Node and Link Analysis

Transcription

1 Node and Link Analysis Leonid E. Zhukov School of Applied Mathematics and Information Science National Research University Higher School of Economics Leonid E. Zhukov (HSE) Lecture / 25

2 Centrality Measures Sociology. Linton Freeman, Most "important"actors: actor location in the social network Actor centrality - involvment with other actors, many ties, source or recepient Actor prestige - recipient (object) of many ties, ties directed to an actor Three graphs: Star graph Circle graph Line graph Leonid E. Zhukov (HSE) Lecture / 25

3 Three graphs Leonid E. Zhukov (HSE) Lecture / 25

4 Degree Centrality Degree centrality C D (i) = k(i) = j A ij = j A ji Normalized degree centrality C D (i) = 1 n 1 C D(i) High centrality degree -direct contact with many other actors Low degree - not active, peripherial Leonid E. Zhukov (HSE) Lecture / 25

5 Closeness Centrality Closeness centrality How close an actor to all the other actors in network C C (i) = d(i, j) j 1 Normalized closenss centrality C C (i) = (n 1)C C (i) Actor in the center can quickly interact with all others, short communication path to others, minimal number of steps to reach others Leonid E. Zhukov (HSE) Lecture / 25

6 Betweenness Centrality Betweenness Centrality Number of shortest paths going through the actor σ st (i) C B (i) = Normalized betweenness centrality s t i σ st (i) σ st C B (i) = 2 (n 1)(n 2) C B(i) Probability that a communication from s to t will go through i (geodesics) Edge betweenness Leonid E. Zhukov (HSE) Lecture / 25

7 Degree Prestige Degree prestige P D (i) = k in (i) = j A ji Normalized degree prestige P D (i) = 1 n 1 P D(i) Prestigious actors recieve many nominations Leonid E. Zhukov (HSE) Lecture / 25

8 Proximity Prestige Proximity prestige Influence domain - set of actors that can reach i directly and indirectly. I i - size of influence domain. Average distance j d(j, i)/i i P p (i) = I i/(n 1) j d(j, i)/i i Leonid E. Zhukov (HSE) Lecture / 25

9 Centralization Centralization (network measure) - how central the most central node in the network in relation to all other nodes. C x - one of the centrality measures N i [C x (p ) C x (p i )] C x = max N i [C x (p ) C x (p i )] Leonid E. Zhukov (HSE) Lecture / 25

10 Link Analysis Graph G(n, m) and adjacency matrix A ij, edge i j Leonid E. Zhukov (HSE) Lecture / 25

11 Status or Rank Prestige Leo Katz, Take into account status (prestige) of directly connected actors p i j N(i) p j = j A ji p j p i = j A ji p j p = A T p (I A T )p = 0 Nontrivial solution only if det(i A T ) = 0. Need to constraint matrix Leonid E. Zhukov (HSE) Lecture / 25

12 Status or Rank Prestige Leo Katz, 1953 An actor gives equal parts of its prestige to all nearest neighbours p i j A ji p j k out (j) = j A ji k out (j) p j p = (D 1 A) T p where D ii = max(k i, 1) D 1 A - stochastic matrix, j (D 1 A) ij = 1, guaranteed λ max = 1 (D 1 A) T p = λp Leonid E. Zhukov (HSE) Lecture / 25

13 Eigenvector Centrality Phillip Bonacich, c i j A ji c j c i = 1 A ji c j κ κc = A T c (κi A T )p = 0 Nontrivial solution only when det(κi A T ) = 0. Eigenvalue problem. Choose eigenvector corresponding do maximum eigenvalue: κ max = κ 1, c = c 1 j Leonid E. Zhukov (HSE) Lecture / 25

14 Status or Rank Prestige. Eigenvector Centrality Phillip Bonacich, Parametrized centrality measure c(α, β) c i = j (α + βc j )A ji c = αa T e + βa T c c = α(i βa T ) 1 A T e α - found from normalizaion c 2 = c 2 i = 1 β - parameter, degree and direction of dependence on others Leonid E. Zhukov (HSE) Lecture / 25

15 Link Analysis Graph G(n, m) zero out degree nodes, k out (i) = 0 zero in degree nodes, k in (i) = 0 Leonid E. Zhukov (HSE) Lecture / 25

16 Real World Bow tie structure of the web Leonid E. Zhukov (HSE) Lecture / 25

17 Perron-Frobenius Theorem Oscar Perron, 1907, Georg Frobenius,1912. Eigenvalue problem: Pp = λp Perron-Frobenius theorem: Real square matrix with positive entries stochastic (non-negative and rows sum up to one) irreducible (strongly connected graph) aperiodic then unique largest eigenvalue λ max = 1, with positive left eigenvector and power iterations coneverges to it. Solution satisfies p 1 = 1 Stationary distribution of Markov chain Leonid E. Zhukov (HSE) Lecture / 25

18 PageRank Sergey Brin and Larry Page, 1998 Transition matrix: P = D 1 A Stochastic matrix: P = P + det n PageRank matrix: P = αp + (1 α) eet n Eigenvalue problem (choose solution with λ = 1): P T p = λp Notations: e - unit column vector, d - absorbing nodes indicator vector (column) Leonid E. Zhukov (HSE) Lecture / 25

19 PageRank computations Eigenvalue problem [ ) ] α ((D 1 A) T + edt + (1 α) eet p = λp n n Power iterations p α(d 1 A) T p + α e n (dt p) + (1 α) e n (et p) p p p Sparse linear system (λ = 1, p 1 = 1) [ )] I α ((D 1 A) T + edt p = (1 α) e n n Leonid E. Zhukov (HSE) Lecture / 25

20 Hubs and Authorities HITS, Jon Kleinberg, 1999 Citation networks. Reviews vs original research (authoritative) papers authorities, contain useful information, a i hubs, contains links to authorities, h i Mutual recursion Good authorities reffered by good hubs a i j A ji h j Good hubs point to good authorities h i j A ij a j Leonid E. Zhukov (HSE) Lecture / 25

21 HITS System of linear equations a = αa T h h = βaa Symmetric eigenvalue problem (A T A)a = λa (AA T )h = λh where eigenvalue λ = (αβ) 1 Leonid E. Zhukov (HSE) Lecture / 25

22 Centrality and Prestige of Florentine Families The Medici family marriage network Leonid E. Zhukov (HSE) Lecture / 25

23 Ranking comparison The Kendall tau rank distance is a metric that counts the number of pairwise disagreements between two ranking lists Kendall rank correlation coefficient, commonly referred to as Kendall s tau coefficient τ = n c n d n(n 1)/2 n c - number of concordant pairs, n d - number of discordant pairs 1 τ 1, perfect agreement τ = 1, reversed τ = 1 Example Rank 1 A B C D E Rank 2 C D A B E τ = 6 4 5(5 1)/2 = 0.2 Leonid E. Zhukov (HSE) Lecture / 25

24 References Centrality in Social Networks. Conceptual Clarification, Linton C. Freeman, Social Networks, 1, , 1979 Power and Centrality: A Family of Measures, Phillip Bonacich, The American Journal of Sociology, Vol. 92, No. 5 (Mar., 1987), Leonid E. Zhukov (HSE) Lecture / 25

25 References A new status index dervided from sociometric analysis, L. Katz, Psychometrika, 19, 39-43, Power and centrality: A family of measures, P. Bonacich, American Journal of Sociology, 92, ,1987. Graph structure in the Web, Andrei Broder et all, Procs of the 9th international World Wide Web conference on Computer networks, The PageRank Citation Ranknig: Bringing Order to the Web, S. Brin, L. Page, R. Motwany, T. Winograd, Stanford Digital Library Technologies Project, 1998 Authoritative Sources in a Hyperlinked Environment, Jon M. Kleinberg, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, Leonid E. Zhukov (HSE) Lecture / 25