Local properties of PageRank and graph limits. Nelly Litvak University of Twente Eindhoven University of Technology, The Netherlands MIPT 2018

Transcription

1 Local properties of PageRank and graph limits Nelly Litvak University of Twente Eindhoven University of Technology, The Netherlands MIPT 2018

2 Centrality in networks Network as a graph G = (V, E)

3 Centrality in networks Network as a graph G = (V, E) Undirected:

4 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks

5 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks Directed:

6 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks Directed: Twitter, bank transactions, food webs, WWW, scientific citations

7 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks Directed: Twitter, bank transactions, food webs, WWW, scientific citations V set of vertices, E set of edges

8 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks Directed: Twitter, bank transactions, food webs, WWW, scientific citations V set of vertices, E set of edges V = n, we can let n

9 Centrality in networks Network as a graph G = (V, E) Undirected: Facebook, roads networks, airline networks, power grids, co-authorship networks Directed: Twitter, bank transactions, food webs, WWW, scientific citations V set of vertices, E set of edges V = n, we can let n Which nodes are most central in a network?

10 Google search Brin and Page 1998 The anatomy of a large-scale hypertextual web search engine. Page, Brin, Motwani and Winograd 1999 The PageRank Citation Ranking: Bringing Order to the Web.

11 Google search Brin and Page 1998 The anatomy of a large-scale hypertextual web search engine. Page, Brin, Motwani and Winograd 1999 The PageRank Citation Ranking: Bringing Order to the Web. Yahoo, AltaVista,...

12 Google search Brin and Page 1998 The anatomy of a large-scale hypertextual web search engine. Page, Brin, Motwani and Winograd 1999 The PageRank Citation Ranking: Bringing Order to the Web. Yahoo, AltaVista,... Directory-based, comparable to telephone books

13 Google search Brin and Page 1998 The anatomy of a large-scale hypertextual web search engine. Page, Brin, Motwani and Winograd 1999 The PageRank Citation Ranking: Bringing Order to the Web. Yahoo, AltaVista,... Directory-based, comparable to telephone books

14 Google PageRank PageRank r i of page i = 1,..., n is defined as: r i = j : j i α d j r j + (1 α)q i, i = 1,..., n d j = # out-links of page j α (0, 1), damping factor originally 0.85 q i 0, i q i = 1, originally, q i = 1/n.

15 Easily bored surfer model r i = j : j i α d j r j + (1 α)q i, i = 1,..., n r i is a stationary distribution of a Markov chain

16 Easily bored surfer model r i = j : j i α d j r j + (1 α)q i, i = 1,..., n r i is a stationary distribution of a Markov chain with probability α follow a randomly chosen outgoing link with probability 1 α random jump (to page i w.p. q i )

17 Easily bored surfer model r i = j : j i α d j r j + (1 α)q i, i = 1,..., n r i is a stationary distribution of a Markov chain with probability α follow a randomly chosen outgoing link with probability 1 α random jump (to page i w.p. q i ) Dangling nodes, d j = 0: Random jump from dangling nodes

18 Easily bored surfer model r i = j : j i α d j r j + (1 α)q i, i = 1,..., n r i is a stationary distribution of a Markov chain with probability α follow a randomly chosen outgoing link with probability 1 α random jump (to page i w.p. q i ) Dangling nodes, d j = 0: Random jump from dangling nodes Stationary distribution π = r/ r 1

19 Easily bored surfer model r i = j : j i α d j r j + (1 α)q i, i = 1,..., n r i is a stationary distribution of a Markov chain with probability α follow a randomly chosen outgoing link with probability 1 α random jump (to page i w.p. q i ) Dangling nodes, d j = 0: Random jump from dangling nodes Stationary distribution π = r/ r 1 The page is important if many important pages link to it!

20 PageRank beyond web search Applications: Topic-sensitive search (Haveliwala 2002); Spam detection (Gyöngyi et al. 2004) Finding related entities (Chakrabarti 2007); Link prediction (Liben-Nowell and Kleinberg 2003; Voevodski, Teng, Xia 2009); Finding local cuts (Andersen, Chung, Lang 2006); Graph clustering (Tsiatas, Chung 2010); Person name disambiguation (Smirnova, Avrachenkov, Trousse 2010); Finding most influential people in Wikipedia (Shepelyansky et al 2010, 2013)

21 PageRank beyond web search Applications: Topic-sensitive search (Haveliwala 2002); Spam detection (Gyöngyi et al. 2004) Finding related entities (Chakrabarti 2007); Link prediction (Liben-Nowell and Kleinberg 2003; Voevodski, Teng, Xia 2009); Finding local cuts (Andersen, Chung, Lang 2006); Graph clustering (Tsiatas, Chung 2010); Person name disambiguation (Smirnova, Avrachenkov, Trousse 2010); Finding most influential people in Wikipedia (Shepelyansky et al 2010, 2013) Ranking in sports (Beggs et al. 2017, Rojas-Mora et al. 2017, Lazova and Basnarkov 2015) Global characteristic of the graph

22 Example: food web

23 Example: food web Allesina and Pascual 2009

24 Matrix expansion r = αrp + (1 α)q r = (1 α)q[i αp] 1 = (1 α)q Computation by matrix iterations: r (0) = (1/n,..., 1/n) r (k) = αr (k 1) P + (1 α)q α t P t. t=0 k 1 = r (0) α k P k + (1 α)q α t P t t=0

25 Matrix expansion r = αrp + (1 α)q r = (1 α)q[i αp] 1 = (1 α)q Computation by matrix iterations: r (0) = (1/n,..., 1/n) r (k) = αr (k 1) P + (1 α)q α t P t. t=0 k 1 = r (0) α k P k + (1 α)q α t P t t=0 Exponentially fast convergence due to α (0, 1)

26 Matrix expansion r = αrp + (1 α)q r = (1 α)q[i αp] 1 = (1 α)q Computation by matrix iterations: r (0) = (1/n,..., 1/n) r (k) = αr (k 1) P + (1 α)q α t P t. t=0 k 1 = r (0) α k P k + (1 α)q α t P t t=0 Exponentially fast convergence due to α (0, 1) Matrix iterations are used to compute PageRank in practice Langville&Meyer 2004, Berkhin 2005

27 Power laws in complex networks Power laws: Internet, WWW, social networks, biological networks, etc...

28 Power laws in complex networks Power laws: Internet, WWW, social networks, biological networks, etc... degree of the node = # (in-/out-) links [fraction nodes degree at least k] = p k, Power law: p k const k γ 1, α > 0. Power law is the model for high variability: some nodes (hubs) have extremely many connections

29 Power laws in complex networks Power laws: Internet, WWW, social networks, biological networks, etc... degree of the node = # (in-/out-) links [fraction nodes degree at least k] = p k, Power law: p k const k γ 1, α > 0. Power law is the model for high variability: some nodes (hubs) have extremely many connections log p k = log(const) (γ + 1) log k

30 Power laws in complex networks Power laws: Internet, WWW, social networks, biological networks, etc... degree of the node = # (in-/out-) links [fraction nodes degree at least k] = p k, Power law: p k const k γ 1, α > 0. Power law is the model for high variability: some nodes (hubs) have extremely many connections log p k = log(const) (γ + 1) log k Straight line on the log-log scale

31 Power law hypothesis for PageRank Pandurangan, Raghavan, Upfal 2002.

32 Power law hypothesis for PageRank Pandurangan, Raghavan, Upfal Power law hypothesis for PageRank: When (in-)degree distribution follows a power law then the PageRank follows power law with the same exponent.

33 Stochastic model for PageRank r i = j : j i α d j r j + (1 α)q i, i = 1,..., n

34 Stochastic model for PageRank r i = j : j i α d j r j + (1 α)q i, i = 1,..., n Rescale: R i = nr i, Q i n(1 α)q i so that E(R) = 1 Idea: model R as a solution of stochastic equation (Volkovich&L 2010): R d = α N j=1 1 D j R j + Q N: in-degree of the randomly chosen page D: out-degree of page that links to the randomly chosen page R j is distributed as R; N, D, R j are independent Denote C j = α/d j.

35 Stochastic model for PageRank r i = j : j i α d j r j + (1 α)q i, i = 1,..., n Rescale: R i = nr i, Q i n(1 α)q i so that E(R) = 1 Idea: model R as a solution of stochastic equation (Volkovich&L 2010): R d = N C j R j + Q j=1 N: in-degree of the randomly chosen page D: out-degree of page that links to the randomly chosen page R j is distributed as R; N, D, R j are independent Denote C j = α/d j.

36 Results for stochastic recursion R d = N C j R j + Q j=1 Theorem (Volkovich&L 2010) If P(Q > x) = o(p(n > x)), then the following are equivalent: P(N > x) L(x)x γ as x, P(R > x) al(x)x γ as x, where a = (E[C]) γ (1 E[N]E[C γ ]) 1 Here a b means a/b 1 Note that E[C] = 1/E(N), the role of out-degrees is minimal!

37 Results for stochastic recursion R d = N C j R j + Q j=1 Olvera-Cravioto, Jelenkovic 2010, 2012 analyzed the recursion under most general assumptions on C j s. For example, R can be heavy-tailed even when N is light-tailed.

38 Results for stochastic recursion N R = d C j R j + Q j=1 Olvera-Cravioto, Jelenkovic 2010, 2012 analyzed the recursion under most general assumptions on C j s. For example, R can be heavy-tailed even when N is light-tailed. However, this does not completely explain the behavior of PageRank in networks because the recursion implicitly assumes an underlying tree structure.

39 Results for stochastic recursion N R = d C j R j + Q j=1 Olvera-Cravioto, Jelenkovic 2010, 2012 analyzed the recursion under most general assumptions on C j s. For example, R can be heavy-tailed even when N is light-tailed. However, this does not completely explain the behavior of PageRank in networks because the recursion implicitly assumes an underlying tree structure.

40 PageRank in a random graph Chen, L, Olvera-Cravioto 2017 M = M(n) R n n is related to the adjacency matrix of the graph: { C i P ij, if there are s ij edges from i to j, M i,j = 0, otherwise. Q R n is a personalization vector. Assume E(Q i ) < H. We are interested in the distribution of one coordinate, R (n) 1, of the vector R (n) R n defined by R (n) = R (n) M + Q

41 Finite approximation of PageRank Assume C i α, E Q i = 1 α. Then R (n,n) R (n, ) 1 We want to bound R (n, ) i Nodes are not identical Q 1 α N+i α N = Q 1 1 α. i=0 R (n,n) i for individual nodes The standard results on mixing times do not help to get rid of the factor n

42 Local approximation of PageRank Idea: Average over a random node [ [ ] E R Vn (n) R (N) 1 n V n (n) = E n i=1 [ 1 n E α k [ ] QP k] n i i=1 k=n+1 = E 1 n n α k Q j Pji k n = 1 n n i=1 k=n+1 j=1 k=n+1 j=1 α k (1 α) k=n+1 n Pji k α N+1, i=1 [ QM k ] i ]

43 Local approximation of PageRank w.h.p. Markov inequality: ([ ] ) P R (n, ) 1 R (n,n) 1 > ε = O ( εα k) Approximation with arbitrary precision in finite number of iterations It is a weaker result than bounding R (n, ) i R (n,k) i, but it is good enough

44 Directed Configuration Model (DCM)

45 Directed Configuration Model (DCM)

46 Directed Configuration Model (DCM) F + v 1 v 1 F v 2 v 2.. v n v n

47 Directed Configuration Model (DCM) F + v 1 v 1 F v 2 v 2.. v n v n

48 Directed Configuration Model (DCM) F + v 1 v 1 F v 2 v 2.. v n v n

49 Directed Configuration Model (DCM) F + v 1 v 1 F v 2 v 2.. v n v n

50 Directed Configuration Model (DCM) F + v 1 v 1 F v 2 v 2.. v n v n We keep self-loops and double edges. The result is a multi-graph

51 Limiting distribution of PageRank in the DCM Chen, L, Olvera-Cravioto 2017 R D = N C j R j + Q, j=1 Let R denote the endogenous solution to the SFPE above. The endogenous solution is the limit of iterations of the recursion starting, say, from R 0 = 1. Main result: R (n) 1 R, n, where denotes weak convergence and R is given by N 0 R := C j R j + Q 0, j=1

52 Methodology 1. Finite approximation. PageRank is accurately approximated by a finite number of matrix iterations.

53 Methodology 1. Finite approximation. PageRank is accurately approximated by a finite number of matrix iterations. 2. Coupling with a tree. Construct a coupling of the DCM graph and a thorny branching tree (TBT). The coupling between the graph and the TBT will hold for a number of generations in the tree that is logarithmic in n.

54 Methodology 1. Finite approximation. PageRank is accurately approximated by a finite number of matrix iterations. 2. Coupling with a tree. Construct a coupling of the DCM graph and a thorny branching tree (TBT). The coupling between the graph and the TBT will hold for a number of generations in the tree that is logarithmic in n. 3. Convergence to a weighted branching process. Show that the rank of the root node of the TBT converges weakly to the stated limit. Chen and Olvera-Cravioto (2014)

55 Coupling with a branching tree We start with random node (node 1) and explore its neighbors, labeling the stubs that we have already seen τ the number of generations of WBP completed before coupling breaks

56 Coupling with branching tree Lemma (The Coupling Lemma) Suppose (N n, D n, C n, Q n ) satisfies WLLN, µ = E(ND)/E(D). Then, for any 1 k h log n with 0 < h < 1/(2 log µ), if µ > 1, for any 1 k n b with b < 1/2, if µ 1, we have as n. O ( (n/µ 2k ) 1/2), µ > 1, P (τ k Ω n ) = O ( (n/k 2 ) 1/2), µ = 1, O ( n 1/2), µ < 1, Remark: µ corresponds to the average number of offspring of a node in TBT.

57 The Coupling Lemma: idea of the proof Ẑ s # individuals in generation s of the tree ˆV s # outbound stubs of all nodes in generation s

58 The Coupling Lemma: idea of the proof Ẑ s # individuals in generation s of the tree ˆV s # outbound stubs of all nodes in generation s Ẑ s, ˆV s are not much larger than their means: ] [ ] E n [Ẑs µ s+1, E n ˆV s λµ s, λ = E[D 2 ]/µ.

59 The Coupling Lemma: idea of the proof Ẑ s # individuals in generation s of the tree ˆV s # outbound stubs of all nodes in generation s Ẑ s, ˆV s are not much larger than their means: ] [ ] E n [Ẑs µ s+1, E n ˆV s λµ s, λ = E[D 2 ]/µ. An inbound stub of a node in the rth generation will be the first one to be paired with a labeled outbound stub with a probability not larger than P r := 1 L n r ˆV s s=0 λµr n(µ 1).

60 The Coupling Lemma: idea of the proof Ẑ s # individuals in generation s of the tree ˆV s # outbound stubs of all nodes in generation s Ẑ s, ˆV s are not much larger than their means: ] [ ] E n [Ẑs µ s+1, E n ˆV s λµ s, λ = E[D 2 ]/µ. An inbound stub of a node in the rth generation will be the first one to be paired with a labeled outbound stub with a probability not larger than P r := 1 L n r ˆV s s=0 λµr n(µ 1). {τ = r} is equivalent to the event that Binomial(Ẑ r, P r ) r.v. is greater or equal than 1.

61 The Coupling Lemma: idea of the proof Ẑ s # individuals in generation s of the tree ˆV s # outbound stubs of all nodes in generation s Ẑ s, ˆV s are not much larger than their means: ] [ ] E n [Ẑs µ s+1, E n ˆV s λµ s, λ = E[D 2 ]/µ. An inbound stub of a node in the rth generation will be the first one to be paired with a labeled outbound stub with a probability not larger than P r := 1 L n r ˆV s s=0 λµr n(µ 1). {τ = r} is equivalent to the event that Binomial(Ẑ r, P r ) r.v. is greater or equal than 1. Markov s inequality: P(τ = r) Ẑr P r = O ( µ 2r n 1), r k.

62 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach:

63 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach: Consider a limit of a graph in a local weak convergence sense

64 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach: Consider a limit of a graph in a local weak convergence sense Prove that PageRank in a random graph converges in distribution to the PageRank of a limit.

65 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach: Consider a limit of a graph in a local weak convergence sense Prove that PageRank in a random graph converges in distribution to the PageRank of a limit. Note: This does not prove the power law hypothesis!

66 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach: Consider a limit of a graph in a local weak convergence sense Prove that PageRank in a random graph converges in distribution to the PageRank of a limit. Note: This does not prove the power law hypothesis! The hope is that in the limiting graph the analysis is easier

67 PageRank and local weak convergence Garavaglia, vd Hofstad, L 2018 Alternative approach: Consider a limit of a graph in a local weak convergence sense Prove that PageRank in a random graph converges in distribution to the PageRank of a limit. Note: This does not prove the power law hypothesis! The hope is that in the limiting graph the analysis is easier Example: if the limit is a tree then the PageRank satisfies the stochastic recursion

68 Local weak convergence (LWC) LWC for undirected graphs (Benjamini&Schramm 2001)

69 Local weak convergence (LWC) LWC for undirected graphs (Benjamini&Schramm 2001) (G n ) n N converges to a rooted graph (G, )

70 Local weak convergence (LWC) LWC for undirected graphs (Benjamini&Schramm 2001) (G n ) n N converges to a rooted graph (G, ) (G, ) is a (possibly) random object

71 Local weak convergence (LWC) LWC for undirected graphs (Benjamini&Schramm 2001) (G n ) n N converges to a rooted graph (G, ) (G, ) is a (possibly) random object One vertex in (G, ) is labeled as a root

72 LWC through finite neighborhoods U k (i), i G n, is a neighborhood of i up to distance k in G n

73 LWC through finite neighborhoods U k (i), i G n, is a neighborhood of i up to distance k in G n (G, ) random rooted graph, is a root

74 LWC through finite neighborhoods U k (i), i G n, is a neighborhood of i up to distance k in G n (G, ) random rooted graph, is a root G n G in distribution in the LWC sense if for any finite rooted graph (H, y), and any k N, 1 n P ( U k (i) = (H, y) ) P ( U k ( ) = (H, y) ) i [n]

75 LWC through finite neighborhoods U k (i), i G n, is a neighborhood of i up to distance k in G n (G, ) random rooted graph, is a root G n G in distribution in the LWC sense if for any finite rooted graph (H, y), and any k N, 1 n P ( U k (i) = (H, y) ) P ( U k ( ) = (H, y) ) i [n] Local properties of the graph can pass to the limit

76 LWC through finite neighborhoods U k (i), i G n, is a neighborhood of i up to distance k in G n (G, ) random rooted graph, is a root G n G in distribution in the LWC sense if for any finite rooted graph (H, y), and any k N, 1 n P ( U k (i) = (H, y) ) P ( U k ( ) = (H, y) ) i [n] Local properties of the graph can pass to the limit V n a uniformly chosen vertex in G n, then lim E [f (G n, V n )] = E [f (G, )] n

77 Rooted directed graphs G = (V (G), E(G)) is a directed graph

78 Rooted directed graphs G = (V (G), E(G)) is a directed graph For every vertex i V (G), di is the in-degree and d i + out-degree is the

79 Rooted directed graphs G = (V (G), E(G)) is a directed graph For every vertex i V (G), di is the in-degree and d i + is the out-degree Exploring the graph using in-bound edges D i + is a mark assigned to i V (G) such that d i + D i + < M(G) = {D + i } i V (G). (G,, M(G)) is a rooted marked directed graph.

80 k-neighborhoods in rooted marked directed graphs

81 k-neighborhoods in rooted marked directed graphs (U k ( ),, M(U k ( ))) k-neighborhood of root

82 k-neighborhoods in rooted marked directed graphs (U k ( ),, M(U k ( ))) k-neighborhood of root Obtained by exploring in-degree only!

83 k-neighborhoods in rooted marked directed graphs (U k ( ),, M(U k ( ))) k-neighborhood of root Obtained by exploring in-degree only! Example of a two-neighborhood

84 Distance for rooted marked directed graphs (G,, M(G)) = (G,, M(G )) isomorphic

85 Distance for rooted marked directed graphs (G,, M(G)) = (G,, M(G )) isomorphic γ : V (G) V (G ) such that (i, j) E(G) if and only if (γ(i), γ(j)) E(G ); γ( ) = ; for every i V (G), D + i = D + γ(i).

86 Distance for rooted marked directed graphs (G,, M(G)) = (G,, M(G )) isomorphic γ : V (G) V (G ) such that (i, j) E(G) if and only if (γ(i), γ(j)) E(G ); γ( ) = ; for every i V (G), D + i = D + γ(i). For directed graphs (G,, M(G)) and (G,, M(G )), where d loc ( (G,, M(G)), (G,, M(G )) ) = κ, κ = inf { k 0 : (U k ( ),, M(U n ( ))) = (U n ( ),, M(U k ( ))) }, and the inf of the empty set is defined as +

87 Distance for rooted marked directed graphs (G,, M(G)) = (G,, M(G )) isomorphic γ : V (G) V (G ) such that (i, j) E(G) if and only if (γ(i), γ(j)) E(G ); γ( ) = ; for every i V (G), D + i = D + γ(i). For directed graphs (G,, M(G)) and (G,, M(G )), where d loc ( (G,, M(G)), (G,, M(G )) ) = κ, κ = inf { k 0 : (U k ( ),, M(U n ( ))) = (U n ( ),, M(U k ( ))) }, and the inf of the empty set is defined as + d loc is not a metric but a pseudometric

88 Distance for rooted marked directed graphs (G,, M(G)) = (G,, M(G )) isomorphic γ : V (G) V (G ) such that (i, j) E(G) if and only if (γ(i), γ(j)) E(G ); γ( ) = ; for every i V (G), D + i = D + γ(i). For directed graphs (G,, M(G)) and (G,, M(G )), where d loc ( (G,, M(G)), (G,, M(G )) ) = κ, κ = inf { k 0 : (U k ( ),, M(U n ( ))) = (U n ( ),, M(U k ( ))) }, and the inf of the empty set is defined as + d loc is not a metric but a pseudometric (G,, M(G)) G is an equivalence class of the relation d loc = 0

89 LWC for marked directed graphs For any graph G define 1 P(G) = V (G) i V (G) δ (G,i,M(G)).

90 LWC for marked directed graphs For any graph G define 1 P(G) = V (G) i V (G) δ (G,i,M(G)). (G n, M(G n )) converges in the LWC sense to a probability P on G if, for any bounded continuous function f : G R, E P(Gn ) [f ] E P [f ]

91 LWC for marked directed graphs For any graph G define 1 P(G) = V (G) i V (G) δ (G,i,M(G)). (G n, M(G n )) converges in the LWC sense to a probability P on G if, for any bounded continuous function f : G R, E P(Gn ) [f ] E P [f ] Theorem. G n P in the LW sense iff k N and finite directed rooted marked graph (H, y, M(H)), 1 1 { U k (i) = n (H, y, M(H)) } i [n] P ( U k ( ) = (H, y, M(H)) ).

92 Main result Theorem (Existence of asymptotic PageRank distribution) Consider a sequence of directed random graphs (G n ) n N. Then: if G n converges in distribution in the local weak sense, then there exists a limiting distribution R, with E[R ] 1, such that d R Vn (n) R ; if G n converges in probability in the local weak sense, then there exists a limiting distribution R, with E[R ] 1, such that, for every r > 0, 1 n 1{R i (n) > r} i [n] P P (R > r).

93 Scheme of the proof R Vn (n) (A) n R (a) finite N (c) N R (N) (n) Vn (b) n R (N)

94 Scheme of the proof R Vn (n) (A) n R (a) finite N (c) N R (N) (n) Vn (b) n R (N) ( ) (a) Matrix iterations: P R Vn (n) R (N) V n (n) ε αn+1 ε

95 Scheme of the proof R Vn (n) (A) n R (a) finite N (c) N R (N) (n) Vn (b) n R (N) ( ) (a) Matrix iterations: P R Vn (n) R (N) V n (n) ε (b) LWC: [ ] ( ) lim E 1{R (N) n V n > r} = lim P R (N) n V n (n) > r αn+1 ε ( ) = P R (N) > r

96 Scheme of the proof R Vn (n) (A) n R (a) finite N (c) N R (N) (n) Vn (b) n R (N) ( ) (a) Matrix iterations: P R Vn (n) R (N) V n (n) ε (b) LWC: [ ] ( ) lim E 1{R (N) n V n > r} = lim P R (N) n V n (n) > r (c) Monotone convergence of (R (N) ) N N αn+1 ε ( ) = P R (N) > r

97 Examples where convergence holds Directed Configuration Model with independent in- and out-degrees Chen, L, Olvera-Cravioto 2017

98 Examples where convergence holds Directed Configuration Model with independent in- and out-degrees Chen, L, Olvera-Cravioto 2017 Directed Configuration Model with dependent in- and out-degrees Garavaglia, vd Hofstad, L 2018

99 Examples where convergence holds Directed Configuration Model with independent in- and out-degrees Chen, L, Olvera-Cravioto 2017 Directed Configuration Model with dependent in- and out-degrees Garavaglia, vd Hofstad, L 2018 Directed Generalized Random Graphs Lee and Olvera-Cravioto 2017

100 Examples where convergence holds Directed Configuration Model with independent in- and out-degrees Chen, L, Olvera-Cravioto 2017 Directed Configuration Model with dependent in- and out-degrees Garavaglia, vd Hofstad, L 2018 Directed Generalized Random Graphs Lee and Olvera-Cravioto 2017 Directed Preferential Attachment Model Garavaglia, vd Hofstad, L 2018

101 Ranking algorithms/centrality measures Boldi and Vigna 2014 Based on distances: (in-)degree: number of nodes on distance 1 Closeness centrality (Bavelas 1950) Harmonic centrality (Boldi and Vigna 2014) Based on paths: Betweenness centrality (Anthonisse 1971) Katz s index (Katz 1953) Based on spectrum: Seeley index (Seeley 1949) HITS (Kleinberg 1997) PageRank (Brin, Page, Motwani and Vinograd 1999)

102 Open Wikipedia ranking

103 Thank you!