The giant component in a random subgraph of a given graph

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1 The giant component in a random subgraph of a given graph Fan Chung, Paul Horn, and Linyuan Lu 2 University of California, San Diego 2 University of South Carolina Abstract We consider a random subgraph G p of a host graph G formed by retaining each edge of G with probability p We address the question of determining the critical value p (as a function of G) for which a giant component emerges Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence We define the second order average degree d to be d = v d2 v/( v dv) where dv denotes the degree of v We prove that for any ɛ > 0, if p > ( + ɛ)/ d then asymptotically almost surely the percolated subgraph G p has a giant component In the other direction, if p < ( ɛ)/ d then almost surely the percolated subgraph G p contains no giant component Introduction Almost all information networks that we observe are subgraphs of some host graphs that often have sizes prohibitively large or with incomplete information A natural question is to deduce the properties that a random subgraph of a given graph must have We are interested in random subgraphs of G p of a graph G, obtained as follows: for each edge in G p we independently decide to retain the edge with probability p, and discard the edge with probability p A natural special case of this process is the Erdős-Rényi graph model G(n, p) which is the special case where the host graph is K n Other examples are the percolation problems that have long been studied [3, 4] in theoretical physics, mainly with the host graph being the lattice graph Z k In this paper, we consider a general host graph, an example of which being a contact graph, consisting of edges formed by pairs of people with possible contact, which is of special interest in the study of the spread of infectious diseases or the identification of community in various social networks A fundamental question is to ask for the critical value of p such that G p has a giant connected component, that is a component whose volume is a positive fraction of the total volume of the graph For the spread of disease on contact networks, the answer to this question corresponds to the problem of finding the epidemic threshold for the disease under consideration, for instance For the case of K n, Erdős and Rényi answered this in their seminal paper []: if p = c n for c <, then almost surely G contains no giant connected component and all components are of size at most O(log n), and if c > then, indeed, there is a giant component of size ɛn For general host This author was supported in part by NSF grant ITR and ONR MURI This author was supported in part by NSF grant DMS 070

2 graphs, the answer has been more elusive Results have been obtained either for very dense graphs or bounded degree graphs Bollobas, Borgs, Chayes and Riordan [4] showed that for dense graphs (where the degrees are of order Θ(n)), the giant component threshold is /ρ where ρ is the largest eigenvalue of the adjacency matrix Frieze, Krivelevich and Martin [2] consider the case where the host graph is d-regular with adjacency eigenvalue λ and they show that the critical probability is close to /d, strengthening earlier results on hypercubes [2, 3] and Cayley graphs [5] For expander graphs with degrees bounded by d, Alon, Benjamini and Stacey [] proved that the percolation threshold is greater than or equal to /(2d) There are several recent papers, mainly in studying percolation on special classes of graphs, which have gone further Their results nail down the precise critical window during which component sizes grow from log(n) vertices to a positive proportion of the graph In [5, 6], Borgs, et al find the order of this critical window for transitive graphs, and cubes Nachmias [6] looks at a similar situation to that of Frieze, Krivelevich and Martin [2] and uses random walk techniques to study percolation within the critical window for quasi-random transitive graphs Percolation within the critical window on random regular graphs is also studied by Nachmias and Peres in [7] Our results differ from these in that we study percolation on graphs with a much more general degree sequence The greater preciseness of these results, however, is quite desirable It is an interesting open question to describe the precise scaling window for percolation for the more general graphs studied here Here, we are interested in percolation on graphs which are not necessarily regular, and can be relatively sparse (ie, o(n 2 ) edges) Compared with earlier results, the main advantage of our results is the ability to handle general degree sequences To state our results, we give a few definitions here For a subset S of vertices, the volume of S, denoted by vol(s) is the sum of degrees of vertices in S The kth order volume of S is the kth moment of the degree sequence, ie vol k (S) = v S dk v We write vol (S) = vol(s) and vol k (G) = vol k (V (G)), where V (G) is the vertex set of G We denote by d = vol 2 (G)/ the second order degree of G, and by σ the spectral gap of the normalized Laplacian, which we fully define in Section 2 Further, recall that f(n) is O(g(n)) if lim sup n f(n) / g(n) <, and f(n) is o(g(n)) if lim n f(n) / g(n) = 0 We will prove the following Theorem Suppose G has the maximum degree satisfying = o( d σ c ) For p d, aas every connected component in G p has volume at most O( vol 2 (G)g(n)), where g(n) is any slowly growing function as n Here, an event occurring aas indicates that it occurs with probability tending to one as n tends to infinity In order to prove the emergence of giant component where p ( + c)/ d, we need to consider some additional conditions Suppose there is a set U satisfying (i) vol 2 (U) ( ɛ)vol 2 (G) (ii) vol 3 (U) Mdvol 2 (G)

3 where ɛ and M are constant independent of n In this case, we say G is (ɛ, M)-admissible and U is an (ɛ, M)-admissible set We note that the admissibility measures of skewness of the degree sequence For example, all regular graphs are (ɛ, )-admissible for any ɛ, but a graph needs not be regular to be admissible We also note that in the case that vol 3 (G) Mdvol 2 (G), G is (ɛ, M)-admissible for any ɛ Theorem 2 Suppose p +c for some c d 20 σ = o(n κ ) for some κ > 0, and G is ( cκ 0 d Suppose G satisfies = o( σ ), = o( d n log n ) and, M)-admissible Then aas there is a unique giant connected component in G p with volume Θ(), and no other component has volume more than max(2d log n, ω(σ )) Here, recall that f(n) = Θ(g(n)) if f(n) = O(g(n)) and g(n) = O(f(n)) In this case, we say that f and g are of the same order Also, f(n) = ω(g(n)) if g(n) = o(f(n)) We note that under the assumption that the maximum degree of G satisfying = o( d σ ), it can be shown that the spectral norm of the adjacency matrix satisfies A = ρ = ( + o()) d Under the assumption in Theorem 2, we observe that the percolation threshold of G is d To examine when the conditions of Theorems and 2 are satisfied, we note that admissibility implies that d = Θ(d), which essentially says that while there can be some vertices with degree much higher than d, there cannot be too many Chung, Lu and Vu [8] show that for random graphs with a given expected degree sequence σ = O( d ), and hence for graphs with average degree n ɛ the spectral condition of Theorem 2 easily holds for random graphs The results here can be viewed as a generalization of the result of Frieze, Krivelevich and Martin [2] with general degree sequences and is also a strengthening of the original results of Erdős and Reyni to general host graphs The paper is organized as follows: In Section 2 we introduce the notation and some basic facts In Section 3, we examine several spectral lemmas which allow us to control the expansion In Section 4, we prove Theorem, and in Section 5, we complete the proof of Theorem 2 2 Preliminaries Suppose G is a connected graph on vertex set V Throughout the paper, G p denotes a random subgraph of G obtained by retaining each edge of G independently with probability p Let A = (a uv ) denote the adjacency matrix of G, defined by { if {u, v} is an edge; a uv = 0 otherwise We let d v = u a uv denote the degree of vertex v Let = max v d v denote the maximum degree of G and δ = min v d v denote the minimum degree For each vertex set S and a positive integer k,

4 we define the k-th volume of G to be vol k (S) = v S d k v The volume is simply the sum of all degrees, ie = vol (G) We define the average degree d = n = vol (G) vol 0 (G) and the second order average degree d = vol 2(G) vol (G) Let D = diag(d v, d v2,, d vn ) denote the diagonal degree matrix Let denote the column vector with all entries and d = D be column vector of degrees The normalized Laplacian of G is defined as L = I D 2 AD 2 The spectrum of the Laplacian is the eigenvalues of L sorted in increasing order 0 = λ 0 λ λ n Many properties of λ i s can be found in [7] For example, the least eigenvalue λ 0 is always equal to 0 We have λ > 0 if G is connected and λ n 2 with equality holding only if G has a bipartite component Let σ = max{ λ, λ n } Then σ < if G is connected and non-bipartite For random graphs with a given expected degree sequence [8], σ = O( d ), and in general for regular graphs it is easy to write σ in terms of the second largest eigenvalue of the adjacency matrix Furthermore, σ is closely related to the mixing rate of random walks on G, see eg [7] The following lemma measures the difference of adjacency eigenvalue and d using σ Lemma The largest eigenvalue of the adjacency matrix of G, ρ, satisfies ρ d σ Proof: Recall that ϕ = D /2 is the unit eigenvector of L corresponding to eigenvalue 0 We have Then, I L ϕϕ σ ρ d dd = A A dd = D /2 (I L ϕϕ )D /2 D /2 I L ϕϕ D /2 σ For any subset of the vertices, S, we let let S denote the complement set of S The vertex boundary of S in G, denoted by Γ G (S) is defined as follows:

5 Γ G (S) = {u S v S such that {u, v} E(G)} When S consists of one vertex v, we simply write Γ G (v) for Γ G ({v}) We also write Γ (S) = Γ G (S) if there is no confusion Similarly, we define Γ Gp (S) to be the set of neighbors of S in our percolated subgraph G p 3 Several spectral lemmas We begin by proving two lemmas, first relating expansion in G to the spectrum of G, then giving a probabilistic bound on the expansion in G p Lemma 2 For two disjoint sets S and T, we have d v Γ (v) S vol(s)vol 2(T ) σ vol(s)vol 3 (T ) d v Γ (v) S 2 vol(s)2 vol 3 (T ) vol(s) 2 σ2 vol(s) max {d2 v} + 2σ 3 vol 5 (T ) Proof: Let S (or T ) be the indicative column vector of the set S (or T ) respectively Note d v Γ (v) S = SAD T vol(s) = Sd vol 2 (T ) = d D T Here S denotes the transpose of S as a row vector We have d v Γ (v) S vol(s)vol 2(T ) = SAD T Sdd D T = SD 2 (D 2 AD 2 D 2 D 2 )D 3 2 T Let ϕ = D /2 denote the eigenvector of I L for the eigenvalue The matrix I L ϕϕ, which is the projection of I L to the hyperspace ϕ, has L 2 -norm σ

6 We have d v Γ (v) S vol(s)vol 2(T ) = SD 2 (I L ϕϕ )D 3 2 T σ D 2 S D 3 2 T σ vol(s)vol 3 (T ) Let e v be the column vector with v-th coordinate and 0 else where Then Γ G (v) S = S Ae v We have d v Γ G (v) S 2 = d v SAe v e va S = SAD T A S Here D T = d ve v e v is the diagonal matrix with degree entry at vertex in T and 0 else where We have d v Γ G (v) S 2 vol(s)2 vol 3 (T ) 2 = SAD T A S 2 Sdd D T dd S SAD T A S Sdd D T A S + Sdd D T A S 2 Sdd D T dd S = S D 2 (I L ϕϕ )D 2 DT A S + Sdd D T D 2 (I L ϕϕ )D 2 S S D 2 (I L ϕ ϕ)d 2 DT D 2 (I L ϕϕ )D 2 S +2 Sdd D T D 2 (I L ϕϕ )D 2 S vol(s) σ 2 vol(s) max {d2 v} + 2σ 3 vol 5 (T ) Lemma 3 Suppose that two disjoint sets S and T satisfy 25σ 2 vol 3 (T ) 2 δ 2 vol 2 (T ) 2 vol 2 (T ) 5p 2δ σ2 max {d2 v} () vol(s) 2δvol 2(T ) 5pvol 3 (T ) (2)

7 Then we have that ( with probability at least exp δ( δ)pvol 2(T )vol(s) 0 vol(s) δ2 vol 2 (T ) 2 25p 2 σ 2 vol 5 (T ) (3) vol(γ Gp (S) T ) > ( δ)p vol 2(T ) vol(s) ) Proof: For any v T, let X v be the indicative random variable for v Γ Gp (S) We have P(X v = ) = ( p) Γ G (v) S Let X = Γ Gp (S) T Then X is the sum of independent random variables X v X = d v X v Note that E(X) = d v E(X v ) = d v ( ( p) Γ G (v) S ) d v (p Γ G (v) S p2 2 Γ G (v) S 2 ) p( vol(s)vol 2(T ) σ vol(s)vol 3 (T )) p2 vol 3 (T ) 2 (vol(s)2 2 + σ 2 vol(s) max {d2 v} + 2σ > ( 4 5 δ)pvol 2(T ) vol(s) by using Lemma 2 and the assumptions on S and T We apply the following Chernoff inequality, see eg [0] P(X E(X) a) e 2 d 2 v E[X2 v ] e a2 a 2 vol(s) 3 vol 5 (T ) ) 2 E(X) We set a = αe(x), with α chosen so that ( α)( 4 5δ) = ( δ) Then P(X ( δ)p vol 2(T ) vol(s)) < P(X ( α)e(x)) ( ) ( exp α2 E(X) < exp α( δ)pvol ) 2(T )vol(s) 2 2 To complete the proof, note α > δ/5

8 4 The range of p with no giant component In this section, we will prove Theorem Proof of Theorem : It suffices to prove the following claim Claim A: If pρ <, where ρ is the largest eigenvalue of the adjacency matrix, with probability at least C 2 ( pρ), all components have volume at most C vol 2 (G) Proof of Claim A: Let x be the probability that there is a component of G p having volume greater than C vol 2 (G) Now we choose two random vertices with the probability of being chosen proportional to their degrees in G Under the condition that there is a component with volume greater than C vol 2 (G), the probability of each vertex in this component is at least C vol 2 (G) Therefore, the probability that the random pair of vertices are in the same component is at least ( C vol 2 (G) x ) 2 = C2 x d (4) On the other hand, for any fixed pair of vertices u and v and any fixed path P of length k in G, the probability that u and v is connected by this path in G p is exactly p k The number of k-paths from u to v is at most ua k d v Since the probabilities of u and v being selected are u and d v respectively, the probability that the random pair of vertices are in the same connected component is at most d u d v n n p k ua k v = 2 pk d A k d We have u,v n k=0 k=0 2 pk d A k d k=0 k=0 p k ρ k vol 2 (G) 2 d ( pρ) Combining with (4), we have C2 x d d ( pρ) which implies x C 2 ( pρ) Claim A is proved, and the theorem follows taking C to be g(n) 5 The emergence of the giant component Lemma 4 Suppose G contains an (ɛ, M)-admissible set U Then we have d M ( ɛ) 2 d 2 For any U U with vol 2 (U ) > ηvol 2 (U), we have vol(u ) η2 ( ɛ) d Md

9 Proof: Since G is (ɛ, M)-admissible, we have a set U satisfying (i) vol 2 (U) ( ɛ)vol 2 (G) (ii) vol 3 (U) Mdvol 2 (G) We have d = vol 2(G) vol 2(G) vol(u) vol 2 (U) ɛ vol(u) ɛ vol 3 (U) vol 2 (U) M ( ɛ) 2 d For any U U with vol 2 (U ) > ηvol 2 (U), we have vol(u ) vol 2(U ) 2 vol 3 (U ) η2 vol 2 (U) 2 vol 3 (U) Proof of Theorem 2: It suffices to assume p = +c d η2 ( ɛ)vol 2 (G) Md for some c < 20 η2 ( ɛ) d Md Let ɛ = cκ 0 be a small constant, and U be a (ɛ, M)-admissible set in G Define U to be the subset of U containing all vertices with degree at least ɛd We have vol 2 (U ) vol 2 (U) d 2 v ( ɛ)vol 2 (G) ɛnd 2 ( 2ɛ)vol 2 (G) d v< ɛd Hence, U is a (2ɛ, M) admissible set We will concentrate on the neighborhood expansion within U Let δ = c 25M 2 and C = Take an initial set S δ 2 ( 4ɛ) 2 0 U with max(cσ 2, ln n) vol(s 0 ) max(cσ 2, ln n) + Let T 0 = U \ S 0 For i, we will recursively define S i = Γ Gp (S i ) U and T i = U \ i j=0 S j until vol 2 (T i ) ( 3ɛ)vol 2 (G) or vol(s i ) 2δvol 2(T i ) 5pvol 3 (T i ) Condition in Lemma 3 is always satisfied 5p 2δ σ2 max i d 2 v 5( + c) 2 dδ σ 2 2 = ( σ d 2 5( + c) ) vol 2 (G) 2δ = o(vol 2 (G)) vol 2 (T i ) Condition 3 in Lemma 3 is also trivial because δ 2 vol 2 (T i ) 2 25p 2 σ 2 vol 5 (T i ) δ 2 vol 2 (T i ) 2 25p 2 σ 2 2 vol 3 (T i ) δ2 ( 3ɛ)vol 2 (G) 25p 2 σ 2 2 Md ( d σ )2 δ2 ( 3ɛ) 25( + c) 2 = ω() M

10 Now we verify condition 2 We have vol(s 0 ) > Cσ 2 = 25M δ 2 ( 4ɛ) 2 σ2 25σ2 vol 3 (T 0 ) 2 δ 2 vol 2 (T 0 ) 2 The conditions of Lemma 3 are all satisfied Then we have that ( with probability at least exp vol(γ Gp (S 0 ) T 0 ) > ( δ)p vol 2(T 0 ) vol(s 0) δ( δ)pvol 2(T 0 )vol(s 0 ) 0 Since ( δ)p vol 2(T i ) ( δ)( 3ɛ)( + c) = β > by our assumption that c is small (noting that ɛ and δ are functions of c), the neighborhood of S i grows exponentially, allowing condition 2 of Lemma 3 to continue to hold and us to continue the process We stop when one of the following two events happens, vol(s i ) 2δvol 2(T i ) 5pvol 3 (T i ) vol 2 (T i ) ( 3ɛ)vol 2 (G) Let us denote the time that this happens by t If the first, but not the second, case occurs we have ) vol(s t ) 2δvol 2(T t ) 5pvol 3 (T t ) 2δ( 3ɛ) 5M( + c) In the second case, we have vol 2 ( t S j ) = vol 2 (U ) vol 2 (T t ) ɛvol 2 (G) ɛvol(u ) j=0 By Lemma 4 with η = ɛ, we have vol( t j=0 S j) ɛ2 ( 2ɛ) d Md On the other hand, note that since vol(s i ) βvol(s i ), we have that vol(s i ) β i t vol(s t ), and hence we have vol( t j=0 S j) t j=0 β j vol(s t ) so vol(s t ) ɛ2 ( 2ɛ) d(β ) Mdβ In either case we have vol(s t ) = Θ() For the moment, we restrict ourselves to the case where Cσ 2 n > ln n Each vertex in S t is in the same component as some vertex in S 0, which has size at most vol(s 0) ɛd C σ 2 n We now combine the k largest components to form a set W () with vol(w () ) > Cσ 2,

11 such that k 2 is minimal If k 2, vol(w () ) 2Cσ 2 Note that since the average size of a component is vol(st) S 0 C σ 2 n, k C σ4 n We grow as before: Let W () 0 = W (), Q () 0 = T t \ W () 0 Note that the conditions for Lemma 3 are satisfied by W () 0 and T () 0 We run the process as before, setting W () t = Γ (W () t ) Q () and Q() t = Q () t stopping when either vol(q () t ) < ( 4ɛ)vol 2 (G) or vol(w () t ) > t 2δvol 2 (Q () t ) 5pvol 3 (Q () t ) t \ W () 2δ( 4ɛ) 5M(+c) () As before, in either case vol(w t ) = Θ() Note that if k =, we are now done as all vertices in W () t lie in the same component of G p Now we iterate Each of the vertices in W () t lies in one of the k components of W () 0 We combine the largest k 2 components to form a set W (2) of size > Cσ 2 If k 2 =, then one more growth finishes us, otherwise vol(w (2) ) < 2Cσ 2, the average size of components is at least C 2 σ 4 n and hence k 2 C 2 σ6 n We iterate, growing W (m) until either vol(q (m) t ) < ( (m + 3)ɛ)vol 2 (G) or vol(w (m) t ) > ), so that W (m) t has volume θ() and then creating W (m+) by combining the 2δvol 2 (Q (m) t 5pvol 3 (Q (m) t ) largest k m+ components to form a W (m+) with volume at least Cσ 2 n Once k m = for some m all vertices in W (m) are in the same component and one more growth round finishes the process, resulting in a giant component in G Note that the average size of a component in W n (m) has size at least C m (that is, components must grow by a factor of at least each iteration) and σ 2(m+) n σ 2 if k m >, we must have k m C mσ 2(m+) n If m = 2κ, this would imply that k m = o() by our condition σ = o(n κ ), so after at most 2κ rounds, we must have k m = and the process will halt with a giant connected component In the case where ln n > Cσ 2 n, we note that S 0 vol(s 0) ɛd of components in S t is at least C d ln n component for n large enough, and the proof goes as above C ln n ɛd, and the average volume = ω( ln n), so we can form W () by taking just one We note that throughout, if we try to expand we have that ( ( ) ) vol(q (m) t ) > ( (m + 3)ɛ)vol 2 (G) > 2κ + 4 ɛ vol 2 (G) > ( 9c ) vol 2 (G) 20 By our choice of c being sufficiently small, ( (m+3)ɛ)( δ)(+c) > at all times, so throughout, noting that vol(s i ) and vol(w (m) i ) are at least ln n, we are guaranteed our exponential growth by Lemma 3 with an error probability bounded by ( exp δ( δ)pvol ) ( 9c 2(T i )vol(s i ) δ( δ)( 20 exp )vol(s ) i) n K 0 2 We run for a constant number of phases, and run for at most a logarithmic number of steps in each growth phase as the sets grow exponentially Thus, the probability of failure is at most

12 C log(n)n K = o() for some constant C, thus completing our argument that G p contains a giant component with high probability Finally, we prove the uniqueness assertion With probability C log(n)n K there is a giant component X Let u be chosen at random; we estimate the probability that u is in a component of volume at least max(2d log n, ω(σ )) Let Y be the component of u Theorem 5 of [7] asserts that if vol(y ) max(2d log n, ω(σ )): e(x, Y ) vol(x)vol(y ) σ vol(x)vol(y ) 5d log n Note that the probability that Y is not connected to X given that vol(y ) = ω(σ ) is ( p) e(x,y ) = o(n ), so with probability o() no vertices are in such a component - proving the uniqueness of large components References N Alon, I Benjamini, and A Stacey, Percolation on finite graphs and isoperimetric inequalities, Annals of Probability, 32, no 3 (2004), M Ajtai, J Komlós and E Szemerédi, Largest random component of a k-cube, Combinatorica 2 (982), 7 3 B Bollobas, Y Kohayakawa and T Luczak, The evolution of random subgraphs of the cube, Random Structures and Algorithms 3(), 55 90, (992) 4 B Bollobas, C Borgs, J Chayes, O Riordan, Percolation on dense graph sequences, preprint 5 C Borgs, J Chayes, R van der Hofstad, G Slade, and J Spencer Random subgraphs of finite graphs I The scaling window under the triangle condition, Random Structures and Algorithms, 27 (2), (2005) 6 C Borgs, J Chayes, R van der Hofstad, G Slade, and J Spencer Random subgraphs of finite graphs III The scaling window under the triangle condition, Combinatorica, 26 (4), (2006) 7 F Chung, Spectral Graph Theory, AMS Publications, F Chung, L Lu, and V Vu, The spectra of random graphs with given expected degrees, Internet Mathematics (2004) F Chung, and L Lu, Connected components in random graphs with given expected degree sequences, Annals of Combinatorics 6, (2002), 25 45, 0 F Chung, and L Lu, Complex Graphs and Networks, AMS Publications, 2006 P Erdős and A Rényi, On Random Graphs I, Publ Math Debrecen 6, (959), A Frieze, M Krivelevich, R Martin, The emergence of a giant component of pseudo-random graphs, Random Structures and Algorithms 24, (2004), G Grimmett, Percolation, Springer, New York, H Kesten, Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics Birkhäuser Boston, Mass, (982) 5 C Malon and I Pak, Percolation on finite Cayley graphs, Lecture Notes in Comput Sci 2483, Springer, Berlin, (2002), A Nachmias, Mean-field conditions for percolation in finite graphs, preprint (2007) 7 A Nachmias and Y Peres, Critical percolation on random regular graphs, preprint (2007)