Spectra of Random Graphs

Transcription

1 Spectra of Random Graphs Linyuan Lu University of South Carolina Selected Topics on Spectral Graph Theory (III) Nankai University, Tianjin, May 29, 2014

2 Five talks Selected Topics on Spectral Graph Theory 1. Graphs with Small Spectral Radius Time: Friday (May 16) 4pm.-5:30p.m. 2. Laplacian and Random Walks on Graphs Time: Thursday (May 22) 4pm.-5:30p.m. 3. Spectra of Random Graphs Time: Thursday (May 29) 4pm.-5:30p.m. 4. Hypergraphs with Small Spectral Radius Time: Friday (June 6) 4pm.-5:30p.m. 5. Laplacian of Random Hypergraphs Time: Thursday (June 12) 4pm.-5:30p.m. Spectra of Random Graphs Linyuan Lu 2 / 68

3 Backgrounds Linear Algebra III I II Graph Theory Probability Theory I: Spectral Graph Theory II: Random Graph Theory III: Random Matrix Theory Spectra of Random Graphs Linyuan Lu 3 / 68

4 Outline Classical random theory: Erdős-Rényi model Spectra of Random Graphs Linyuan Lu 4 / 68

5 Outline Classical random theory: Erdős-Rényi model Power law graphs Spectra of Random Graphs Linyuan Lu 4 / 68

6 Outline Classical random theory: Erdős-Rényi model Power law graphs Chung-Lu model Spectra of Random Graphs Linyuan Lu 4 / 68

7 Outline Classical random theory: Erdős-Rényi model Power law graphs Chung-Lu model Edge-independent random graphs Spectra of Random Graphs Linyuan Lu 4 / 68

8 Preliminary A graph consists of two sets V and E. - V is the set of vertices (or nodes). - E is the set of edges, where each edge is a pair of vertices. The degree of a vertex is the number of edges, which are incident to that vertex. Diameter: the maximum distance d(u,v), where u and v are in the same connected component. Spectra of Random Graphs Linyuan Lu 5 / 68

9 Preliminary A graph consists of two sets V and E. - V is the set of vertices (or nodes). - E is the set of edges, where each edge is a pair of vertices. The degree of a vertex is the number of edges, which are incident to that vertex. Diameter: the maximum distance d(u,v), where u and v are in the same connected component. Average distance: the average among all distance d(u, v) for pairs of u and v in the same connected component. Spectra of Random Graphs Linyuan Lu 5 / 68

10 Random graphs A random graph is a set of graphs together with a probability distribution on that set. Spectra of Random Graphs Linyuan Lu 6 / 68

11 Random graphs A random graph is a set of graphs together with a probability distribution on that set. Example: A random graph on 3 vertices and 2 edges with the uniform distribution on it. Probability 1 3 Probability 1 3 Probability 1 3 Spectra of Random Graphs Linyuan Lu 6 / 68

12 Random graphs A random graph is a set of graphs together with a probability distribution on that set. Example: A random graph on 3 vertices and 2 edges with the uniform distribution on it. Probability 1 3 Probability 1 3 Probability 1 3 A random graph G almost surely satisfies a property P, if Pr(G satisfies P) = 1 o n (1). Spectra of Random Graphs Linyuan Lu 6 / 68

13 Erdős-Rényi model G(n,p) - n nodes Spectra of Random Graphs Linyuan Lu 7 / 68

14 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. Spectra of Random Graphs Linyuan Lu 7 / 68

15 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. - The graph with e edges has the probability p e (1 p) (n 2) e. Spectra of Random Graphs Linyuan Lu 7 / 68

16 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. - The graph with e edges has the probability p e (1 p) (n 2) e. Spectra of Random Graphs Linyuan Lu 7 / 68

17 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. - The graph with e edges has the probability p e (1 p) (n 2) e. p p p p Spectra of Random Graphs Linyuan Lu 7 / 68

18 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. - The graph with e edges has the probability p e (1 p) (n 2) e. p 1 p p p p 1 p Spectra of Random Graphs Linyuan Lu 7 / 68

19 Erdős-Rényi model G(n,p) - n nodes - For each pair of vertices, create an edge independently with probability p. - The graph with e edges has the probability p e (1 p) (n 2) e. p 1 p p p p 1 p The probability of this graph is p 4 (1 p) 2. Spectra of Random Graphs Linyuan Lu 7 / 68

20 Evolution of G(n,p) Erdős-Rényi 1960s: p c/n for 0 < c < 1: The largest connected component of G n,p is a tree and has about 1 α (logn 5 2 loglogn) vertices, where α = c 1 logc. Spectra of Random Graphs Linyuan Lu 8 / 68

21 Evolution of G(n,p) Erdős-Rényi 1960s: p c/n for 0 < c < 1: The largest connected component of G n,p is a tree and has about 1 α (logn 5 2 loglogn) vertices, where α = c 1 logc. p 1/n+c/n 4/3, the largest connected component is Θ(n 2/3 ). Double jump: Θ(logn) Θ(n 2/3 ) Θ(n). Spectra of Random Graphs Linyuan Lu 8 / 68

22 Evolution of G(n,p) Erdős-Rényi 1960s: p c/n for 0 < c < 1: The largest connected component of G n,p is a tree and has about 1 α (logn 5 2 loglogn) vertices, where α = c 1 logc. p 1/n+c/n 4/3, the largest connected component is Θ(n 2/3 ). Double jump: Θ(logn) Θ(n 2/3 ) Θ(n). p c/n for c > 1: Except for one giant component, all the other components are relatively small. The giant component has approximately f(c)n vertices, where f(c) = 1 1 c k=1 k k 1 k! (ce c ) k. Spectra of Random Graphs Linyuan Lu 8 / 68

23 Diameter of G(n,p) Bollobás (1985): (denser graph) logn diam(g(n, p)) = or lognp logn lognp if np logn. Spectra of Random Graphs Linyuan Lu 9 / 68

24 Diameter of G(n,p) Bollobás (1985): (denser graph) logn diam(g(n, p)) = or lognp logn lognp if np logn. Chung Lu, (2000) (Sparser graph) diam(g(n, p)) = { (1+o(1)) logn lognp Θ( logn lognp if np ) if > np > 1. Spectra of Random Graphs Linyuan Lu 9 / 68

25 Wigner s semicircle law (Wigner, 1958) - A is a real symmetric n n matrix. - Entries a ij are independent random variables. - E(aij 2k+1 ) = 0. - E(a 2 ij ) = m2. - E(a 2k ij ) < M. The distribution of eigenvalues of A converges into a semicircle distribution of radius 2m n. Spectra of Random Graphs Linyuan Lu 10 / 68

26 Spectra of G(n,p) The eigenvalues of an Erdős-Rényi random graph follow the semicircle law. ( Füredi and Komlós, 1981) Laplacian eigenvalues also follow the semicircle law. Spectra of Random Graphs Linyuan Lu 11 / 68

27 Challenge Erdős-Rényi model G(n, p) is classical, simple, beautiful..., but not suitable to model complex graphs. Spectra of Random Graphs Linyuan Lu 12 / 68

28 Challenge Erdős-Rényi model G(n, p) is classical, simple, beautiful..., but not suitable to model complex graphs. What are complex graphs? Spectra of Random Graphs Linyuan Lu 12 / 68

29 Challenge Erdős-Rényi model G(n, p) is classical, simple, beautiful..., but not suitable to model complex graphs. What are complex graphs? How to model these complex graphs by random graphs? Spectra of Random Graphs Linyuan Lu 12 / 68

30 Challenge Erdős-Rényi model G(n, p) is classical, simple, beautiful..., but not suitable to model complex graphs. What are complex graphs? How to model these complex graphs by random graphs? How to deduce the graph properties of these general random graph models? Spectra of Random Graphs Linyuan Lu 12 / 68

31 Examples of complex graphs WWW Graphs Call Graphs Collaboration Graphs Gene Regulatory Graphs Graph of U.S. Power Grid Costars Graph of Actors. Spectra of Random Graphs Linyuan Lu 13 / 68

32 A subgraph of the Collaboration Graph Spectra of Random Graphs Linyuan Lu 14 / 68

33 Collaboration Graph at USC Spouge P.L. Erdős Wormald Johnson de Caen Biró Steel Dankelmann Füredi Griggs Czabarka Székely Lu Bokal Toroczkai Miklós Katona Sali Chudak Howard Karolyi Wagner WangShahrokhi Entriger Smith Jin Ho Anstee Kleitman Yang Mohr Yeh Liu Dove Ouyang Wu Li Liang Lan Man Peng Lin Zhu Chen Martin Zhao Sun Jonas Sandberg Jordan Odlyzko Milan West Johnston Chin Horn Chung Graham Pralat Narayan Boehnlein Walters Kay Tetali Fenner Cooper Dutle Faculty, Ph.D. students, Postdocs, and visitors to the Combinatorics Group at the University of South Carolina. Spectra of Random Graphs Linyuan Lu 15 / 68

34 An IP Graph (by Bill Cheswick) Spectra of Random Graphs Linyuan Lu 16 / 68

35 BGP Graph Vertex: AS (autonomous system) Edges: AS pairs in BGP routing table. Spectra of Random Graphs Linyuan Lu 17 / 68

36 Large BGP subgraph Only a portion of 6400 vertices and edges is drawn. Spectra of Random Graphs Linyuan Lu 18 / 68

37 Hollywood Graph Vertex: actors and actress Edges: co-playing in the same movie Only 10,000 out of 225,000 are shown. Spectra of Random Graphs Linyuan Lu 19 / 68

38 Protein-interaction network Snel, Bork & Huynen, PNAS 99, 5890 (2002) Spectra of Random Graphs Linyuan Lu 20 / 68

39 A subgraph of the Collaboration Graph Spectra of Random Graphs Linyuan Lu 21 / 68

40 Folklore of Erdős numbers Erdős has Erdős number 0. Erdős coauthor has Erdős number 1. Erdős coauthor s coauthor has Erdős number 2.. Spectra of Random Graphs Linyuan Lu 22 / 68

41 Folklore of Erdős numbers Erdős has Erdős number 0. Erdős coauthor has Erdős number 1. Erdős coauthor s coauthor has Erdős number 2.. My Erdős number is 2. Spectra of Random Graphs Linyuan Lu 22 / 68

42 Folklore of Erdős numbers Erdős has Erdős number 0. Erdős coauthor has Erdős number 1. Erdős coauthor s coauthor has Erdős number 2.. My Erdős number is 2. Erdős number is the graph distance to Erdős in the Collaboration graph. Spectra of Random Graphs Linyuan Lu 22 / 68

43 Collaboration Graph Spectra of Random Graphs Linyuan Lu 23 / 68

44 Characteristics Large Spectra of Random Graphs Linyuan Lu 24 / 68

45 Characteristics Large Sparse Spectra of Random Graphs Linyuan Lu 24 / 68

46 Characteristics Large Sparse Power law degree distribution Spectra of Random Graphs Linyuan Lu 24 / 68

47 Characteristics Large Sparse Power law degree distribution Small world phenomenon Spectra of Random Graphs Linyuan Lu 24 / 68

48 The power law The number of vertices of degree k is approximately proportional to k β for some positive β. Spectra of Random Graphs Linyuan Lu 25 / 68

49 The power law The number of vertices of degree k is approximately proportional to k β for some positive β. A power law graph is a graph whose degree sequence satisfies the power law. Spectra of Random Graphs Linyuan Lu 25 / 68

50 Power law distribution Left: The collaboration graph follows the power law degree distribution with exponent β 3.0 Spectra of Random Graphs Linyuan Lu 26 / 68

51 Power law distribution Left: The collaboration graph follows the power law degree distribution with exponent β 3.0 Right: An IP graph follows the power law degree distribution with exponent β 2.4 Spectra of Random Graphs Linyuan Lu 26 / 68

52 Power law graphs Left: Part of the collaboration graph (authors with Erdős number 2) Right: An IP graph (by Bill Cheswick) Spectra of Random Graphs Linyuan Lu 27 / 68

53 Robustness of Power Law size degree distribution 25, ,186 Spectra of Random Graphs Linyuan Lu 28 / 68

54 Basic questions How to model power law graphs? Spectra of Random Graphs Linyuan Lu 29 / 68

55 Basic questions How to model power law graphs? What graph properties can be derived from the model? Spectra of Random Graphs Linyuan Lu 29 / 68

56 Model G(w 1,w 2,...,w n ) Random graph model with given expected degree sequence (Chung-Lu model) - n nodes with weights w 1,w 2,...,w n. Spectra of Random Graphs Linyuan Lu 30 / 68

57 Model G(w 1,w 2,...,w n ) Random graph model with given expected degree sequence (Chung-Lu model) - n nodes with weights w 1,w 2,...,w n. - For each pair (i,j), create an edge independently with probability p ij = w i w j ρ, where ρ = 1 n i=1 w i. Spectra of Random Graphs Linyuan Lu 30 / 68

58 Model G(w 1,w 2,...,w n ) Random graph model with given expected degree sequence (Chung-Lu model) - n nodes with weights w 1,w 2,...,w n. - For each pair (i,j), create an edge independently with probability p ij = w i w j ρ, where ρ = 1 n i=1 w i. - The graph H has probability p ij ij E(H) ij E(H) (1 p ij ). Spectra of Random Graphs Linyuan Lu 30 / 68

59 Model G(w 1,w 2,...,w n ) Random graph model with given expected degree sequence (Chung-Lu model) - n nodes with weights w 1,w 2,...,w n. - For each pair (i,j), create an edge independently with probability p ij = w i w j ρ, where ρ = 1 n i=1 w i. - The graph H has probability p ij ij E(H) ij E(H) (1 p ij ). - The expected degree of vertex i is w i. Spectra of Random Graphs Linyuan Lu 30 / 68

60 An example: G(w 1,w 2,w 3,w 4 ) Spectra of Random Graphs Linyuan Lu 31 / 68

61 An example: G(w 1,w 2,w 3,w 4 ) w 3 w 4 w 1 w 2 Spectra of Random Graphs Linyuan Lu 31 / 68

62 An example: G(w 1,w 2,w 3,w 4 ) w 1 w 3 ρ w 3 w 4 w 2 w 3 ρ w 1 w 4 ρ w 1 w 2 w 1 w 2 ρ Spectra of Random Graphs Linyuan Lu 31 / 68

63 An example: G(w 1,w 2,w 3,w 4 ) w 1 w 3 ρ 1 w 3 w 4 ρ w 3 w 4 w 2 w 3 ρ w 1 w 4 ρ w 1 w 2 w 1 w 2 ρ 1 w 2 w 4 ρ Spectra of Random Graphs Linyuan Lu 31 / 68

64 An example: G(w 1,w 2,w 3,w 4 ) 1 w 3 w 4 ρ w 3 w 4 w 2 w 3 ρ w 1 w 4 ρ w 1 w 2 w 1 w 2 ρ w 1 w 3 ρ 1 w3 2ρ 1 w4ρ 2 1 w 2 1ρ 1 w 2 w 4 ρ 1 w 2 2ρ Spectra of Random Graphs Linyuan Lu 31 / 68

65 An example: G(w 1,w 2,w 3,w 4 ) 1 w 3 w 4 ρ w 3 w 4 w 2 w 3 ρ w 1 w 4 ρ w 1 w 2 w 1 w 2 ρ w 1 w 3 ρ 1 w 2 1ρ The probability of the graph is 1 w3 2ρ 1 w4ρ 2 1 w 2 w 4 ρ w 3 1w 2 2w 2 3w 4 ρ 4 (1 w 2 w 4 ρ) (1 w 3 w 4 ρ) 1 w 2 2ρ 4 (1 wiρ). 2 Spectra of Random Graphs Linyuan Lu 31 / 68 i=1

66 Chung-Lu model For G = G(w 1,...,w n ), let - d = 1 n n i=1 w i n - d = i=1 w2 i n i=1 w. i - The volume of S: Vol(S) = i S w i. Spectra of Random Graphs Linyuan Lu 32 / 68

67 Chung-Lu model For G = G(w 1,...,w n ), let - d = 1 n n i=1 w i n - d = i=1 w2 i n i=1 w. i - The volume of S: Vol(S) = i S w i. We have d d = holds if and only if w 1 = = w n. Spectra of Random Graphs Linyuan Lu 32 / 68

68 Chung-Lu model For G = G(w 1,...,w n ), let - d = 1 n n i=1 w i n - d = i=1 w2 i n i=1 w. i - The volume of S: Vol(S) = i S w i. We have d d = holds if and only if w 1 = = w n. A connected component S is called a giant component if vol(s) = Θ(vol(G)). Spectra of Random Graphs Linyuan Lu 32 / 68

69 Connected components Chung and Lu (2001) For G = G(w 1,...,w n ), If d < 1 ǫ, then almost surely, all components have volume at most O( nlogn). Spectra of Random Graphs Linyuan Lu 33 / 68

70 Connected components Chung and Lu (2001) For G = G(w 1,...,w n ), If d < 1 ǫ, then almost surely, all components have volume at most O( nlogn). If d > 1+ǫ, then almost surely there is a unique giant component of volume Θ(Vol(G)). All other components have size at most { logn 1 d 1 log d ǫd if 1 ǫ < d < 2 logn 1+logd log4+2log(1 ǫ) if d > 4 e(1 ǫ). 2 1 ǫ Spectra of Random Graphs Linyuan Lu 33 / 68

71 Volume of Giant Component Chung and Lu (2004) If the average degree is strictly greater than 1, then almost surely the giant component in a graph G in G(w) has volume (λ 0 +O ( n log 3.5 n Vol(G) )) Vol(G), where λ 0 is the unique positive root of the following equation: n i=1 w i e w iλ = (1 λ) n w i. i=1 Spectra of Random Graphs Linyuan Lu 34 / 68

72 A real application Apply to the Collaboration Graph (2002 data): The size of giant component is predicted to be about 177,400 by our theory. This is rather close to the actual value 176,000, within an error bound of less than 1%. Spectra of Random Graphs Linyuan Lu 35 / 68

73 G(n,p) versus G(w 1,...,w n ) Question: Does the random graph with equal expected degrees generates the smallest giant component among all possible degree distribution with the same volume? Spectra of Random Graphs Linyuan Lu 36 / 68

74 G(n,p) versus G(w 1,...,w n ) Question: Does the random graph with equal expected degrees generates the smallest giant component among all possible degree distribution with the same volume? Chung Lu (2004) Yes, for 1 < d e e 1. Spectra of Random Graphs Linyuan Lu 36 / 68

75 G(n,p) versus G(w 1,...,w n ) Question: Does the random graph with equal expected degrees generates the smallest giant component among all possible degree distribution with the same volume? Chung Lu (2004) Yes, for 1 < d e e 1. No, for sufficiently large d. Spectra of Random Graphs Linyuan Lu 36 / 68

76 G(n,p) versus G(w 1,...,w n ) Question: Does the random graph with equal expected degrees generates the smallest giant component among all possible degree distribution with the same volume? Chung Lu (2004) Yes, for 1 < d e e 1. No, for sufficiently large d. When d 4 e, almost surely the giant component of G(w 1,...,w n ) has volume at least ( 1 ( ) ) +o(1) Vol(G). 2 de This is asymptotically best possible. Spectra of Random Graphs Linyuan Lu 36 / 68

77 Diameter of G(w 1,...,w n ) Chung Lu (2002) For a random graph G with admissible expected degree sequence (w 1,...,w n ), the average distance is almost surely (1+o(1)) logn log d. Spectra of Random Graphs Linyuan Lu 37 / 68

78 Diameter of G(w 1,...,w n ) Chung Lu (2002) For a random graph G with admissible expected degree sequence (w 1,...,w n ), the average distance is almost surely (1+o(1)) logn log d. For a random graph G with strongly admissible expected degree sequence (w 1,...,w n ), the diameter is almost surely Θ( logn ). log d Spectra of Random Graphs Linyuan Lu 37 / 68

79 Diameter of G(w 1,...,w n ) Chung Lu (2002) For a random graph G with admissible expected degree sequence (w 1,...,w n ), the average distance is almost surely (1+o(1)) logn log d. For a random graph G with strongly admissible expected degree sequence (w 1,...,w n ), the diameter is almost surely Θ( logn ). log d These results apply to G(n,p) and random power law graph with β > 3. Spectra of Random Graphs Linyuan Lu 37 / 68

80 Non-admissible graph versus admissible graph A random subgraph of the Collaboration Graph. A Connected component of G(n, p) with n = 500 and p = Spectra of Random Graphs Linyuan Lu 38 / 68

81 Non-admissible graph versus admissible graph A random subgraph of the Collaboration Graph. - Dense core for non-admissible graphs. - No dense core for admissible graphs. A Connected component of G(n, p) with n = 500 and p = Spectra of Random Graphs Linyuan Lu 38 / 68

82 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. Spectra of Random Graphs Linyuan Lu 39 / 68

83 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. - Non-admissible. Spectra of Random Graphs Linyuan Lu 39 / 68

84 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. - Non-admissible. - Containing a dense core, with diameter loglogn. Spectra of Random Graphs Linyuan Lu 39 / 68

85 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. - Non-admissible. - Containing a dense core, with diameter loglogn. - Mostly vertices are within the distance of O(loglogn) from the core. Spectra of Random Graphs Linyuan Lu 39 / 68

86 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. - Non-admissible. - Containing a dense core, with diameter loglogn. - Mostly vertices are within the distance of O(loglogn) from the core. - There are some vertices at the distance of O(logn). Spectra of Random Graphs Linyuan Lu 39 / 68

87 Power law graphs with β (2,3) Chung, Lu (2002) - Examples: the WWW graph, Collaboration graph, etc. - Non-admissible. - Containing a dense core, with diameter loglogn. - Mostly vertices are within the distance of O(loglogn) from the core. - There are some vertices at the distance of O(logn). The diameter is Θ(log n), while the average distance is O(log log n). Spectra of Random Graphs Linyuan Lu 39 / 68

88 Experimental results Faloutsos et al. (1999) The eigenvalues of the Internet graph do not follow the semicircle law. Farkas et. al. (2001), Goh et. al. (2001) The spectrum of a power law graph follows a triangular-like distribution. Mihail and Papadimitriou (2002) They showed that the large eigenvalues are determined by the large degrees. Thus, the significant part of the spectrum of a power law graph follows the power law. µ i d i. Spectra of Random Graphs Linyuan Lu 40 / 68

89 Eigenvalues of G(w 1,...,w n ) Chung, Vu, and Lu (2003) Suppose w 1 w 2... w n. Let µ i be i-th largest eigenvalue of G(w 1,w 2,...,w n ). Let m = w 1 and d = n i=1 w2 iρ. Almost surely we have: (1 o(1))max{ m, d} µ 1 7 logn max{ m, d}. Spectra of Random Graphs Linyuan Lu 41 / 68

90 Eigenvalues of G(w 1,...,w n ) Chung, Vu, and Lu (2003) Suppose w 1 w 2... w n. Let µ i be i-th largest eigenvalue of G(w 1,w 2,...,w n ). Let m = w 1 and d = n i=1 w2 iρ. Almost surely we have: (1 o(1))max{ m, d} µ 1 7 logn max{ m, d}. µ 1 = (1+o(1)) d, if d > mlogn. Spectra of Random Graphs Linyuan Lu 41 / 68

91 Eigenvalues of G(w 1,...,w n ) Chung, Vu, and Lu (2003) Suppose w 1 w 2... w n. Let µ i be i-th largest eigenvalue of G(w 1,w 2,...,w n ). Let m = w 1 and d = n i=1 w2 iρ. Almost surely we have: (1 o(1))max{ m, d} µ 1 7 logn max{ m, d}. µ 1 = (1+o(1)) d, if d > mlogn. µ 1 = (1+o(1)) m, if m > dlog 2 n. Spectra of Random Graphs Linyuan Lu 41 / 68

92 Eigenvalues of G(w 1,...,w n ) Chung, Vu, and Lu (2003) Suppose w 1 w 2... w n. Let µ i be i-th largest eigenvalue of G(w 1,w 2,...,w n ). Let m = w 1 and d = n i=1 w2 iρ. Almost surely we have: (1 o(1))max{ m, d} µ 1 7 logn max{ m, d}. µ 1 = (1+o(1)) d, if d > mlogn. µ 1 = (1+o(1)) m, if m > dlog 2 n. µ k w k and µ n+1 k w k, if w k > dlog 2 n. Spectra of Random Graphs Linyuan Lu 41 / 68

93 Random power law graphs The first k and last k eigenvalues of the random power law graph with β > 2.5 follows the power law distribution with exponent 2β 1. It results a triangular-like shape. Spectra of Random Graphs Linyuan Lu 42 / 68

94 Laplacian spectrum Random walks on a graph G: π k+1 = AD 1 π k. AD 1 D 1/2 AD 1/2. 1 d v v 1 d v 1 d v Spectra of Random Graphs Linyuan Lu 43 / 68

95 Laplacian spectrum Random walks on a graph G: π k+1 = AD 1 π k. AD 1 D 1/2 AD 1/2. Laplacian spectrum 1 d v v 1 d v 1 d v 0 = λ 0 λ 1 λ n 1 2 are the eigenvalues of L = I D 1/2 AD 1/2. Spectra of Random Graphs Linyuan Lu 43 / 68

96 Laplacian spectrum Random walks on a graph G: π k+1 = AD 1 π k. AD 1 D 1/2 AD 1/2. Laplacian spectrum 1 d v v 1 d v 1 d v 0 = λ 0 λ 1 λ n 1 2 are the eigenvalues of L = I D 1/2 AD 1/2. The eigenvalues of AD 1 are 1,1 λ 1,...,1 λ n 1. Spectra of Random Graphs Linyuan Lu 43 / 68

97 Let Laplacian Spectral Radius - w min = min{w 1,...,w n }, - d = 1 n n i=1 w i, - g(n) a function tending to infinity arbitrarily slowly. Chung, Vu, and Lu (2003) If w min log 2 n, then almost surely the Laplacian spectrum λ i s of G(w 1,...,w n ) satisfy max 1 λ i (1+o(1)) 4 + g(n)log2 n. i 0 d w min Spectra of Random Graphs Linyuan Lu 44 / 68

98 Let Laplacian Spectral Radius - w min = min{w 1,...,w n }, - d = 1 n n i=1 w i, - g(n) a function tending to infinity arbitrarily slowly. Chung, Vu, and Lu (2003) If w min log 2 n, then almost surely the Laplacian spectrum λ i s of G(w 1,...,w n ) satisfy max 1 λ i (1+o(1)) 4 + g(n)log2 n. i 0 d w min If w min d, the Laplacian spectrum follows the semi-circle distribution with radius r 2 d. Spectra of Random Graphs Linyuan Lu 44 / 68

99 General random graphs General edge-independent random graphs: n: the number of vertices. p ij : a probability for ij being an edge. Edges are mutually independent. Question: What can we say about the spectrum of the adjacency matrix and the Laplacian matrix? Spectra of Random Graphs Linyuan Lu 45 / 68

100 Notation - A: adjacency matrix - Ā := (p ij ): the expectation of A - : the maximum expected degree - δ: the minimum expected degree - D: the diagonal matrix of degrees - D: the expectation of D - L := I D 1/2 AD 1/2 : the normalized Laplacian - L := I D 1/2 Ā D 1/2 : the Laplacian of Ā Spectra of Random Graphs Linyuan Lu 46 / 68

101 Known results Oliveira [2010]: For Clnn, with high probability we have λ i (A) λ i (Ā) 4 lnn. For δ Clnn, with high probability we have λ i (L) λ i ( L) 14 ln(4n)/δ. Spectra of Random Graphs Linyuan Lu 47 / 68

102 Known results Oliveira [2010]: For Clnn, with high probability we have λ i (A) λ i (Ā) 4 lnn. For δ Clnn, with high probability we have λ i (L) λ i ( L) 14 ln(4n)/δ. Chung-Radcliffe [2011] reduces the constant coefficient using a new matrix Chernoff inequality. Spectra of Random Graphs Linyuan Lu 47 / 68

103 Our results Lu-Peng [2012+]: If ln 4 n, then almost surely λ i (A) λ i (Ā) (2+o(1)). Spectra of Random Graphs Linyuan Lu 48 / 68

104 Our results Lu-Peng [2012+]: If ln 4 n, then almost surely λ i (A) λ i (Ā) (2+o(1)). Lu-Peng [2012+]: Let Λ := {λ i ( L): 1 λ i ( L) = ω(1/ lnn)}. If δ max{ Λ,ln 4 n}, then almost surely λ i (L) λ i ( L) 2+ (1 λ) 2 +o(1) 1. δ λ Λ Spectra of Random Graphs Linyuan Lu 48 / 68

105 Our results Lu-Peng [2012+]: If ln 4 n, then almost surely λ i (A) λ i (Ā) (2+o(1)). Lu-Peng [2012+]: Let Λ := {λ i ( L): 1 λ i ( L) = ω(1/ lnn)}. If δ max{ Λ,ln 4 n}, then almost surely λ i (L) λ i ( L) 2+ (1 λ) 2 +o(1) 1. δ λ Λ In both case, we remove the multiplicative factor lnn. Spectra of Random Graphs Linyuan Lu 48 / 68

106 Random symmetric matrices B = (b ij ) is a random symmetric matrix satisfying: - b ij : independent, but not necessary identical, - b ij K, - E(b ij ) = 0, - Var(b ij ) σ 2. Spectra of Random Graphs Linyuan Lu 49 / 68

107 Random symmetric matrices B = (b ij ) is a random symmetric matrix satisfying: - b ij : independent, but not necessary identical, - b ij K, - E(b ij ) = 0, - Var(b ij ) σ 2. Füredi-Komlós [1981]: B 2σ n+cn 1/3 lnn. Spectra of Random Graphs Linyuan Lu 49 / 68

108 Random symmetric matrices B = (b ij ) is a random symmetric matrix satisfying: - b ij : independent, but not necessary identical, - b ij K, - E(b ij ) = 0, - Var(b ij ) σ 2. Füredi-Komlós [1981]: Vu [2007]: B 2σ n+cn 1/3 lnn. B 2σ n+c(kσ) 1/2 n 1/4 lnn. Spectra of Random Graphs Linyuan Lu 49 / 68

109 Our result Lu-Peng [2012+]: We further assume Var(b ij ) σ 2 ij. Let := max 1 i n n j=1 σ2 ij. If C K 2 ln 4 n, then asymptotically almost surely B 2 +C K 1/4 lnn. It generalizes Vu s theorem. This result is asymptotically tight. Spectra of Random Graphs Linyuan Lu 50 / 68

110 Graph percolation G: a connected graph on n vertices p: a probability (0 p 1) G p : a random spanning subgraph of G, obtained as follows: for each edge f of G, independently, Pr(f is an edge of G p ) = p. Spectra of Random Graphs Linyuan Lu 51 / 68

111 Graph percolation G: a connected graph on n vertices p: a probability (0 p 1) G p : a random spanning subgraph of G, obtained as follows: for each edge f of G, independently, Pr(f is an edge of G p ) = p. Spectra of Random Graphs Linyuan Lu 51 / 68

112 Spectrum of G p Lu-Peng [2012+]: If p ln4 n, then almost surely we have λ i (A(G p )) pλ i (A(G)) (2+o(1)) p. Spectra of Random Graphs Linyuan Lu 52 / 68

113 Spectrum of G p Lu-Peng [2012+]: If p ln4 n, then almost surely we have λ i (A(G p )) pλ i (A(G)) (2+o(1)) p. Suppose that all but k Laplacian eigenvalues λ of G satisfies 1 λ = o( 1 lnn ). If δ max{k,ln 4 n}, then for p max{ k δ, ln4 n δ }, almost surely we have λ i (L(G p )) λ i (L(G)) (2+ k i=1 (1 λ i) 2 +o(1))) 1 pδ. Spectra of Random Graphs Linyuan Lu 52 / 68

114 Method We will illustrate Wigner s trace method through the sketch proof of the following result. Spectra of Random Graphs Linyuan Lu 53 / 68

115 Method We will illustrate Wigner s trace method through the sketch proof of the following result. Lu-Peng [2012+]: If B = (b ij ) is a random symmetric matrix satisfying: - b ij : independent, but not necessary identical, - b ij K, - E(b ij ) = 0, - Var(b ij ) σ 2 ij. then almost surely B 2 +C K 1/4 lnn, where := max 1 i n n j=1 σ2 ij. Spectra of Random Graphs Linyuan Lu 53 / 68

116 Sketch proof WLOG, we can assume K = 1 and b ii = 0. Using Wigner s trace method, we have E ( Trace(B k ) ) = = i 1,i 2,...,i k E(b i1 i 2 b i2 i 3...b ik 1 i k b ik i 1 ) k/2 +1 p=2 w G(n,k,p) e E(w) E(b q e e ). Here G(n,k,p) is the set of good closed walks w in K n of length k on p vertices, where each edge in w appears more than once (q e 2). Spectra of Random Graphs Linyuan Lu 54 / 68

117 Continue Let G(k,p) be the set of good closed walks w of length k on the complete graph K p where vertices first appear in w in the order 1,2,...,p. Spectra of Random Graphs Linyuan Lu 55 / 68

118 Continue Let G(k,p) be the set of good closed walks w of length k on the complete graph K p where vertices first appear in w in the order 1,2,...,p. All walks in G(n,k,p) can be coded by a walk in G(k,p) plus the ordered p distinct vertices. Let [n] p := {(v 1,v 2,...,v p ) [n] p : v 1,v 2,...,v p are distinct}. Spectra of Random Graphs Linyuan Lu 55 / 68

119 Continue Let G(k,p) be the set of good closed walks w of length k on the complete graph K p where vertices first appear in w in the order 1,2,...,p. All walks in G(n,k,p) can be coded by a walk in G(k,p) plus the ordered p distinct vertices. Let [n] p := {(v 1,v 2,...,v p ) [n] p : v 1,v 2,...,v p are distinct}. Define a rooted tree T(w) so that the edge i j i j+1 E(T(w)) if it brings in a new vertex i j+1 when it occurs first time. Spectra of Random Graphs Linyuan Lu 55 / 68

120 w G(n,k,p) e E(w) σ 2 e = w G(k,p) = w G(k,p) continue w G(k,p) n v 1 =1 n v 1 =1 w G(k,p) n v 2 =1 n v 2 =1 n v 1 =1 (v 1,...,v p ) [n] p n v 2 =1 n p 1 G(k,p). n v p =1 n v p 1 =1 xy E(T) p 1 n y=2 v p 1 =1 xy E( w) σ 2 v x v y σ 2 v η(y) v y p 1 y=2 σ 2 v η(y) v y σ 2 v x v y n v p =1 σ 2 v η(p) v p Spectra of Random Graphs Linyuan Lu 56 / 68

121 Continue Vu [2007] proved ( ) k G(k,p) 2 2k 2p+3 p k 2p+2 (k 2p+4) k 2p+2. 2p 2 Spectra of Random Graphs Linyuan Lu 57 / 68

122 Continue Vu [2007] proved ( ) k G(k,p) 2 2k 2p+3 p k 2p+2 (k 2p+4) k 2p+2. 2p 2 We get ( E Trace(B k ) ) n := n k/2+1 p=2 k/2+1 p=2 w G(n,k) e E(w) σ 2 e k/2+1 Spectra of Random Graphs Linyuan Lu 57 / 68 p=2 n p 1 G(k,p) ( ) k p 1 2 2k 2p+3 p k 2p+2 (k 2p+4) k 2p+2 2p 2 S(n,k,p).

123 Continue One can show S(n,k,p 1) 16k4 S(n,k,p). Spectra of Random Graphs Linyuan Lu 58 / 68

124 Continue One can show S(n,k,p 1) 16k4 S(n,k,p). For any even integer k such that k 4 32, we get ( E Trace(B k ) ) k/2+1 p=2 S(n,k,p) S(n,k,k/2+1) < 2S(n,k,k/2+1) = n2 k+2 k/2. k/2+1 p=2 ( ) k/2+1 p 1 2 Spectra of Random Graphs Linyuan Lu 58 / 68

125 For even k, we have continue Pr( B 2 +C 1/4 lnn) = Pr( B k (2 +C 1/4 lnn) k ) Pr(Trace(B k ) (2 +C 1/4 lnn) k ) E(Trace(B k )) (2 (Markov s inequality) +C 1/4 lnn)) k n2 k+2 k/2 (2 +C 1/4 lnn)) k = 4ne (1+o(1))C 2 k 1/4lnn. Setting k = ( 1/4, 32) this probability is o(1) for sufficiently large C. Spectra of Random Graphs Linyuan Lu 59 / 68

126 For even k, we have continue Pr( B 2 +C 1/4 lnn) = Pr( B k (2 +C 1/4 lnn) k ) Pr(Trace(B k ) (2 +C 1/4 lnn) k ) E(Trace(B k )) (2 (Markov s inequality) +C 1/4 lnn)) k n2 k+2 k/2 (2 +C 1/4 lnn)) k = 4ne (1+o(1))C 2 k 1/4lnn. Setting k = ( 1/4, 32) this probability is o(1) for sufficiently large C. Spectra of Random Graphs Linyuan Lu 59 / 68

127 Percolation threshold p c For p < p c, almost surely there is no giant component For p > p c, almost surely there is a giant component. p c Spectra of Random Graphs Linyuan Lu 60 / 68

128 Motivations Graph theory: random graphs Theoretical physics: crystals melting Sociology: the spread of disease on contact networks Spectra of Random Graphs Linyuan Lu 61 / 68

129 Percolation of Z d Spectra of Random Graphs Linyuan Lu 62 / 68

130 Percolation of Z d Spectra of Random Graphs Linyuan Lu 63 / 68

131 Percolation of Z d Kesten (1980): p c (Z 2 ) = 1 2. Spectra of Random Graphs Linyuan Lu 63 / 68

132 Percolation of Z d Kesten (1980): p c (Z 2 ) = 1 2. Lorenz and Ziff (1997, simulation): p c (Z 3 ) ± if it exists. Spectra of Random Graphs Linyuan Lu 63 / 68

133 Percolation of Z d Kesten (1980): p c (Z 2 ) = 1 2. Lorenz and Ziff (1997, simulation): p c (Z 3 ) ± if it exists. Kesten (1990): p c (Z d ) 1 2d as d. Spectra of Random Graphs Linyuan Lu 63 / 68

134 d-regular graphs Alon, Benjamini, Stacey (2004): Suppose d 2 and let (G n ) be a sequence of d-regular expanders with girth(g n ), then p c = 1 d 1 +o(1). Spectra of Random Graphs Linyuan Lu 64 / 68

135 Percolation of dense graphs Bollobás, Borgs, Chayes, and Riordan (2008): Suppose that G is a dense graph (i.e., average degree d = Θ(n)). Let µ be the largest eigenvalue of the adjacency matrix of G. Then p c 1 µ. Spectra of Random Graphs Linyuan Lu 65 / 68

136 Percolation of dense graphs Bollobás, Borgs, Chayes, and Riordan (2008): Suppose that G is a dense graph (i.e., average degree d = Θ(n)). Let µ be the largest eigenvalue of the adjacency matrix of G. Then p c 1 µ. Remark: The requirement of dense graph is essential. Their methods can not be extended to sparse graphs. Spectra of Random Graphs Linyuan Lu 65 / 68

137 Percolation of sparse graphs Chung, Lu, Horn [2008]: If p < 1 µ, then G p has no giant component. Spectra of Random Graphs Linyuan Lu 66 / 68

138 Percolation of sparse graphs Chung, Lu, Horn [2008]: If p < 1 µ, then G p has no giant component. The condition p > 1 µ in general does not imply that G p has a giant component. Spectra of Random Graphs Linyuan Lu 66 / 68

139 Percolation of sparse graphs Chung, Lu, Horn [2008]: If p < 1 µ, then G p has no giant component. The condition p > 1 µ in general does not imply that G p has a giant component. If p > 1 1 µ, = O(d), and σ = o( logn ), then G p has a giant component. Spectra of Random Graphs Linyuan Lu 66 / 68

140 Percolation of G(w) Bhamidi-van der Hofstad-van Leeuwaarden [2012]: Consider G(w), where w = (w 1,...,w n ) follows the power law of exponent β. If E( n i=1 w2 i ) converges and is bounded, then the percolation threshed is (1+o(1)) 1 d. For β > 4, E( n i=1 w3 i ) converges. The largest component has the size Θ(n 2/3 ) at the critical window. Spectra of Random Graphs Linyuan Lu 67 / 68

141 Percolation of G(w) Bhamidi-van der Hofstad-van Leeuwaarden [2012]: Consider G(w), where w = (w 1,...,w n ) follows the power law of exponent β. If E( n i=1 w2 i ) converges and is bounded, then the percolation threshed is (1+o(1)) 1 d. For β > 4, E( n i=1 w3 i ) converges. The largest component has the size Θ(n 2/3 ) at the critical window. For 2 < β < 3, E( n i=1 w3 i ) diverges. The largest component has the size Θ(n β 2 β 1 ) at the critical window. Spectra of Random Graphs Linyuan Lu 67 / 68

142 References 1. Fan Chung, Linyuan Lu, and Van Vu, Eigenvalues of random power law graphs, Annals of Combinatorics, 7 (2003), Fan Chung, Linyuan Lu and Van Vu, The spectra of random graphs with given expected degrees, Proceedings of National Academy of Sciences, 100, No. 11, (2003), Fan Chung, Linyuan Lu, and Van Vu, Eigenvalues of random power law graphs, Internet Mathematics, 1 No. 3, (2004), Fan Chung and Linyuan Lu, The volume of the giant component for a random graph with given expected degrees, SIAM J. Discrete Math., 20 (2006), No. 2, Linyuan Lu and Xing Peng, Spectra of edge-independent random graphs, Electronic Journal of Combinatorics, 20 (4), (2013) P27. Homepage: lu/ Thank You Spectra of Random Graphs Linyuan Lu 68 / 68