A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
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A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks

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Many, if not most network analysis algorithms have been designed specifically for single-relational networks; that is, networks in which all edges are of the same type. For example, edges may either ...

Many, if not most network analysis algorithms have been designed specifically for single-relational networks; that is, networks in which all edges are of the same type. For example, edges may either represent "friendship," "kinship," or "collaboration," but not all of them together. In contrast, a multi-relational network is a network with a heterogeneous set of edge labels which can represent relationships of various types in a single data structure. While multi-relational networks are more expressive in terms of the variety of relationships they can capture, there is a need for a general framework for transferring the many single-relational network analysis algorithms to the multi-relational domain. It is not sufficient to execute a single-relational network analysis algorithm on a multi-relational network by simply ignoring edge labels. This article presents an algebra for mapping multi-relational networks to single-relational networks, thereby exposing them to single-relational network analysis algorithms.

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A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks Presentation Transcript

  • A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks Marko A. Rodriguez marko@lanl.gov T-7: Mathematical Modeling and Analysis Group CNLS: Center for Nonlinear Studies Los Alamos National Laboratory June 26, 2008
  • Single- and Multi-Relational Networks Human-D Human-B Human-F Human-C Human-A Human-E Publisher-A Article-A publishedBy authored editorOf Human-B Journal-A containedIn authored Human-A authored Article-B Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Presentation Article Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to Single-Relational Network Analysis Algorithms”, LA-UR-08-03931, May 2008, http://arxiv.org/abs/0806.2274. Acknowledgements: • Ideas inspired by the MESUR problem space [Bollen et al., 2007]. • Vadas Gintautas aided in reviewing drafts of the article and presentation. • Michael Ham aided in reviewing drafts of the presentation. • Razvan Teodorescu taught me FoilTex and this is his template (I’m sorry, I just don’t know how to change the color scheme!) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Problem Statement Think about all of the known network analysis algorithms: • geodesics: diameter, eccentricity [Harary & Hage, 1995], closeness [Bavelas, 1950], betweenness [Freeman, 1977], ... • spectral: PageRank [Brin & Page, 1998], eigenvector centrality [Bonacich, 1987], ... • community detection: leading eigenvector [Newman, 2006], edge betweenness [Girvan & Newman, 2002], ... • mixing pattens: scalar and discrete assortativity [Newman, 2003], ... • on and on and on... These algorithms have been developed for directed or undirected single-relational networks. What do you do when you have a multi-relational network? Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Problem Statement Now think about all of the known network analysis packages: • Java Universal Network/Graph Framework (JUNG) [O’Madadhain et al., 2005] • iGraph: Package for Complex Network Research [Csardi, 2006] • Pajek • NetworkX [Hagberg et al., n.d.] • on and on and on... These packages (for the most part) have been developed for directed or undirected single-relational networks. What do you do when you have a multi-relational network? Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Problem Statement • Do you reimplement all of the known algorithms to support a multi- relational network? • Even if you do, what do these algorithms look like? Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Solution Statement • You map your multi-relational network to a “meaningful” single-relational network and re-use existing algorithms, packages, and theorems from the single-relational domain. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • An Undirected Single-Relational Network Human-D Human-B Human-F Human-C Human-A Human-E All edges have a single homogenous meaning (e.g. co-author). G = (V, E ⊆ {V × V }) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A Directed Single-Relational Network Article-D Article-B Article-F Article-C Article-A Article-E All edges have a single homogenous meaning (e.g. citation). G = (V, E ⊆ (V × V )) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A Multi-Relational Network Publisher-A Article-A publishedBy authored editorOf Human-B Journal-A containedIn authored Human-A authored Article-B Edges are heterogenous in meaning. M = (V, E = {E0 ⊆ (V × V ), E1, . . . , Em}) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Flatten the Multi-Relational Network • Suppose you have a multi-relational network, where there exists only two edge sets defined as coauthor and friend. M = (V, E = {E0, E1}) and you want to determine the most central “scholar” in this network. • It is not sufficient to simply ignore edge labels (flatten the multi- relational network to a single-relational network) and execute a centrality algorithm on the network. You will confuse central friendship with central scholarship. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Extract a Single-Relational Network Component • You could simply pull out the coauthor single relational network G = (V, E0) and calculate a centrality algorithm on that network to get your result. • That works, but for more complex situations with “richer semantics”, this mechanism will not work. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Execute a Grammar-Based Walker • A walker obeys a “grammar” that specifies the way in which the walker should move through the network [Rodriguez, 2008]. coauthor primary eigenvector grammar Publisher-A while(true) incr vertex counter publishedBy go authored go authored -1 but don't go back editorOf Human-B to previous vertex Journal-A containedIn authored Human-A authored Article-B coauthor • Problem – this solution mixes the analysis algorithm and the traversed implicit network. • Solution – an algebra that is agnostic to the final executing algorithm. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • An Adjacency Matrix Representation of a Single-Relational Network n = |V | 0 1 0 0 1 Article-A Article-D 0 0 0 0 0 Article-B n = |V | Article-B 0 0 0 1 1 Article-C Article-F Article-C 0 0 0 0 0 Article-D 0 0 0 0 0 Article-E Article-A Article-C Article-D Article-A Article-B Article-E Article-E * NOTE: Sorry about missing the vertex Article-F in the adjacency matrix. Too lazy to redo diagrams. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • An Adjacency Matrix Representation of a Single-Relational Network A single-relational network defined as G = (V, E ⊆ (V × V )) can be represented as the adjacency matrix A ∈ {0, 1}n×n, where 1 if (i, j) ∈ E Ai,j = 0 otherwise. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A Three-Way Tensor Representation of a Multi-Relational Network 0 1 1 0 0 Human-A Article-A 0 0 0 0 0 n = |V | Publisher-A Article-A 0 0 0 0 0 Article-B publishedBy 0 0 1 0 0 Human-B In authored Human-B d editorOf ne Journal-A 0 0 Journal-A f 0 0 0 rO y ai dB ito nt m ed co he ed is Article-A Article-B Journal-A Human-A Human-B = or bl authored th pu containedIn au Human-A |E authored Article-B | n = |V | * NOTE: Sorry about missing the vertex Publisher-A in the tensor. Too lazy to redo diagrams. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A Three-Way Tensor Representation of a Multi-Relational Network A three-way tensor can be used to represent a multi-relational network [Kolda et al., 2005]. If M = (V, E = {E0, E1, . . . , Em ⊆ (V × V )}) is a multi-relational network, then A ∈ {0, 1}n×n×m and 1 if (i, j) ∈ Em Am i,j = 0 otherwise. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The General Purpose of the Path Algebra • Map a multi-relational tensor A ∈ {0, 1}n×n×m to a single-relational path matrix Z ∈ Rn×n . + • By performing operations on A, a single-relational path matrix is created whose “edges” are loaded with meaning. • For example, you can create a coauthorship network, a social science journal citation network, a coauthorship network for scholars from the same university who have not been on the same project in the last 10 years, but are in the same department, etc. • The theorems of the algebra can be used to manipulate your mapping operation to a smaller/more efficient form (i.e. how a composition is spoken in words can differ from its reduced form). 0 1 1 0 0 24 1 0 0 0 0 0 0 0 0 0 72 0 4 0 0 0 0 0 0 23 0 0 0 0 0 0 1 0 0 0 0 15.3 0 0 0 0 0 0 0 0 0 0 0 12 A ∈ {0, 1}n×n×m Z ∈ Rn×n + Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Elements of the Path Algebra • A ∈ {0, 1}n×n×m: a three-way tensor representation of a multi-relational network. • Z ∈ Rn×n: a path matrix derived by means of operations applied to A. + —————————————————————————————— • Cj ∈ {0, 1}n×n: “to” path filters. • Ri ∈ {0, 1}n×n: “from” path filters. • I ∈ {0, 1}n×n: the identity matrix as a self-loop filter. • 1 ∈ 1n×n: a matrix in which all entries are equal to 1. • 0 ∈ 0n×n: a matrix in which all entries are equal to 0. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i Example Scholarly Tensor Used in the Remainder of the Presentation • A1: authored : human → article • A2: cites : article → article • A3: contains : journal → article • A4: category : journal → subject category • A5: developed : human → program/software. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Operations of the Path Algebra • A · B: ordinary matrix multiplication determines the number of (A, B)- paths between vertices. • A : matrix transpose inverts path directionality. • A ◦ B: Hadamard, entry-wise multiplication applies a filter to selectively exclude paths. • n(A): not generates the complement of a {0, 1}n×n matrix. • c(A): clip generates a {0, 1}n×n matrix from a Rn×n matrix. + • v ±(A): vertex generates a {0, 1}n×n matrix from a Rn×n matrix, where + only certain rows or columns contain non-zero values. • λA: scalar multiplication weights the entries of a matrix. • A + B: matrix addition merges paths. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Traverse Operation • An interesting aspect of the single-relational adjacency matrix A ∈ {0, 1}n×n is that when it is raised (k) to the kth power, the entry Ai,j is equal to the number of paths of length k that connect vertex i to vertex j [Chartrand, 1977]. (1) • Given, by definition, that Ai,j (i.e. Ai,j ) represents the number of paths that go from i to j of length 1 (i.e. a single edge) and by the rules of ordinary matrix multiplication, (k) (k−1) Ai,j = Ai,l · Al,j : k ≥ 2. l∈V a b c a b c a b c a b c a 0 1 0 a 0 1 0 a 0 0 1 b 0 0 1 · b 0 0 1 = b 0 0 0 c 0 0 0 c 0 0 0 c 0 0 0 there is a path of length 2 from a to c Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Traverse Operation Z = A1 · A2 · A1 , Zi,j defines the number of paths from vertex i to vertex j such that a path goes from author i to one the articles he or she has authored, from that article to one of the articles it cites, and finally, from that cited article to its author j . Semantically, Z is an author-citation single-relational path matrix. A2 Article-A cites Article-B A1 authored A1 authored Human-A author-citation Human-B Z * NOTE: All diagrams are with respect to a “source” vertex (the blue vertex) in order to preserve clarity. In reality, the operations operate on all vertices in parallel. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Filter Operation Various path filters can be defined and applied using the entry-wise Hadamard matrix product denoted ◦, where   A1,1 · B1,1 · · · A1,m · B1,m A◦B= . . ... . . . An,1 · Bn,1 · · · An,m · Bn,m 24 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 72 0 4 0 0 1 0 0 0 0 72 0 0 0 23 0 0 0 0 ◦ 1 0 0 0 0 = 23 0 0 0 0 0 0 15.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 Path Matrix Path Filter Filtered Path Matrix Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Filter Operation • A◦1=A • A◦0=0 • A◦B=B◦A • A ◦ (B + C) = (A ◦ B) + (A ◦ C) • A ◦ B = (A ◦ B) . Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Not Filter The not filter is useful for excluding a set of paths to or from a vertex. n : {0, 1}n×n → {0, 1}n×n with a function rule of 1 if Ai,j = 0 n(A)i,j = 0 otherwise. 0 0 1 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 n 0 1 1 1 1 = 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Not Filter If A ∈ {0, 1}n×n, then • n(n(A)) = A • A ◦ n(A) = 0 • n(A) ◦ n(A) = n(A). Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Not Filter A coauthorship path matrix is Z = A1 · A1 ◦ n(I) Article-A A1 authored A1 authored Human-A coauthor Human-B Z n(I) coauthor Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Clip Filter The general purpose of clip is to take a path matrix and “clip”, or normalize, it to a {0, 1}n×n matrix. c : Rn×n → {0, 1}n×n + 1 if Zi,j > 0 c(Z)i,j = 0 otherwise. 24 1 0 0 0 1 1 0 0 0 0 72 0 4 0 0 1 0 1 0 c 23 0 0 0 0 = 1 0 0 0 0 0 0 15.3 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 1 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Clip Filter If A, B ∈ {0, 1}n×n and Y, Z ∈ Rn×n, then + • c(A) = A • c(n(A)) = n(c(A)) = n(A) • c(Y ◦ Z) = c(Y) ◦ c(Z) • n(A ◦ B) = c (n(A) + n(B)) • n(A + B) = n(A) ◦ n(B) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Clip Filter Suppose we want to create an author citation path matrix that does not allow self citation or coauthor citations. „ « „ „ «« 1 2 1 1 1 Z= A ·A ·A ◦n c A · A ◦ n(I) ◦ n(I) |{z} | {z } | {z } no self cites no coauthors Z author-citation Human-C authored 2 A A1 Article-A cites Article-B A 1 A1 authored authored authored Human-A coauthor Human-B n c A1 · A1 ◦ n(I) self n(I) Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Clip Filter However, using various theorems of the algebra, Z = A1 · A2 · A1 ◦ n c A1 · A1 ◦ n(I) ◦ n(I) no self cites no coauthors becomes Z = A1 · A2 · A1 ◦ n c A1 · A1 ◦ n(I). Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Vertex Filter In many cases, it is important to filter out particular paths to and from a vertex. v − : Rn×n × N → {0, 1}n×n, + − 1 if k∈V Zi,k > 0 v (Z)i,j = 0 otherwise turns a non-zero column into an all 1-column and v + : Rn×n × N → {0, 1}n×n, + + 1 if k∈V Zk,j > 0 v (Z)i,j = 0 otherwise turns a non-zero row into an all 1-row. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Vertex Filter 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 v− 0 2 0 32 0 = 0 1 0 1 0 0 23 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 v + not diagrammed, but acts the same except for makes 1-rows. Two import filters are the column and row filters, C ∈ {0, 1}n×n and R ∈ {0, 1}n×n , respectively. 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 C2 = 0 1 0 0 0 R3 = 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Vertex Filter • v −(Ci) = Ci • v +(Rj ) = Rj • v −(Z) = v +(Z ) • v +(Z) = v −(Z ) . Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Vertex Filter Assume that vertex 1 is the social science subject category vertex and we want to create a journal citation network for social science journals only. » „ «– + 4 3 2 3 − 4 h “ ” i Z= v C1 ◦ A ◦ A ·A · A ◦v R1 ◦ A . | {z } | {z } soc.sci. journal articles articles in soc.sci. journals social-science journal citation Z 1 Social Science category category A2 Article-B contains Journal-B A3 cites A3 v − R1 ◦ A4 Journal-A contains Article-A cites + 4 v C1 ◦ A 2 Article-C contains Journal-C A A3 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Vertex Filter + 4 3 h “ ” i v C1 ◦ A ◦A | {z } soc.sci. journal articles S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S 1 0 0 0 0 0 0 S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 J-A 1 0 0 0 0 0 0 J-A 1 0 0 0 0 0 0 J-A 1 0 0 0 0 0 0 J-B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ◦ 0 J-B 0 J-B 0 = 1 1 1 J-C 1 0 0 0 0 0 0 J-C 0 0 0 0 0 0 0 J-C 0 0 0 0 0 0 0 A-A 1 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-B 1 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-C 1 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 C1 A4 C1 ◦ A4 S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 J-A 1 1 1 1 1 1 1 J-A 0 0 0 0 1 0 0 J-A 0 0 0 0 1 0 0 J-B 1 1 1 1 1 1 1 J-B 0 0 0 0 0 1 0 J-B 0 0 0 0 0 1 0 J-C 0 0 0 0 0 0 0 ◦ J-C 0 0 0 0 0 0 1 = J-C 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 v + (C1 ◦ A4 ) A3 v + (C1 ◦ A4 ) ◦ A3 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Vertex Filter Z = v + C1 ◦ A4 ◦ A3 ·A2 · A3 ◦ v − R1 ◦ A4 . soc.sci. journal articles articles in soc.sci. journals However, v − R1 ◦ A4 = v− C1 ◦ A4 Cx = Rx = v + C1 ◦ A4 v +(Z) = v −(Z ) . Therefore, because A ◦ B = (A ◦ B) , Z = v + C1 ◦ A4 ◦ A3 ·A2 · v + C1 ◦ A4 ◦ A3 . reused reused Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Weight and Merge Filter • λZ: scalar multiplication weights paths. • Y + Z: matrix addition merges paths. 24 1 0 0 0 0 1 0 0 0 24 2 0 0 0 0 72 0 4 0 0 10 0 0 0 0 82 0 4 0 23 0 0 0 0 + 1 0 34 0 0 = 24 0 34 0 0 0 0 15.3 0 0 0 0 0 0 0 0 0 15.3 0 0 0 0 0 0 12 0 0 0 0 2 0 0 0 0 14 Center for Non-Linear Studies Public Lecture - June 26, 2008
  • A1 : authored A2 : cites A3 : contains A4 : category A5 : developed h ih ih ih ih i The Weight and Merge Filter Z = 0.6 A1 · A1 ◦ n(I) + 0.4 A5 · A5 ◦ n(I) coauthorship co-development merges the article and software program collaboration path matrices as specified by their respective weights of 0.6 and 0.4. The semantics of the resultant is a software program and article collaboration path matrix that favors article collaboration over software program collaboration. A simplification of the previous composition is Z = 0.6 A1 · A1 + 0.4 A5 · A5 ◦ n(I). Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Outline • Formalizing Single- and Multi-Relational Networks • Background on Multi-Relational Network Analysis • The Elements of the Path Algebra • The Operations of the Path Algebra • Multi-Relational Network Analysis Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Application to the Real-World • A can be represented in a standard matrix manipulation package. • Z can be constructed with the same matrix manipulation package. • The path matrix Z has a weighted network representation. Z = (V, E ⊆ (V × V ), λ), where λ : E → R+ • Z can be used in standard network analysis packages. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Page Rank Tensor In the matrix form of PageRank, there exist two adjacency matrices in [0, 1]n×n denoted 1 1 |Γ+ (i)| if (i, j) ∈ E Ai,j = 0 otherwise. and 1 A2 = i,j . |V | A1 is a row-stochastic adjacency matrix and A2 is a fully connected adjacency matrix known as the teleportation matrix. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • The Page Rank Tensor The purpose of PageRank is to identify the primary eigenvector of a weighted merged path matrix of the form Z = δ · A1 + (1 − δ) · A2 . Z is guaranteed to be a strongly connected single-relational path matrix because there is some probability (defined by 1 − δ) that every vertex is reachable by every other vertex. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • Conclusion • Most of graph and network theory is concerned with the design of theorems and algorithms for single-relational networks. • Given a multi-relational network, you can manipulate a tensor representation of it to yield a “semantically-rich” single-relational network. • Thus, a multi-relational network can be exposed to the concepts of the single-relational domain. Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to Single-Relational Network Analysis Algorithms”, LA-UR-08-03931, May 2008, http://arxiv.org/abs/0806.2274. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • References [Bavelas, 1950] Bavelas, A. 1950. Communication Patterns in Task Oriented Groups. The Journal of the Acoustical Society of America, 22, 271–282. [Bollen et al., 2007] Bollen, Johan, Rodriguez, Marko A., & Van de Sompel, Herbert. 2007. MESUR: usage-based metrics of scholarly impact. In: Joint Conference on Digital Libraries (JCDL07). Vancouver, Canada: IEEE/ACM. [Bonacich, 1987] Bonacich, Phillip. 1987. Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170–1182. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • [Brin & Page, 1998] Brin, Sergey, & Page, Lawrence. 1998. The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30(1–7), 107–117. [Chartrand, 1977] Chartrand, Gary. 1977. Introductory Graph Theory. Dover. [Csardi, 2006] Csardi, Gabor. 2006. The igraph software package for complex network research. InterJournal Complex Systems. [Freeman, 1977] Freeman, L. C. 1977. A set of measures of centrality based on betweenness. Sociometry, 40(35–41). [Girvan & Newman, 2002] Girvan, Michelle, & Newman, M. E. J. 2002. Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99, 7821. Center for Non-Linear Studies Public Lecture - June 26, 2008
  • [Hagberg et al., n.d.] Hagberg, Aric, Schult, Daniel A., & Swart, Pieter J. NetworkX. https://networkx.lanl.gov. [Harary & Hage, 1995] Harary, Frank, & Hage, Per. 1995. Eccentricity and centrality in networks. Social Networks, 17, 57–63. [Kolda et al., 2005] Kolda, Tamara G., Bader, Brett W., & Kenny, Joseph P. 2005. Higher-Order Web Link Analysis Using Multilinear Algebra. In: Proceedings of the Fifth IEEE International Conference on Data Mining ICDM’05. IEEE. [Newman, 2003] Newman, M. E. J. 2003. Mixing patterns in networks. Physical Review E, 67(2), 026126. [Newman, 2006] Newman, M. E. J. 2006. Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74(May). Center for Non-Linear Studies Public Lecture - June 26, 2008
  • [O’Madadhain et al., 2005] O’Madadhain, Joshua, Fisher, Danyel, Nelson, Tom, & Krefeldt, Jens. 2005. JUNG: Java Universal Network/Graph Framework. [Rodriguez, 2008] Rodriguez, Marko A. 2008. Grammar-Based Random Walkers in Semantic Networks. Knowledge-Based Systems, [in press]. Center for Non-Linear Studies Public Lecture - June 26, 2008