. p.1. Mathematical Models of the WWW and related networks. Alan Frieze

Transcription

1 . p.1 Mathematical Models of the WWW and related networks Alan Frieze

2 The WWW is an example of a large real-world network.. p.2

3 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design). p.2

4 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples:

5 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples: Internet

6 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples: Internet Metabolic Networks

7 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples: Internet Metabolic Networks Social networks

8 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples: Internet Metabolic Networks Social networks Neural Networks

9 . p.2 The WWW is an example of a large real-world network. It grows unpredictably (at least not by formal design) Other examples: Internet Metabolic Networks Social networks Neural Networks Peer to Peer Networks

10 We assume that they arise via some random process.. p.3

11 . p.3 We assume that they arise via some random process. We model them as random graphs

12 . p.3 We assume that they arise via some random process. We model them as random graphs Classical Erdős and Rényi Model: G n,m

13 . p.3 We assume that they arise via some random process. We model them as random graphs Classical Erdős and Rényi Model: G n,m Vertex set {1, 2,..., n}, N = ( n) 2 Each of the ( N m) graphs with m edges is equally likely.

14 Problem. p.4

15 . p.4 Problem Suppose that m = cn, c constant. For small k, the number n k of vertices of degree k satisfies n k ck e c k! n whp

16 . p.4 Problem Suppose that m = cn, c constant. For small k, the number n k of vertices of degree k satisfies n k ck e c k! n whp In many real world cases n k A k α n power law for some constants A, α e.g. Faloutsos,Faloutsos,Faloutsos

17 Fixed (expected) degree sequence models.. p.5

18 . p.5 Fixed (expected) degree sequence models. Choose a degree sequence d 1, d 2,..., d n and then choose a graph G uniformly from graphs with this degree sequence.

19 . p.5 Fixed (expected) degree sequence models. Choose a degree sequence d 1, d 2,..., d n and then choose a graph G uniformly from graphs with this degree sequence. Bender and Canfield McKay,Wormald Bollobás Molloy and Reed

20 . p.5 Fixed (expected) degree sequence models. Choose a degree sequence d 1, d 2,..., d n and then choose a graph G uniformly from graphs with this degree sequence. Bender and Canfield McKay,Wormald Bollobás Molloy and Reed Let Θ = i d i(d i 2). Θ < 0 implies all components of G are small whp Θ > 0 implies G contains a giant component (size Ω(n)) whp

21 . p.6 Digraphs with a fixed degree sequence: Cooper and Frieze Consider a random digraph D with n vertices and l i,j vertices of in-degree i and out-degree j.

22 . p.6 Digraphs with a fixed degree sequence: Cooper and Frieze Consider a random digraph D with n vertices and l i,j vertices of in-degree i and out-degree j. θn = i,j il i,j is the number of arcs in D. d = i,j ijl i,j/(θn) is the expected out-degree of a randomly chosen arc.

23 . p.6 Digraphs with a fixed degree sequence: Cooper and Frieze Consider a random digraph D with n vertices and l i,j vertices of in-degree i and out-degree j. θn = i,j il i,j is the number of arcs in D. d = i,j ijl i,j/(θn) is the expected out-degree of a randomly chosen arc. Then whp θ < 0 implies all strong components of G are small θ > 0 implies G has a giant strong component S (size Ω(n))

24 . p.7 More on d > 1. Let L + be the set of vertices with a giant fan-out and L be the set of vertices with a giant fan-in. Then whp S = L + L

25 . p.7 More on d > 1. Let L + be the set of vertices with a giant fan-out and L be the set of vertices with a giant fan-in. Then whp S = L + L temp.gif (GIF Image, 522x394 pixels) file:///home/alan/texfiles/webgraph/talk/temp.gif

26 Eigenvalues Mihail and Papadimitriou Chung,Lu and Vu.. p.8

27 . p.8 Eigenvalues Mihail and Papadimitriou Chung,Lu and Vu. Faloutsos,Faloutsos,Faloutsos observed that the largest eigenvalues of the adjacency matrix followed a power law.

28 . p.8 Eigenvalues Mihail and Papadimitriou Chung,Lu and Vu. Faloutsos,Faloutsos,Faloutsos observed that the largest eigenvalues of the adjacency matrix followed a power law. Expected degree model Fix a 1, a 2,..., a n. Let A = a a n. Then put an edge between i and j with probability a i a j /A.

29 . p.8 Eigenvalues Mihail and Papadimitriou Chung,Lu and Vu. Faloutsos,Faloutsos,Faloutsos observed that the largest eigenvalues of the adjacency matrix followed a power law. Expected degree model Fix a 1, a 2,..., a n. Let A = a a n. Then put an edge between i and j with probability a i a j /A. Let λ 1 λ 2 be the largest eigenvalues of the adjacency matrix of the graph produced.

30 Suppose that a i = a 1 i α, 1/2 < α < 1 for 1 i n β, β sufficiently small.. p.9

31 . p.9 Suppose that a i = a 1 i α, 1/2 < α < 1 for 1 i n β, β sufficiently small. for i n β. (1 o(1)) a i λ i (1 + o(1)) a i whp

32 . p.9 Suppose that a i = a 1 i α, 1/2 < α < 1 for 1 i n β, β sufficiently small. for i n β. (1 o(1)) a i λ i (1 + o(1)) a i whp ements Star of degree d: largest eigenvalue d 1/2

33 Dynamic models: Preferential Attachment Model (PAM). Barabasi and Albert. p.10

34 . p.10 Dynamic models: Preferential Attachment Model (PAM). Barabasi and Albert We build the graph dynamically: At time t: Add a new vertex v t. Connect v t to m randomly chosen vertices u 1,..., u m in {v 1,..., v t 1 }.

35 . p.10 Dynamic models: Preferential Attachment Model (PAM). Barabasi and Albert We build the graph dynamically: At time t: Add a new vertex v t. Connect v t to m randomly chosen vertices u 1,..., u m in {v 1,..., v t 1 }. Pr(u i = u) = deg t 1(u) 2m(t 1)

36 . p.10 Dynamic models: Preferential Attachment Model (PAM). Barabasi and Albert We build the graph dynamically: At time t: Add a new vertex v t. Connect v t to m randomly chosen vertices u 1,..., u m in {v 1,..., v t 1 }. Pr(u i = u) = deg t 1(u) 2m(t 1) The rich get richer in the WWW world.

37 . p.10 Dynamic models: Preferential Attachment Model (PAM). Barabasi and Albert We build the graph dynamically: At time t: Add a new vertex v t. Connect v t to m randomly chosen vertices u 1,..., u m in {v 1,..., v t 1 }. Pr(u i = u) = deg t 1(u) 2m(t 1) The rich get richer in the WWW world. Preferential attachment also used in models by Yule 1925 and Simon 1955.

38 Let d k (t) denote the expected number of vertices of degree k at time t.. p.11

39 . p.11 Let d k (t) denote the expected number of vertices of degree k at time t. d k (t+1) = d k (t)+m (k 1)d k 1(t) 2mt m kd k(t) 2mt +1 k=m+error terms.

40 . p.11 Let d k (t) denote the expected number of vertices of degree k at time t. d k (t+1) = d k (t)+m (k 1)d k 1(t) 2mt m kd k(t) 2mt +1 k=m+error terms. Assume that d k (t) d k t. Then d k ( k ) d k 1 k k=m

41 . p.11 Let d k (t) denote the expected number of vertices of degree k at time t. d k (t+1) = d k (t)+m (k 1)d k 1(t) 2mt m kd k(t) 2mt +1 k=m+error terms. Assume that d k (t) d k t. Then d k ( k ) d k 1 k k=m d k 2m(m + 1) t for k m. (k + 2)(k + 1)k Bollobás,Riordan,Spencer,Tusnady

42 . p.12 Diameter: Bollobás,Riordan. Diameter log t log log t

43 . p.12 Diameter: Bollobás,Riordan. Diameter log t log log t Expected degree of vertex v s at time t is m t/s.

44 . p.12 Diameter: Bollobás,Riordan. Diameter log t log log t Expected degree of vertex v s at time t is m t/s. If i < j, Pr(Edge (v i, v j ) exists) ω where ω slowly. j/i 2mj ω ij

45 . p.12 Diameter: Bollobás,Riordan. Diameter Let k = (1 ɛ) log t/ log log t Pr( P ath length k, t t 1) log t log log t t 0 =t,...,t k =t 1 ω k 1 t(t 1) = o(1). k i=1 ( k i=1 ω ti 1 t i ) k 1 i

46 . p.12 Diameter: Bollobás,Riordan. Diameter log t log log t Upper bound is harder

47 . p.13 Copying Model Communities: A large dense bipartite sub-graph of the WWW indicates a community. Experiments indicate a larger number of communities than you would get say from the simple model PAM. The next model does give many. It is due to Kumar,Raghavan,Sivakumar,Upfal

48 . p.13 Copying Model Communities: A large dense bipartite sub-graph of the WWW indicates a community. Experiments indicate a larger number of communities than you would get say from the simple model PAM. The next model does give many. It is due to Kumar,Raghavan,Sivakumar,Upfal The edges of these bipartite cliques are oriented the same way.

49 As in PAM, at each stage we add a new vertex v t and give it m edges. Its construction rests on a parameter 0 < α, 1.. p.14

50 . p.14 As in PAM, at each stage we add a new vertex v t and give it m edges. Its construction rests on a parameter 0 < α, 1. A vertex u is chosen uniformly at random from {v 1,..., v t 1 } and then for i = 1,..., m we

51 . p.14 As in PAM, at each stage we add a new vertex v t and give it m edges. Its construction rests on a parameter 0 < α, 1. A vertex u is chosen uniformly at random from {v 1,..., v t 1 } and then for i = 1,..., m we 1. With probability α we create edge (v t, x) where x is chosen uniformly at random.

52 . p.14 As in PAM, at each stage we add a new vertex v t and give it m edges. Its construction rests on a parameter 0 < α, 1. A vertex u is chosen uniformly at random from {v 1,..., v t 1 } and then for i = 1,..., m we 1. With probability α we create edge (v t, x) where x is chosen uniformly at random. 2. With probability 1 α we create edge (v t, y) where y is the ith choice of u.

53 Whp the degree sequence has a power law with exponent 2 α 1 α.. p.15

54 . p.15 Whp the degree sequence has a power law with exponent 2 α 1 α. Whp the number of copies of K r,r, r m is Ω(te r )

55 . p.15 Whp the degree sequence has a power law with exponent 2 α 1 α. Whp the number of copies of K r,r, r m is Ω(te r ) This contrasts with PAM which has O(1) in expectation.

56 Deletions: Bollobás and Riordan Robustness: Suppose we build our PAM graph and then delete the first ct vertices. What remains has a giant (Ω(t) size) component iff c < m 1 m p.16

57 . p.16 Deletions: Bollobás and Riordan Robustness: Suppose we build our PAM graph and then delete the first ct vertices. What remains has a giant (Ω(t) size) component iff c < m 1 m + 1. If vertices are deleted randomly with probability p < 1 constant. Then there is a giant component for any p.

58 . p.16 Deletions: Bollobás and Riordan Robustness: Suppose we build our PAM graph and then delete the first ct vertices. What remains has a giant (Ω(t) size) component iff c < m 1 m + 1. If vertices are deleted randomly with probability p < 1 constant. Then there is a giant component for any p. What about deletions during the growing phase?

59 . p.17 Random deletion in a scale free random graph We consider a model where edges are added using preferential attachment and vertices/edges are deleted randomly. Flaxman, Frieze, Vera

60 . p.17 Random deletion in a scale free random graph We consider a model where edges are added using preferential attachment and vertices/edges are deleted randomly. Flaxman, Frieze, Vera Same model considered by Chung and Lu.

61 . p.18 G t = (V t, E t ) denotes the graph at time t. Initialisation: Start with G 1 with vertices x 1 and no edges. Step t 2

62 . p.18 G t = (V t, E t ) denotes the graph at time t. Initialisation: Start with G 1 with vertices x 1 and no edges. Step t 2 A Probability 1 α α 0 : delete a randomly chosen vertex.

63 . p.18 G t = (V t, E t ) denotes the graph at time t. Initialisation: Start with G 1 with vertices x 1 and no edges. Step t 2 A Probability 1 α α 0 : delete a randomly chosen vertex. B Probability α 0 we delete m randomly chosen edges.

64 . p.18 G t = (V t, E t ) denotes the graph at time t. Initialisation: Start with G 1 with vertices x 1 and no edges. Step t 2 A Probability 1 α α 0 : delete a randomly chosen vertex. B Probability α 0 we delete m randomly chosen edges. C Probability α 1 : add new vertex x t and m random neighbours w 1,..., w m. Pr(w i = w) = d(w, t 1) 2e t 1. (-3)

65 . p.18 G t = (V t, E t ) denotes the graph at time t. Initialisation: Start with G 1 with vertices x 1 and no edges. Step t 2 A Probability 1 α α 0 : delete a randomly chosen vertex. B Probability α 0 we delete m randomly chosen edges. C Probability α 1 : add new vertex x t and m random neighbours w 1,..., w m. Pr(w i = w) = d(w, t 1) 2e t 1. (-3) D Probability α α 1 : Add m random edges with endpoints chosen independently as in (-3).

66 We skip over details of what to do if there are no vertices to delete or what to do with multiple edges etc.. p.19

67 . p.20 D k (t) is the number of vertices of degree k in G t and D k (t) = E(D k (t)). β = 2(α α 0 ) 3α 1 α 1 α 0 Theorem Under natural restrictions on the parameters, there exists a constant C = C(m, α, α 0, α 1 ) such that for k 1, D k (t) t Ck 1 β = O ( t ɛ) + O ( k 2 β).

68 Suppose that v t, e t denote the number of vertices and edges in G t.. p.21

69 . p.21 Suppose that v t, e t denote the number of vertices and edges in G t. D k (t + 1) = D k (t)+ ( (2α α 1 )me kd k(t) + (k 1)D ) k 1(t) 2e t 2e t e t > 0 Pr(e t > 0) ( ) Dk+1 (t) D k (t) + (1 α)(k + 1)E e t > 0 Pr(e t > 0) v t + α 1 1 k=m + error terms.

70 . p.22 Let We show ν = α + α 0 + α 1 1 > 0 and η = m(α α 0)ν 1 + α 1 α α 0.

71 . p.22 Let We show ν = α + α 0 + α 1 1 > 0 and η = m(α α 0)ν 1 + α 1 α α 0. v t νt t 1/2 log t, qs.

72 . p.22 Let We show ν = α + α 0 + α 1 1 > 0 and η = m(α α 0)ν 1 + α 1 α α 0. v t νt t 1/2 log t, qs. Pr( e t ηt t 1 ɛ ) = O(t ɛ ).

73 . p.22 Let We show ν = α + α 0 + α 1 1 > 0 and η = m(α α 0)ν 1 + α 1 α α 0. v t νt t 1/2 log t, qs. Pr( e t ηt t 1 ɛ ) = O(t ɛ ). We used Chebychef to handle e t. Chung and Lu modify Azuma s inequality, avoids constraint on edge deletion prob.

74 . p.23 As a consequence we can write D k (t + 1) = D k (t) + (A 2 (k + 1) + B 2 ) D k+1(t) t (A 1 k+b 1 +1) D k(t) t +(A 0 (k 1)+B 0 ) D k 1(t) t + +α 1 1 k=m +O(t ɛ ). A 2 = 1 α α 0 ν + mα 0 η A 1 = (2α α 1 + 2α 0 )m 2η A 0 = (2α α 1)m 2η 1 α α 0 ν B 2 = 0 B 1 = 1 1 α α 0 ν B 0 = 0

75 . p.24 Assume D k (t) d k t. d 1 = 0 and for k 1, (A 2 (k + 2) + B 2 )d k+2 + (A 1 (k + 1) + B 1 )d k+1 + (A 0 k + B 0 )d k = α 1 1 k=m 1. (-2)

76 . p.24 Assume D k (t) d k t. d 1 = 0 and for k 1, (A 2 (k + 2) + B 2 )d k+2 + (A 1 (k + 1) + B 1 )d k+1 + (A 0 k + B 0 )d k = α 1 1 k=m 1. (-2) We first tackle homogeneous equation: for k 1 (A 2 (k +2)+B 2 )e k+2 +(A 1 (k +1)+B 1 )e k+1 +(A 0 k +B 0 )e k = 0,

77 . p.24 Assume D k (t) d k t. d 1 = 0 and for k 1, (A 2 (k + 2) + B 2 )d k+2 + (A 1 (k + 1) + B 1 )d k+1 + (A 0 k + B 0 )d k = α 1 1 k=m 1. (-2) We first tackle homogeneous equation: for k 1 (A 2 (k +2)+B 2 )e k+2 +(A 1 (k +1)+B 1 )e k+1 +(A 0 k +B 0 )e k = 0, We use Laplace s method and make the substitution e k = t=b t=a t k 1 v(t)dt for a, b, v(t) to be determined.

78 . p.25 Integrating by parts ke k = [t k v(t)] b a t=b t=a t k v (t)dt

79 . p.25 Integrating by parts Let ke k = [t k v(t)] b a t=b t=a t k v (t)dt φ 1 (t) = A 2 t 2 + A 1 t + A 0, φ 0 (t) = B 2 t 2 + B 1 t + B 0

80 . p.25 Integrating by parts Let ke k = [t k v(t)] b a t=b t=a t k v (t)dt φ 1 (t) = A 2 t 2 + A 1 t + A 0, φ 0 (t) = B 2 t 2 + B 1 t + B 0 Substituting gives [t k φ 1 (t)v(t)] b a b a t k φ 1 (t)v (t)dt + b a t k 1 φ 0 (t)v(t)dt = 0.

81 . p.26 So, v(t) will give a solution to the homogeneous equation if [t k v(t)φ 1 (t)] b a = 0 and v (t) v(t) = φ 0(t) tφ 1 (t).

82 . p.26 So, v(t) will give a solution to the homogeneous equation if [t k v(t)φ 1 (t)] b a = 0 and v (t) v(t) = φ 0(t) tφ 1 (t). The differential equation is homogeneous and can be integrated to give, where A > 1 and C 0 0. v(t) = C 0 (t 1) β (t A) β

83 We take a = 0, b = 1 and satisfy [t k v(t)φ 1 (t)] b a = 0.. p.27

84 . p.27 We take a = 0, b = 1 and satisfy [t k v(t)φ 1 (t)] b a = 0. Substituting we get the following solution to the homogeneous equation: valid for k 1, u 1 (k) = 1 0 ( ) β 1 t t k 1 dt 1 γt where γ < 1.

85 . p.28 u 1 (k) = = = = = t k 1 (1 t) β 1 (1 γt) β dt t k 1 (1 t) β j=0 ( β + j 1 ) (γt) j dt j ( β + j 1 1 )γ j t k+j 1 (1 t) β dt j j=0 ( β + j 1 ) j Γ(k + j)γ(β + 1) γ j Γ(k + j + β + 1) j=0 j=0 γ j Γ(β + j) Γ(j + 1)Γ(β) 0 Γ(k + j)γ(β + 1) Γ(k + j + β + 1) assuming k is large, using Stirling for Γ(k + j), Γ(k + j + β + 1), we get = (1 + O(k 1 j Γ(j + β) ))β γ (k + β + j) β 1 Γ(j + 1) j=0 = (1 + O(k 1 ))C 1 k 1 β

86 We need to consider the non-homogeneous equation.. p.29

87 . p.29 We need to consider the non-homogeneous equation. d 1 = 0 and for k 1, (A 2 (k + 2) + B 2 )d k+2 + (A 1 (k + 1) + B 1 )d k+1 + (A 0 k + B 0 )d k = α 1 1 k=m 1. (-2)

88 . p.29 We need to consider the non-homogeneous equation. d 1 = 0 and for k 1, (A 2 (k + 2) + B 2 )d k+2 + (A 1 (k + 1) + B 1 )d k+1 + (A 0 k + B 0 )d k = α 1 1 k=m 1. (-2) We show there is a solution such that d k = Cu 1 (k) for k m.

89 Adversarial deletion in a scale free random graph: Flaxman, Frieze, Vera. p.30

90 . p.30 Adversarial deletion in a scale free random graph: Flaxman, Frieze, Vera What if an adversary is allowed to delete ɛn vertices as the the graph grows.

91 . p.30 Adversarial deletion in a scale free random graph: Flaxman, Frieze, Vera What if an adversary is allowed to delete ɛn vertices as the the graph grows. We show that if m is sufficiently large then the final graph has a giant component (size Θ(n)) whp.

92 . p.30 Adversarial deletion in a scale free random graph: Flaxman, Frieze, Vera What if an adversary is allowed to delete ɛn vertices as the the graph grows. We show that if m is sufficiently large then the final graph has a giant component (size Θ(n)) whp. Bollobás and Riordan allow the adversary to delete vertices at the end only.

93 . p.30 Adversarial deletion in a scale free random graph: Flaxman, Frieze, Vera What if an adversary is allowed to delete ɛn vertices as the the graph grows. We show that if m is sufficiently large then the final graph has a giant component (size Θ(n)) whp. Bollobás and Riordan allow the adversary to delete vertices at the end only. On the other hand, we use their idea of coupling with G n,p.

94 A geometric scale free random graph: Flaxman, Frieze, Vera.. p.31

95 . p.31 A geometric scale free random graph: Flaxman, Frieze, Vera. V n = {X 1, X 2,..., X n } are chosen uniformly from the surface of the unit sphere in R 3

96 . p.31 A geometric scale free random graph: Flaxman, Frieze, Vera. V n = {X 1, X 2,..., X n } are chosen uniformly from the surface of the unit sphere in R 3 Step t: Add m random edges from where X t to Y 1, Y 2,..., Y m {X 1, X 2,..., X t 1 } Y i X t r = n ɛ 1/2 and the Y i s are chosen via preferential attachment.

97 . p.32 Theorem If m is a sufficiently large constant then there exists a constant c > 0 such that d k (n) cn k(k + 1)(k + 2) where d k (n) is the number of vertices of degree k at time n.

98 . p.32 Theorem If m is a sufficiently large constant then there exists a constant c > 0 such that d k (n) cn k(k + 1)(k + 2) where d k (n) is the number of vertices of degree k at time n. Whp V n can be partitioned into T, T such that T, T n/2, and there are at most 4 πrnm edges between T and T.

99 . p.32 Theorem If m is a sufficiently large constant then there exists a constant c > 0 such that d k (n) cn k(k + 1)(k + 2) where d k (n) is the number of vertices of degree k at time n. Whp V n can be partitioned into T, T such that T, T n/2, and there are at most 4 πrnm edges between T and T. If m K log n and K is sufficiently large then whp G n is connected.

100 Heuristically Optimized Trade-offs Carlson and Doyle; Fabrikant, Koutsoupias and Papadimitriou. p.33

101 Heuristically Optimized Trade-offs Carlson and Doyle; Fabrikant, Koutsoupias and Papadimitriou X 1, X 2,..., X n are chosen uniformly at random in the unit square [0, 1] 2.. p.33

102 Heuristically Optimized Trade-offs Carlson and Doyle; Fabrikant, Koutsoupias and Papadimitriou X 1, X 2,..., X n are chosen uniformly at random in the unit square [0, 1] 2. After X 1,..., X i have beeen generated there is a tree T i on them.. p.33

103 . p.33 Heuristically Optimized Trade-offs Carlson and Doyle; Fabrikant, Koutsoupias and Papadimitriou X 1, X 2,..., X n are chosen uniformly at random in the unit square [0, 1] 2. After X 1,..., X i have beeen generated there is a tree T i on them. Then at step i + 1, X i+1 is joined by an edge to the vertex X j which minimises α X i+1 X j + h j where h j is the number of edges from X j to X 1 in T i.

104 Theorem If α < 1/ 2 then T n is a star.. p.34

105 . p.34 Theorem If α < 1/ 2 then T n is a star. If α = o(n 1/2 /(log n) 2 ) then whp T n has n o(n) leaves.

106 . p.34 Theorem If α < 1/ 2 then T n is a star. If α = o(n 1/2 /(log n) 2 ) then whp T n has n o(n) leaves. If α/(n 1/2 log n) then A 1 e c 1k ρ k A 2 e c 2k where ρ k is the proportion of vertices of degree k

107 . p.34 Theorem If α < 1/ 2 then T n is a star. If α = o(n 1/2 /(log n) 2 ) then whp T n has n o(n) leaves. If α/(n 1/2 log n) then A 1 e c 1k ρ k A 2 e c 2k where ρ k is the proportion of vertices of degree k If α 4 and α = o(n 1/2 ) then E(ρ k ) c(k/n) β.

108 . p.34 Theorem If α < 1/ 2 then T n is a star. If α = o(n 1/2 /(log n) 2 ) then whp T n has n o(n) leaves. If α/(n 1/2 log n) then A 1 e c 1k ρ k A 2 e c 2k where ρ k is the proportion of vertices of degree k If α 4 and α = o(n 1/2 ) then E(ρ k ) c(k/n) β. Some of these results are from Berger,Bollobás,Borgs,Chayes,Riordan: Also Berger,Borgs,Chayes,D Souza,Kleinberg

109 Crawling on web graphs Cooper and Frieze. p.35

110 . p.35 Crawling on web graphs Cooper and Frieze Sequence of random graphs G(t) : G(t) = G(t 1) plus vertex t and m random edges {t, v i }, i = 1, 2,..., m.

111 . p.35 Crawling on web graphs Cooper and Frieze Sequence of random graphs G(t) : G(t) = G(t 1) plus vertex t and m random edges {t, v i }, i = 1, 2,..., m. Model 1: The vertices v 1, v 2,..., v m are chosen uniformly with replacement from [t 1].

112 . p.35 Crawling on web graphs Cooper and Frieze Sequence of random graphs G(t) : G(t) = G(t 1) plus vertex t and m random edges {t, v i }, i = 1, 2,..., m. Model 1: The vertices v 1, v 2,..., v m are chosen uniformly with replacement from [t 1]. Model 2 The vertices v 1, v 2,..., v m are chosen proportional to their degree after step t 1.

113 . p.35 Crawling on web graphs Cooper and Frieze Sequence of random graphs G(t) : G(t) = G(t 1) plus vertex t and m random edges {t, v i }, i = 1, 2,..., m. Model 1: The vertices v 1, v 2,..., v m are chosen uniformly with replacement from [t 1]. Model 2 The vertices v 1, v 2,..., v m are chosen proportional to their degree after step t 1. Spider S sits at vertex X t 1 of G(t 1). After the addition of vertex t, and before step t + 1, spider makes a random walk of length l

114 ν l,m (t) is the expected number of vertices not visited by S at the end of step t.. p.36

115 . p.36 ν l,m (t) is the expected number of vertices not visited by S at the end of step t. Theorem In either model, if m is sufficiently large then, ν l,m (t) E t s=1 t τ=s ( 1 d(s, τ) 2mτ ) l where d(s, τ) denotes the degree of s in G(τ). Error in expression of order m 1 omitted.

116 . p.37 Let η l = lim m lim t Eν l,m (t) t.

117 . p.37 Let (a) For Model 1, η l = lim m lim t Eν l,m (t) t. η l = 2 l e(l+2)2 /(4l) (l+2)/ 2l e y2 /2 dy where Ψ(x) is the standardized Normal cumulate for the interval (, x]. In particular, η 1 = 0.57 and η l 2/l as l.

118 . p.37 Let (b) For Model 2 η l = lim m lim t Eν l,m (t) t. η l = e l 2l 2 l y 3 e y dy. In particular, η 1 = and η l 2/l as l.

119 . p.38 Further work Try other models of a random web-graph. Try non-uniform random walks. Prove concentration of the number of vertices visited. The proof technique is robust enough to handle other models and walks, once one has established rapid mixing. The calculations for various non-uniform walks can get tedious. It should be possible to estimate the variance of the number of unvisited vertices and apply Chebychef. Stronger concentration seems more challenging.

120 Cover time of preferential attachment graph. p.39

121 . p.39 Cover time of preferential attachment graph G = (V, E) is a connected graph. ( V = n, E = m). For v V let C v be the expected time taken for a simple random walk W on G starting at v, to visit every vertex of G.

122 . p.39 Cover time of preferential attachment graph G = (V, E) is a connected graph. ( V = n, E = m). For v V let C v be the expected time taken for a simple random walk W on G starting at v, to visit every vertex of G. The cover time C G of G is defined as C G = max v V C v.

123 . p.39 Cover time of preferential attachment graph G = (V, E) is a connected graph. ( V = n, E = m). For v V let C v be the expected time taken for a simple random walk W on G starting at v, to visit every vertex of G. The cover time C G of G is defined as C G = max v V C v. (1 o(1))n log n C G (1 + o(1)) 4 27 n3 : Feige

124 . p.39 Cover time of preferential attachment graph G = (V, E) is a connected graph. ( V = n, E = m). For v V let C v be the expected time taken for a simple random walk W on G starting at v, to visit every vertex of G. The cover time C G of G is defined as C G = max v V C v. Cooper and Frieze If G is preferential attachment graph then whp C G 2m m 1 n log n

125 Effect of search engines: Chakrabarti, Frieze, Vera. p.40

126 Effect of search engines: Chakrabarti, Frieze, Vera The model has parameters p, N. p.40

127 . p.40 Effect of search engines: Chakrabarti, Frieze, Vera The model has parameters p, N At time step t we add vertex v t and m randomly chosen edges to {u 1,..., u m }. For each i, u i is chosen preferentially from With probability p we choose u i from N vertices of largest degree.. With probability q = 1 p we choose u i from all vertices.

128 . p.40 Effect of search engines: Chakrabarti, Frieze, Vera The model has parameters p, N At time step t we add vertex v t and m randomly chosen edges to {u 1,..., u m }. For each i, u i is chosen preferentially from With probability p we choose u i from N vertices of largest degree.. With probability q = 1 p we choose u i from all vertices. Supposed to model surfer who obtains links from first N given by search engine.

129 . p.41 Theorem (a) For i N there exists constant α i > 0 such that E [deg n (x i )] = α i n + O (n 1/2) (b) There is an absolute constant A 1 such that for every k m, d k (n) = (1 + o(1)) A 1 n k 1+2/(1 p). where d k (n) is the expected number of vertices of degree k outside the N largest vertices.