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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
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
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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  • 1. 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
  • 2. 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
  • 3. 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
  • 4. 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
  • 5. 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
  • 6. 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
  • 7. 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
  • 8. 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
  • 9. 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
  • 10. 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
  • 11. 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
  • 12. 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
  • 13. 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
  • 14. 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
  • 15. 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
  • 16. 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
  • 17. 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
  • 18. 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
  • 19. 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
  • 20. 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
  • 21. 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
  • 22. 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
  • 23. 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
  • 24. 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
  • 25. 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
  • 26. 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
  • 27. 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
  • 28. 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
  • 29. 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
  • 30. 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
  • 31. 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
  • 32. 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
  • 33. 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
  • 34. 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
  • 35. 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
  • 36. 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
  • 37. 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
  • 38. 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
  • 39. 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
  • 40. 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
  • 41. 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
  • 42. 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
  • 43. 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
  • 44. 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
  • 45. 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
  • 46. 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
  • 47. 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
  • 48. 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
  • 49. 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
  • 50. 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
  • 51. 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
  • 52. [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
  • 53. [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
  • 54. [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

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