimonial bicomponent. The concept of the (boy) and fille (girl). In its present version, P-graph nucleus is a restriction of the definition of the core by notation uses the same letters as positional notation White and Jorion (1996; cf. Houseman and White, (but applies them to lines while positional notation 1996) as the union of all matrimonial bicomponents applies them to vertices). and their single-link connections (equating with 14 This is a variant of the ahnentafel genealogical 2-core as defined by Seidman, 1983). 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