Lecture 4: Phase Transition in Random Graphs
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1 Lecture 4: Phase Transition in Random Graphs Radu Balan February 21, 2017
2 Distributions Today we discuss about phase transition in random graphs. Recall on the Erdöd-Rényi class G n,p of random graphs, the probability mass function on G, P : G [0, 1], is obtained by assuming that, as random variables, edges are independent from one another, and each edge occurs with probability p [0, 1]. Thus a graph G G with m vertices will have probability P(G) given by ( ) n m P(G) = p m 2 (1 p).
3 Distributions Today we discuss about phase transition in random graphs. Recall on the Erdöd-Rényi class G n,p of random graphs, the probability mass function on G, P : G [0, 1], is obtained by assuming that, as random variables, edges are independent from one another, and each edge occurs with probability p [0, 1]. Thus a graph G G with m vertices will have probability P(G) given by ( ) n m P(G) = p m 2 (1 p). Recall the expected number of q-cliques X q is ( ) n E[X q ] = q p q(q 1)/2
4 Distributions We shall also use gnm the set of all graphs on n vertices with m edges. The set Γ n,m has cardinal ( ) n 2. m In Γ n,m each graph is equally probable.
5 Cliques The case of 3-cliques: E[X 3 ] = θn 3 p 3 (θ 1 6 ). The case of 4-cliques: E[X 4 ] = θn 4 p 6 (θ 1 24 ). The first problem we consider is thesize of the largest clique of a random graph. Note, finding the size of the largest clique (called the clique number) is a NP-hard problem.
6 Cliques The case of 3-cliques: E[X 3 ] = θn 3 p 3 (θ 1 6 ). The case of 4-cliques: E[X 4 ] = θn 4 p 6 (θ 1 24 ). The first problem we consider is thesize of the largest clique of a random graph. Note, finding the size of the largest clique (called the clique number) is a NP-hard problem. Idea: Analyze p so that E[X q ] 1. For p > 1 n and large n we expect that graphs will have a 3-clique; For p > 1 and large n, we expect that graphs will have a 4-clique; n 2/3
7 Cliques The case of 3-cliques: E[X 3 ] = θn 3 p 3 (θ 1 6 ). The case of 4-cliques: E[X 4 ] = θn 4 p 6 (θ 1 24 ). The first problem we consider is thesize of the largest clique of a random graph. Note, finding the size of the largest clique (called the clique number) is a NP-hard problem. Idea: Analyze p so that E[X q ] 1. For p > 1 n and large n we expect that graphs will have a 3-clique; For p > 1 and large n, we expect that graphs will have a 4-clique; n 2/3 Question: How sharp are these thresholds?
8 3-Cliques Theorem Let p = p(n) be the edge probability in G n,p. 1 If p 1 n (i.e. lim n np = ) then lim n Prob[G G n,p has a 3 clique] 1. 2 If p 1 n (i.e. lim n np = 0) then lim n Prob[G G n,p has a 3 clique] 0.
9 3-Cliques Theorem Let p = p(n) be the edge probability in G n,p. 1 If p 1 n (i.e. lim n np = ) then lim n Prob[G G n,p has a 3 clique] 1. 2 If p 1 n (i.e. lim n np = 0) then lim n Prob[G G n,p has a 3 clique] 0. Theorem Let m = m(n) be the number of edges in Γ n,m. 1 m If m n (i.e. lim n n = ) then lim n Prob[G Γ n,m has a 3 clique] 1. 2 m If m n (i.e. lim n n = 0) then lim n Prob[G Γ n,m has a 3 clique] 0.
10 4-Cliques Theorem Let p = p(n) be the edge probability in G n,p. 1 If p 1 (i.e. lim n 2/3 n n 2/3 p = ) then lim n Prob[G G n,p has a 4 clique] 1. 2 If p 1 (i.e. lim n 2/3 n n 2/3 p = 0) then lim n Prob[G G n,p has a 4 clique] 0.
11 4-Cliques Theorem Let p = p(n) be the edge probability in G n,p. 1 If p 1 (i.e. lim n 2/3 n n 2/3 p = ) then lim n Prob[G G n,p has a 4 clique] 1. 2 If p 1 (i.e. lim n 2/3 n n 2/3 p = 0) then lim n Prob[G G n,p has a 4 clique] 0. Theorem Let m = m(n) be the number of edges in Γ n,m. 1 If m n 4/3 m (i.e. lim n = ) then n 4/3 lim n Prob[G Γ n,m has a 4 clique] 1. 2 If m n 4/3 m (i.e. lim n = 0) then n 4/3 lim n Prob[G Γ n,m has a 4 clique] 0.
12 q-cliques Theorem Let p = p(n) be the edge probability in G n,p. Let q 3 be and integer. 1 If p 1 (i.e. lim n 2/(q 1) n n 2/(q 1) p = ) then lim n Prob[G G n,p has a q clique] 1. 2 If p 1 (i.e. lim n 2/(q 1) n n 2/(q 1) p = 0) then lim n Prob[G G n,p has a q clique] 0.
13 q-cliques Theorem Let p = p(n) be the edge probability in G n,p. Let q 3 be and integer. 1 If p 1 (i.e. lim n 2/(q 1) n n 2/(q 1) p = ) then lim n Prob[G G n,p has a q clique] 1. 2 If p 1 (i.e. lim n 2/(q 1) n n 2/(q 1) p = 0) then lim n Prob[G G n,p has a q clique] 0. Theorem Let m = m(n) be the number of edges in Γ n,m. Let q 3 be and integer. 1 If m n 2(q 2)/(q 1) m (i.e. lim n = ) then n 2(q 2)/(q 1) lim n Prob[G Γ n,m has a q clique] 1. 2 If m n 2(q 2)/(q 1) m (i.e. lim n = 0) then n 2(q 1)/(q 1) lim n Prob[G Γ n,m has a q clique] 0.
14 Markov and Chebyshev Inequalities We want to control probabilities of the random event X 3 (G) > 0. Two important tools: 1 (Markov s Inequality) Assume X is a non-negative random variable. Then Prob[X t] E[X] t. 2 (Chebyshev s Inequality) For any random variable X, Prob[ X E[X] t] Var[X] t 2. where E[X] is the mean of X, and Var[X] = E[X 2 ] E[X] 2 is the variance of X.
15 Markov and Chebyshev Inequalities We want to control probabilities of the random event X 3 (G) > 0. Two important tools: 1 (Markov s Inequality) Assume X is a non-negative random variable. Then Prob[X t] E[X] t. 2 (Chebyshev s Inequality) For any random variable X, Prob[ X E[X] t] Var[X]. t 2 where E[X] is the mean of X, and Var[X] = E[X 2 ] E[X] 2 is the variance of X. Quick Proof: Prob[X t] = t p X (x)dx 1 t t xp X (x)dx E[X]. t Prob[ X E[X] t] = P[ X E[X] 2 t 2 ] E[ X E[X] 2 ] t 2 = Var[X] t 2.
16 Proofs for the 3-clique case For small probability: We shall use Markov s inequality to show Prob[X 3 > 0] 0 when p 1 n : Prob[X 3 > 0] = Prob[X 3 1] E[X 3] 1 = n(n 1)(n 2) p 3 = θn 3 p
17 Proofs for the 3-clique case For small probability: We shall use Markov s inequality to show Prob[X 3 > 0] 0 when p 1 n : Prob[X 3 > 0] = Prob[X 3 1] E[X 3] 1 = n(n 1)(n 2) p 3 = θn 3 p For large probability: Since E[X 3 ] it follows that Prob[X 3 > 0] > 0. We need to show that Prob[X 3 = 0] 0. By Chebyshev s inequality: Prob[X 3 = 0] Prob[ X 3 E[X 3 ] E[X 3 ]] Var[X 3] E[X 3 ] 2
18 Proofs for the 3-clique case For small probability: We shall use Markov s inequality to show Prob[X 3 > 0] 0 when p 1 n : Prob[X 3 > 0] = Prob[X 3 1] E[X 3] 1 = n(n 1)(n 2) p 3 = θn 3 p For large probability: Since E[X 3 ] it follows that Prob[X 3 > 0] > 0. We need to show that Prob[X 3 = 0] 0. By Chebyshev s inequality: Prob[X 3 = 0] Prob[ X 3 E[X 3 ] E[X 3 ]] Var[X 3] E[X 3 ] 2 Need the variance of X 3 = (i,j,k) S 3 1 i,j,k, X3 2 = 1 i,j,k 1 i,j,k. (i,j,k ) S 3 (i,j,k) S 3
19 Proofs for the 3-clique case X3 2 = 1 i,j,k + (1 i,j,k 1 i,j,l +1 i,j,k 1 j,k,l +1 i,j,k 1 k,i,l )+ (i,j,k) S 3 (n) + (i,j,k) S 3 (n) u,v S 2 (n 3) + (i,j,k) S 3 (n) l S 1 (n 3) (1 i,j,k 1 i,u,v + 1 i,j,k 1 j,u,v 1 i,j,k 1 k,u,v )+ (i,j,k) S 3 (n) (i,j,k ) S 3 (n 3) 1 i,j,k 1 i,j,k
20 Proofs for the 3-clique case X 2 3 = (i,j,k) S 3 (n) + 1 i,j,k + (i,j,k) S 3 (n) u,v S 2 (n 3) + (i,j,k) S 3 (n) l S 1 (n 3) (1 i,j,k 1 i,j,l +1 i,j,k 1 j,k,l +1 i,j,k 1 k,i,l )+ (1 i,j,k 1 i,u,v + 1 i,j,k 1 j,u,v 1 i,j,k 1 k,u,v )+ (i,j,k) S 3 (n) (i,j,k ) S 3 (n 3) E[X 2 3 ] = S 3 p S 3 (n 3)p S 3 Thus ( n i,j,k 1 i,j,k ) p 6 + S 3 ( n 3 3 Var[X 3 ] = E[X 2 3 ] E[X 3 ] 2 =... = θ(n 3 p 3 + n 4 p 5 + n 5 p 6 ). ) p 6.
21 Proofs for the 3-clique case and: Prob[X 3 = 0] θ(n3 p 3 + n 4 p 5 + n 5 p 6 ) θ(n 6 p 6 ) 1 (np) n 0.
22 Proofs for the 3-clique case and: Prob[X 3 = 0] θ(n3 p 3 + n 4 p 5 + n 5 p 6 ) θ(n 6 p 6 ) 1 (np) n 0. Similar proofs for the other cases (4-cliques and q-cliques).
23 Behavior at the threshold In general we obtain a coarse threshold. Recall a Poisson process X with parameter λ has p.m.f. Prob[X = k] = e Theorem In G n,p, λ λk k!. 1 For p = c n, X 3 is asymptotically Poisson with parameter λ = c 3 /6. 2 For p = c n 2/3, X 4 is asymptotically Poisson with parameter λ = c 6 /24.
24 Behavior at the threshold In general we obtain a coarse threshold. Recall a Poisson process X with parameter λ has p.m.f. Prob[X = k] = e Theorem In G n,p, λ λk k!. 1 For p = c n, X 3 is asymptotically Poisson with parameter λ = c 3 /6. 2 For p = c n 2/3, X 4 is asymptotically Poisson with parameter λ = c 6 /24. Theorem In Γ n,m, 1 For m = cn, X 3 is asymptotically Poisson with parameter λ = 4c 3 /3. 2 For m = cn 4/3, X 4 is asymptotically Poisson with parameter λ = 8c 6 /3.
25 Connected Components G n,p class of random graphs has a remarkable property in regards to the largest connected component. We shall express the result in the class Γ n,m.
26 Connected Components Theorem 1 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 0 2 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 1
27 Connected Components Theorem 1 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 0 2 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 1 3 Assume m = 1 2n log(n) + tn + o(n), where o(n) n. Then lim n Prob[G Γn,m is connected] = e e 2t
28 Connected Components Theorem 1 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 0 2 Let m = m(n) satisfies m 1 2n log(n). Then lim Prob[G n Γn,m is connected] = 1 3 Assume m = 1 2n log(n) + tn + o(n), where o(n) n. Then lim n Prob[G Γn,m is connected] = e e 2t In this case 1 2n log(n) is known as a strong threshold.
29 3-cliques & Connectivity results Results for n = 1000 vertices. 1 3-cliques. Recall E[X 3 ] m 3 2 Connectivity. Recall the connectivity threshold is 1 2n log(n) = 3454.
30 3-cliques
31 3-cliques
32 Connectivity
33 Connectivity
34 Connectivity
35 Connectivity
36 References B. Bollobás, Graph Theory. An Introductory Course, Springer-Verlag (25), (2002). F. Chung, Spectral Graph Theory, AMS F. Chung, L. Lu, The average distances in random graphs with given expected degrees, Proc. Nat.Acad.Sci R. Diestel, Graph Theory, 3rd Edition, Springer-Verlag P. Erdös, A. Rényi, On The Evolution of Random Graphs G. Grimmett, Probability on Graphs. Random Processes on Graphs and Lattices, Cambridge Press J. Leskovec, J. Kleinberg, C. Faloutsos, Graph Evolution: Densification and Shrinking Diameters, ACM Trans. on Knowl.Disc.Data, 1(1) 2007.
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