DOCSLIB.ORG

Assortativity and Mixing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Assortativity and Mixing Assortativity and Assortativity and Mixing General mixing between node categories Mixing Assortativity and Mixing Definition Definition I Assume types of nodes are countable, and are Complex Networks General mixing General mixing Assortativity by assigned numbers 1, 2, 3, . Assortativity by CSYS/MATH 303, Spring, 2011 degree degree I Consider networks with directed edges. Contagion Contagion References an edge connects a node of type µ References e = Pr Prof. Peter Dodds µν to a node of type ν Department of Mathematics & Statistics Center for Complex Systems aµ = Pr(an edge comes from a node of type µ) Vermont Advanced Computing Center University of Vermont bν = Pr(an edge leads to a node of type ν) ~ I Write E = [eµν], ~a = [aµ], and b = [bν]. I Requirements: X X X eµν = 1, eµν = aµ, and eµν = bν. µ ν ν µ Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. 1 of 26 4 of 26 Assortativity and Assortativity and Outline Mixing Notes: Mixing Definition Definition General mixing General mixing Assortativity by I Varying eµν allows us to move between the following: Assortativity by degree degree Definition Contagion 1. Perfectly assortative networks where nodes only Contagion References connect to like nodes, and the network breaks into References subnetworks. General mixing P Requires eµν = 0 if µ 6= ν and µ eµµ = 1. 2. Uncorrelated networks (as we have studied so far) Assortativity by degree For these we must have independence: eµν = aµbν . 3. Disassortative networks where nodes connect to nodes distinct from themselves. Contagion I Disassortative networks can be hard to build and may require constraints on the eµν. References I Basic story: level of assortativity reflects the degree to which nodes are connected to nodes within their group. 2 of 26 5 of 26 Assortativity and Assortativity and Basic idea: Mixing Correlation coefficient: Mixing Definition Definition Random networks with arbitrary degree distributions I General mixing General mixing cover much territory but do not represent all Assortativity by I Quantify the level of assortativity with the following Assortativity by networks. degree [5] degree Contagion assortativity coefficient : Contagion I Moving away from pure random networks was a key References P P References first step. eµµ − aµbµ Tr E − ||E2|| = µ µ = 1 r P 2 I We can extend in many other directions and a 1 − µ aµbµ 1 − ||E ||1 natural one is to introduce correlations between different kinds of nodes. where || · ||1 is the 1-norm = sum of a matrix’s entries. I Node attributes may be anything, e.g.: I Tr E is the fraction of edges that are within groups. 1. degree 2 I ||E ||1 is the fraction of edges that would be within 2. demographics (age, gender, etc.) groups if connections were random. 3. group affiliation 2 I 1 − ||E ||1 is a normalization factor so rmax = 1. I We speak of mixing patterns, correlations, biases... I When Tr eµµ = 1, we have r = 1. I Networks are still random at base but now have more X global structure. I When eµµ = aµbµ, we have r = 0. X [4, 5] I Build on work by Newman , and Boguñá and [1] Serano. 3 of 26 6 of 26 Assortativity and Assortativity and Correlation coefficient: Mixing Degree-degree correlations Mixing Definition Definition General mixing General mixing Assortativity by Assortativity by Notes: degree degree I Notation reconciliation for undirected networks: Contagion Contagion I r = −1 is inaccessible if three or more types are P References j k(ejk − Rj Rk ) References present. r = j k σ2 I Disassortative networks simply have nodes R connected to unlike nodes—no measure of how where, as before, Rk is the probability that a unlike nodes are. randomly chosen edge leads to a node of degree I Minimum value of r occurs when all links between k + 1, and non-like nodes: Tr eµµ = 0. 2 I 2 2 X 2 X −||E ||1 σ = j Rj − jRj . r = R min 2 j j 1 − ||E ||1 where −1 ≤ rmin < 0. 7 of 26 10 of 26 Assortativity and Assortativity and Scalar quantities Mixing Degree-degree correlations Mixing Definition Definition Now consider nodes defined by a scalar integer I General mixing General mixing quantity. Assortativity by Assortativity by degree degree I Examples: age in years, height in inches, number of friends, ... Contagion Error estimate for r: Contagion References References I ejk = Pr (a randomly chosen edge connects a node I Remove edge i and recompute r to obtain ri . with value j to a node with value k). I Repeat for all edges and compute using the jackknife [2] I aj and bk are defined as before. method () Can now measure correlations between nodes I 2 X 2 based on this scalar quantity using standard Pearson σr = (ri − r) . correlation coefficient ( ): i MIXING PATTERNS IN NETWORKS PHYSICAL REVIEW E 67, 026126 ͑2003͒ P I Mildly sneaky as variables need to be independent j k j k(ejk − aj bk ) hjki − hjiahkib TABLE II. Size n, degree assortativity coefficient r, and expected error ␴r on the assortativity, for a r = = q q numberfor of social, us technological, to be truly and biological happy networks, and both edges directed are and undirected. correlated... Social networks: σa σb 2 2 2 2 coauthorship networks of ͑a͒ physicists and biologists ͓46͔ and ͑b͒ mathematicians ͓47͔, in which authors are hj ia − hjia hk ib − hkib connected if they have coauthored one or more articles in learned journals; ͑c͒ collaborations of film actors in which actors are connected if they have appeared together in one or more movies ͓5,7͔; ͑d͒ directors of fortune 1000 companies for 1999, in which two directors are connected if they sit on the board of directors of the same company ͓48͔; ͑e͒ romantic ͑not necessarily sexual͒ relationships between students at a U.S. high I This is the observed normalized deviation from school ͓49͔; ͑f͒ network of email address books of computer users on a large computer system, in which an edge from user A to user B indicates that B appears in A’s address book ͓50͔. Technological networks: ͑g͒ randomness in the product jk. network of high voltage transmission lines in the Western States Power Grid of the United States ͓5͔; ͑h͒ 8 of 26 network of direct peering relationships between autonomous systems on the Internet, April 2001 ͓51͔; ͑i͒ 11 of 26 network of hyperlinks between pages in the World Wide Web domain nd.edu, circa 1999 ͓52͔; ͑j͒ network of dependencies between software packages in the GNU/Linux operating system, in which an edge from pack- age A to package B indicates that A relies on components of B for its operation. Biological networks: ͑k͒ Assortativity and protein-protein interaction network in the yeast S. Cerevisiae ͓53͔; ͑l͒ metabolic network of the bacterium E. Assortativity and Degree-degree correlations Mixing MeasurementsColi ͓54͔; ͑m͒ neural network of the of nematode degree-degree worm C. Elegans ͓5,55͔; tropic interactions correlations between species Mixing in the food webs of ͑n͒ Ythan Estuary, Scotland ͓56͔ and ͑o͒ Little Rock Lake, Wisconsin ͓57͔. Definition Definition General mixing Group Network Type Size n Assortativity r Error ␴ General mixing I Natural correlation is between the degrees of r Assortativity by a Physics coauthorship undirected 52 909 0.363 0.002 Assortativity by connected nodes. degree a Biology coauthorship undirected 1 520 251 0.127 0.0004 degree b Mathematics coauthorship undirected 253 339 0.120 0.002 Contagion Contagion I Now define ejk with a slight twist: Social c Film actor collaborations undirected 449 913 0.208 0.0002 References d Company directors undirected 7 673 0.276 0.004 References e Student relationships undirected 573 Ϫ0.029 0.037 an edge connects a degree j + 1 node f Email address books directed 16 881 0.092 0.004 ejk = Pr to a degree k + 1 node g Power grid undirected 4 941 Ϫ0.003 0.013 Technological h Internet undirected 10 697 Ϫ0.189 0.002 i World Wide Web directed 269 504 Ϫ0.067 0.0002 an edge runs between a node of in-degree j j Software dependencies directed 3 162 Ϫ0.016 0.020 = Pr and a node of out-degree k k Protein interactions undirected 2 115 Ϫ0.156 0.010 l Metabolic network undirected 765 Ϫ0.240 0.007 Biological m Neural network directed 307 Ϫ0.226 0.016 n Marine food web directed 134 Ϫ0.263 0.037 I Useful for calculations (as per Rk ) o Freshwater food web directed 92 Ϫ0.326 0.031 I Important: Must separately define P0 as the {ejk } Ϫm) almost never is. In this paper, therefore, we take an ͓58,59͔. The algorithm is as follows. contain no information about isolated nodes. alternative approach,I Social making use networks of computer simulation. tend to be assortative͑1͒ Given the desired (homophily) edge distribution e jk , we first cal- We would like to generate on a computer a random net- culate the corresponding distribution of excess degrees qk I Directed networks still fine but we will assume from work having, forI instance,Technological a particular value and of the biological matrix fromnetworks Eq. ͑23͒, and then tend invert to Eq. be͑22͒ to find the degree e jk . ͓This also fixes the degree distribution, via Eq. ͑23͒.͔ In distribution: here on that ejk = ekj . Ref. ͓22͔ we discusseddisassortative one possible way of doing this using an algorithm similar to that of Sec. II C. One would draw qkϪ1 /k pkϭ . ͑27͒ 9 of 26 edges from the desired distribution e jk and then join the de- 12 of 26 q jϪ1 / j gree k ends randomly in groups of k to create the network.
Load more

Recommended publications
  • Networkx: Network Analysis with Python
    NetworkX: Network Analysis with Python Salvatore Scellato Full tutorial presented at the XXX SunBelt Conference “NetworkX introduction: Hacking social networks using the Python programming language” by Aric Hagberg & Drew Conway Outline 1. Introduction to NetworkX 2. Getting started with Python and NetworkX 3. Basic network analysis 4. Writing your own code 5. You are ready for your project! 1. Introduction to NetworkX. Introduction to NetworkX - network analysis Vast amounts of network data are being generated and collected • Sociology: web pages, mobile phones, social networks • Technology: Internet routers, vehicular flows, power grids How can we analyze this networks? Introduction to NetworkX - Python awesomeness Introduction to NetworkX “Python package for the creation, manipulation and study of the structure, dynamics and functions of complex networks.” • Data structures for representing many types of networks, or graphs • Nodes can be any (hashable) Python object, edges can contain arbitrary data • Flexibility ideal for representing networks found in many different fields • Easy to install on multiple platforms • Online up-to-date documentation • First public release in April 2005 Introduction to NetworkX - design requirements • Tool to study the structure and dynamics of social, biological, and infrastructure networks • Ease-of-use and rapid development in a collaborative, multidisciplinary environment • Easy to learn, easy to teach • Open-source tool base that can easily grow in a multidisciplinary environment with non-expert users
    [Show full text]
  • Network Analysis of the Multimodal Freight Transportation System in New York City
    Network Analysis of the Multimodal Freight Transportation System in New York City Project Number: 15 – 2.1b Year: 2015 FINAL REPORT June 2018 Principal Investigator Qian Wang Researcher Shuai Tang MetroFreight Center of Excellence University at Buffalo Buffalo, NY 14260-4300 Network Analysis of the Multimodal Freight Transportation System in New York City ABSTRACT The research is aimed at examining the multimodal freight transportation network in the New York metropolitan region to identify critical links, nodes and terminals that affect last-mile deliveries. Two types of analysis were conducted to gain a big picture of the region’s freight transportation network. First, three categories of network measures were generated for the highway network that carries the majority of last-mile deliveries. They are the descriptive measures that demonstrate the basic characteristics of the highway network, the network structure measures that quantify the connectivity of nodes and links, and the accessibility indices that measure the ease to access freight demand, services and activities. Second, 71 multimodal freight terminals were selected and evaluated in terms of their accessibility to major multimodal freight demand generators such as warehousing establishments. As found, the most important highways nodes that are critical in terms of connectivity and accessibility are those in and around Manhattan, particularly the bridges and tunnels connecting Manhattan to neighboring areas. Major multimodal freight demand generators, such as warehousing establishments, have better accessibility to railroad and marine port terminals than air and truck terminals in general. The network measures and findings in the research can be used to understand the inventory of the freight network in the system and to conduct freight travel demand forecasting analysis.
    [Show full text]
  • Networkx Reference Release 1.9.1
    NetworkX Reference Release 1.9.1 Aric Hagberg, Dan Schult, Pieter Swart September 20, 2014 CONTENTS 1 Overview 1 1.1 Who uses NetworkX?..........................................1 1.2 Goals...................................................1 1.3 The Python programming language...................................1 1.4 Free software...............................................2 1.5 History..................................................2 2 Introduction 3 2.1 NetworkX Basics.............................................3 2.2 Nodes and Edges.............................................4 3 Graph types 9 3.1 Which graph class should I use?.....................................9 3.2 Basic graph types.............................................9 4 Algorithms 127 4.1 Approximation.............................................. 127 4.2 Assortativity............................................... 132 4.3 Bipartite................................................. 141 4.4 Blockmodeling.............................................. 161 4.5 Boundary................................................. 162 4.6 Centrality................................................. 163 4.7 Chordal.................................................. 184 4.8 Clique.................................................. 187 4.9 Clustering................................................ 190 4.10 Communities............................................... 193 4.11 Components............................................... 194 4.12 Connectivity..............................................
    [Show full text]
  • Strongly Connected Components and Biconnected Components
    Strongly Connected Components and Biconnected Components Daniel Wisdom 27 January 2017 1 2 3 4 5 6 7 8 Figure 1: A directed graph. Credit: Crash Course Coding Companion. 1 Strongly Connected Components A strongly connected component (SCC) is a set of vertices where every vertex can reach every other vertex. In an undirected graph the SCCs are just the groups of connected vertices. We can find them using a simple DFS. This works because, in an undirected graph, u can reach v if and only if v can reach u. This is not true in a directed graph, which makes finding the SCCs harder. More on that later. 2 Kernel graph We can travel between any two nodes in the same strongly connected component. So what if we replaced each strongly connected component with a single node? This would give us what we call the kernel graph, which describes the edges between strongly connected components. Note that the kernel graph is a directed acyclic graph because any cycles would constitute an SCC, so the cycle would be compressed into a kernel. 3 1,2,5,6 4,7 8 Figure 2: Kernel graph of the above directed graph. 3 Tarjan's SCC Algorithm In a DFS traversal of the graph every SCC will be a subtree of the DFS tree. This subtree is not necessarily proper, meaning that it may be missing some branches which are their own SCCs. This means that if we can identify every node that is the root of its SCC in the DFS, we can enumerate all the SCCs.
    [Show full text]
  • Robustness and Assortativity for Diffusion-Like Processes in Scale
    epl draft Robustness and Assortativity for Diffusion-like Processes in Scale- free Networks G. D’Agostino1, A. Scala2,3, V. Zlatic´4 and G. Caldarelli5,3 1 ENEA - CR ”Casaccia” - Via Anguillarese 301 00123, Roma - Italy 2 CNR-ISC and Department of Physics, University of Rome “Sapienza” P.le Aldo Moro 5 00185 Rome, Italy 3 London Institute of Mathematical Sciences, 22 South Audley St Mayfair London W1K 2NY, UK 4 Theoretical Physics Division, Rudjer Boˇskovi´cInstitute, P.O.Box 180, HR-10002 Zagreb, Croatia 5 IMT Lucca Institute for Advanced Studies, Piazza S. Ponziano 6, Lucca, 55100, Italy PACS 89.75.Hc – Networks and genealogical trees PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics PACS 87.23.Ge – Dynamics of social systems Abstract – By analysing the diffusive dynamics of epidemics and of distress in complex networks, we study the effect of the assortativity on the robustness of the networks. We first determine by spectral analysis the thresholds above which epidemics/failures can spread; we then calculate the slowest diffusional times. Our results shows that disassortative networks exhibit a higher epidemi- ological threshold and are therefore easier to immunize, while in assortative networks there is a longer time for intervention before epidemic/failure spreads. Moreover, we study by computer sim- ulations the sandpile cascade model, a diffusive model of distress propagation (financial contagion). We show that, while assortative networks are more prone to the propagation of epidemic/failures, degree-targeted immunization policies increases their resilience to systemic risk. Introduction. – The heterogeneity in the distribu- propagation of an epidemic [13–15].
    [Show full text]
  • Analyzing Social Media Network for Students in Presidential Election 2019 with Nodexl
    ANALYZING SOCIAL MEDIA NETWORK FOR STUDENTS IN PRESIDENTIAL ELECTION 2019 WITH NODEXL Irwan Dwi Arianto Doctoral Candidate of Communication Sciences, Airlangga University Corresponding Authors: [email protected] Abstract. Twitter is widely used in digital political campaigns. Twitter as a social media that is useful for building networks and even connecting political participants with the community. Indonesia will get a demographic bonus starting next year until 2030. The number of productive ages that will become a demographic bonus if not recognized correctly can be a problem. The election organizer must seize this opportunity for the benefit of voter participation. This study aims to describe the network structure of students in the 2019 presidential election. The first debate was held on January 17, 2019 as a starting point for data retrieval on Twitter social media. This study uses data sources derived from Twitter talks from 17 January 2019 to 20 August 2019 with keywords “#pilpres2019 OR #mahasiswa since: 2019-01-17”. The data obtained were analyzed by the communication network analysis method using NodeXL software. Our Analysis found that Top Influencer is @jokowi, as well as Top, Mentioned also @jokowi while Top Tweeters @okezonenews and Top Replied-To @hasmi_bakhtiar. Jokowi is incumbent running for re-election with Ma’ruf Amin (Senior Muslim Cleric) as his running mate against Prabowo Subianto (a former general) and Sandiaga Uno as his running mate (former vice governor). This shows that the more concentrated in the millennial generation in this case students are presidential candidates @jokowi. @okezonenews, the official twitter account of okezone.com (MNC Media Group).
    [Show full text]
  • Ollivier Ricci Curvature of Directed Hypergraphs Marzieh Eidi1* & Jürgen Jost1,2
    www.nature.com/scientificreports OPEN Ollivier Ricci curvature of directed hypergraphs Marzieh Eidi1* & Jürgen Jost1,2 Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general defnition of Ricci curvature on directed hypergraphs and explore the consequences of that defnition. The defnition generalizes Ollivier’s defnition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the defnition looks somewhat complex, in the end we shall be able to express our curvature in a very simple formula, κ = µ0 − µ2 − 2µ3 . This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their curvature. Principles of network analysis. Network analysis constitutes one of the success stories in the study of complex systems3,6,11. For the mathematical analysis, a network is modelled as a (perhaps weighted and/or directed) graph. One can then look at certain graph theoretical properties of an empirical network, like its degree or motiv distribution, its assortativity or clustering coefcient, the spectrum of its Laplacian, and so on. One can also compare an empirical network with certain deterministic or random theoretical models. Successful as this analysis clearly is, we nevertheless see two important limitations. One is that many of the prominent concepts and quantities used in the analysis of empirical networks are node based, like the degree sequence.
    [Show full text]
  • Maximum and Minimum Degree in Iterated Line Graphs by Manu
    Maximum and minimum degree in iterated line graphs by Manu Aggarwal A thesis submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama August 3, 2013 Keywords: iterated line graphs, maximum degree, minimum degree Approved by Dean Hoffman, Professor of Mathematics Chris Rodger, Professor of Mathematics Andras Bezdek, Professor of Mathematics Narendra Govil, Professor of Mathematics Abstract In this thesis we analyze two papers, both by Dr.Stephen G. Hartke and Dr.Aparna W. Higginson, on maximum [2] and minimum [3] degrees of a graph G under iterated line graph operations. Let ∆k and δk denote the minimum and the maximum degrees, respectively, of the kth iterated line graph Lk(G). It is shown that if G is not a path, then, there exist integers A and B such that for all k > A, ∆k+1 = 2∆k − 2 and for all k > B, δk+1 = 2δk − 2. ii Table of Contents Abstract . ii List of Figures . iv 1 Introduction . .1 2 An elementary result . .3 3 Maximum degree growth in iterated line graphs . 10 4 Minimum degree growth in iterated line graphs . 26 5 A puzzle . 45 Bibliography . 46 iii List of Figures 1.1 ............................................1 2.1 ............................................4 2.2 : Disappearing vertex of degree two . .5 2.3 : Disappearing leaf . .7 3.1 ............................................ 11 3.2 ............................................ 12 3.3 ............................................ 13 3.4 ............................................ 14 3.5 ............................................ 15 3.6 : When CD is not a single vertex . 17 3.7 : When CD is a single vertex . 18 4.1 ...........................................
    [Show full text]
  • Degrees & Isomorphism: Chapter 11.1 – 11.4
    “mcs” — 2015/5/18 — 1:43 — page 393 — #401 11 Simple Graphs Simple graphs model relationships that are symmetric, meaning that the relationship is mutual. Examples of such mutual relationships are being married, speaking the same language, not speaking the same language, occurring during overlapping time intervals, or being connected by a conducting wire. They come up in all sorts of applications, including scheduling, constraint satisfaction, computer graphics, and communications, but we’ll start with an application designed to get your attention: we are going to make a professional inquiry into sexual behavior. Specifically, we’ll look at some data about who, on average, has more opposite-gender partners: men or women. Sexual demographics have been the subject of many studies. In one of the largest, researchers from the University of Chicago interviewed a random sample of 2500 people over several years to try to get an answer to this question. Their study, published in 1994 and entitled The Social Organization of Sexuality, found that men have on average 74% more opposite-gender partners than women. Other studies have found that the disparity is even larger. In particular, ABC News claimed that the average man has 20 partners over his lifetime, and the av- erage woman has 6, for a percentage disparity of 233%. The ABC News study, aired on Primetime Live in 2004, purported to be one of the most scientific ever done, with only a 2.5% margin of error. It was called “American Sex Survey: A peek between the sheets”—raising some questions about the seriousness of their reporting.
    [Show full text]
  • A Faster Parameterized Algorithm for PSEUDOFOREST DELETION
    A faster parameterized algorithm for PSEUDOFOREST DELETION Citation for published version (APA): Bodlaender, H. L., Ono, H., & Otachi, Y. (2018). A faster parameterized algorithm for PSEUDOFOREST DELETION. Discrete Applied Mathematics, 236, 42-56. https://doi.org/10.1016/j.dam.2017.10.018 Document license: Unspecified DOI: 10.1016/j.dam.2017.10.018 Document status and date: Published: 19/02/2018 Document Version: Accepted manuscript including changes made at the peer-review stage Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • Group, Graphs, Algorithms: the Graph Isomorphism Problem
    Proc. Int. Cong. of Math. – 2018 Rio de Janeiro, Vol. 3 (3303–3320) GROUP, GRAPHS, ALGORITHMS: THE GRAPH ISOMORPHISM PROBLEM László Babai Abstract Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity status in the P / NP theory: not expected to be NP-complete, yet not known to be solvable in polynomial time. Arguably, the GI problem boils down to filling the gap between symmetry and regularity, the former being defined in terms of automorphisms, the latter in terms of equations satisfied by numerical parameters. Recent progress on the complexity of GI relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinato- rial structures called coherent configurations. Group theory provides the tools to infer either global symmetry or global irregularity from local information, eliminating the symmetry/regularity gap in the relevant scenario; the resulting global structure is the subject of combinatorial analysis. These structural studies are melded in a divide- and-conquer algorithmic framework pioneered in the GI context by Eugene M. Luks (1980). 1 Introduction We shall consider finite structures only; so the terms “graph” and “group” will refer to finite graphs and groups, respectively. 1.1 Graphs, isomorphism, NP-intermediate status. A graph is a set (the set of ver- tices) endowed with an irreflexive, symmetric binary relation called adjacency. Isomor- phisms are adjacency-preseving bijections between the sets of vertices. The Graph Iso- morphism (GI) problem asks to determine whether two given graphs are isomorphic. It is known that graphs are universal among explicit finite structures in the sense that the isomorphism problem for explicit structures can be reduced in polynomial time to GI (in the sense of Karp-reductions1) Hedrlı́n and Pultr [1966] and Miller [1979].
    [Show full text]
  • Navigating Networks by Using Homophily and Degree
    Navigating networks by using homophily and degree O¨ zgu¨r S¸ ims¸ek* and David Jensen Department of Computer Science, University of Massachusetts, Amherst, MA 01003 Edited by Peter J. Bickel, University of California, Berkeley, CA, and approved July 11, 2008 (received for review January 16, 2008) Many large distributed systems can be characterized as networks We show that, across a wide range of synthetic and real-world where short paths exist between nearly every pair of nodes. These networks, EVN performs as well as or better than the best include social, biological, communication, and distribution net- previous algorithms. More importantly, in the majority of cases works, which often display power-law or small-world structure. A where previous algorithms do not perform well, EVN synthesizes central challenge of distributed systems is directing messages to whatever homophily and degree information can be exploited to specific nodes through a sequence of decisions made by individual identify much shorter paths than any previous method. nodes without global knowledge of the network. We present a These results have implications for understanding the func- probabilistic analysis of this navigation problem that produces a tioning of current and prospective distributed systems that route surprisingly simple and effective method for directing messages. messages by using local information. These systems include This method requires calculating only the product of the two social networks routing messages, referral systems for finding measures widely used to summarize all local information. It out- informed experts, and also technological systems for routing performs prior approaches reported in the literature by a large messages on the Internet, ad hoc wireless networking, and margin, and it provides a formal model that may describe how peer-to-peer file sharing.
    [Show full text]