The emergence of a giant component in random subgraphs of pseudo-random graphs
aa r X i v : . [ m a t h . C O ] M a y The emergence of a giant component in random subgraphsof pseudo-random graphs
Alan Frieze ∗ Michael Krivelevich † Ryan Martin ‡ Abstract
Let G be a d -regular graph G on n vertices. Suppose that the adjacency matrix of G issuch that the eigenvalue l which is second largest in absolute value satisfies l = o ( d ). Let G p with p = αd be obtained from G by including each edge of G independently with probability p . We show that if α < whp the maximum component size of G p is O (log n ) and if α > G p contains a unique giant component of size Ω( n ), with all other componentsof size O (log n ). Pseudo-random graphs (sometimes also called quasi-random graphs) can be in-formally defined as graphs whose edge distribution resembles closely that of trulyrandom graphs on the same number of vertices and with the same edge density.Pseudo-random graphs, their constructions and properties have been a subject ofintensive study for the last fifteen years (see [2], [7], [11], [10], [12], to mention justa few).For the purposes of this paper, a pseudo-random graph is a d -regular graph G =( V, E ) with vertex set V = [ n ] = { , . . . , n } , all of whose eigenvalues but the firstone are significantly smaller than d in their absolute values. More formally, let A = A ( G ) be the adjacency matrix of G . This is an n -by- n matrix such that A ij = 1 if ( i, j ) ∈ E ( G ) and A ij = 0 otherwise. Then A is a real symmetricmatrix with non-negative values of its entries. Let λ ≥ λ ≥ · · · ≥ λ n be theeigenvalues of A , also called the eigenvalues of G . It follows from the Perron-Frobenius theorem that λ = d and | λ i | ≤ d for all 2 ≤ i ≤ n . We thus denote λ = λ ( G ) = max ≤ i ≤ n | λ i | . The reader is referred to a monograph of Chung [6] forfurther information on spectral graph theory. ∗ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15213, U.S.A. Supportedin part by NSF grant CCR-9818411. † Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported in part by USA-Israel BSF Grant99-0013, by grant 64-01 from the Israel Science Foundation and by a Bergmann Memorial Grant. ‡ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. Supported in partby NSF VIGRE grant DMS-9819950. t is known (see, e.g. [1]) that the greater is the so-called spectral gap (i.e. thedifference between d and λ ) the more tightly the distribution of the edges of G approaches that of the random graph G ( n, d/n ). We will cite relevant quantitativeresults later in the text (see Lemma 1), for now we just state informally that aspectral gap ensures pseudo-randomness.In this paper we study certain properties of a random subgraph of a pseudo-randomgraph. Given a graph G = ( V, E ) and an edge probability 0 ≤ p = p ( n ) ≤
1, the random subgraph G p is formed by choosing each edge of G independently and withprobability p . We will also need to consider the related random graph G m whoseedge set is a random m -subset of E .The most studied random graph is the so called binomial random graph G ( n, p ),formed by choosing the edges of the complete graph on n labeled vertices inde-pendently with probability p . Here rather than studying random subgraphs of oneparticular graph, we investigate the properties of random subgraphs of graphs froma wide class of regular pseudo-random graphs. As we will see, all such subgraphsviewed as probability spaces share certain common features.Our concern here is with the existence of a giant component in the case p = αd or m = αn where α = 1 is an absolute constant. These two models are sufficientlysimilar so that the results we prove in G p immediately translate to G m and vice-versa. The needed formal relations in the case where G = K n are given in [5] or[9] and they generalise easily to our case.As customary when studying random graphs, asymptotic conventions and nota-tions apply. In particular, we assume where necessary the number of vertices n ofthe base graph G to be as large as needed. Also, we say that a graph property A holds with high probability , or whp for brevity, in G p if the probability that G p has A tends to 1 as n tends to infinity. Monographs [5], [9] provide a necessarybackground and reflect the state of affairs in the theory of random graphs.For α > α < α ) of the equation xe − x = αe − α . We assume from now on that d → ∞ and l = o ( d ) . (1)These requirements are quite minimal.In analogy to the classical case G = K n , studied already by Erd˝os and R´enyi [8], Theorem 1.
Assume that (1) holds.(a) If α < then whp the maximum component size is O (log n ) .(b) If α > then whp there is a unique giant component of asymptotic size (cid:0) − ¯ αα (cid:1) n and the remaining components are of size O (log n ) . One can also prove tighter results on the size and structure of the small compo-nents. They correspond nicely to the case where G = K n . e will use the notation f ( n ) ≫ g ( n ) to mean f ( n ) /g ( n ) → ∞ with n . Similarly, f ( n ) ≪ g ( n ) means that f ( n ) /g ( n ) → Theorem 2.
Assume that (1) holds. Let ω = ω ( n ) → ∞ with n .(a) If d ≫ (log n ) then whp G p contains no isolated trees of size ζ (log n − log log n ) + ω , where ζ − = α − − log α > .(b) If d ≫ log n then whp G p contains an isolated tree of size at least ζ (log n − log log n ) − ω .(c) If d = Ω( n ) then whp G p contains ≤ ω vertices on unicyclic components.(d) Let d ≫ √ n . If α < then whp G p contains no component with k verticesand with more than k edges.(e) Let d ≫ √ n . If α > then whp G p contains no component with k = o ( n ) vertices and with more than k edges. d -regular graphs In this section we put together those properties needed to prove Theorem 1. For
B, C ⊆ V let e ( B, C ) denote the number of ordered pairs ( u, v ) such that u ∈ B , v ∈ C and { u, v } ∈ E . Lemma 1.
Suppose
B, C ⊆ V and | B | = bn and | C | = cn . Then | e ( B, C ) − bcdn | ≤ λn √ bc. This is Corollary 9.2.8 of [3]. Note that B = C is allowed here. Then e ( B, B ) istwice the number of edges of G in the graph induced by B .Now let t k denote the number of k -vertex trees that are contained in G . Lemma 2. n k k − ( d − k ) k − k ! ≤ t k ≤ n k k − d k − k !This is Lemma 2 of [4]. Let p = αd and let C k denote the number of vertices of V that are contained incomponents of size k in G p and let T k ≤ C k denote the number of vertices whichare contained in isolated trees of size k . emma 3. (a) E C k ≤ n k k − k ! α k − e − αk (1 − ξ k ) where ξ k = min (cid:26) kd , kn + λd (cid:27) . (b) For k ≪ d , E T k ≥ n k k − k ! α k − e − αk (1+ η k ) where η k = 2 kd + 2 kαd + αd . Proof (a) Let T k denote the set of trees of size k in G . Then E C k ≤ X T ∈T k kp k − (1 − p ) e T where e T = e (cid:16) V ( T ) , V ( T ) (cid:17) . Now Lemma 1 implies that e T = kd − e ( V ( T ) , V ( T )) ≥ a k def = kd − k dn − lk (2)and we also have the simple inequality e T ≥ b k def = kd − k ( k − d -regular graph.Thus, E C k ≤ kt k p k − (1 − p ) max { a k ,b k } (3)and (a) follows from Lemma 2 and some straightforward estimations.(b) Similarly, E T k ≥ X T ∈T k kp k − (1 − p ) e T + k where we crudely bound by k , the number of edges contained in V ( T ) whichmust be absent to make T an isolated tree component of G p . Now we cansimply use e T ≤ kd nd Lemma 2. We also use 1 − p ≥ e − p − p for p small and( d − k ) k − > d k − (cid:18) − kd (cid:19) k ≥ d k − exp (cid:26) − k d − k d (cid:27) , for k/d small and make some straightforward estimations. ✷ Now choose γ = γ ( α ) such that αe − α +2 αγ = 1 . (Note that αe − α < α = 1.) Lemma 4. Whp , C k = 0 for k ∈ I = h αγ log n, γn i Proof
First assume that k ≤ γd and observe that ξ k ≤ γ in this range. Thenfrom Lemma 3(a) and k ! ≥ (cid:0) ke (cid:1) k we see that γd X k = αγ log n E C k ≤ nα γd X k = αγ log n k − ( αe − α + αξ k ) k ≤ nα γd X k = αγ log n k − ( αe − α + αγ ) k = nα γd X k = αγ log n k − e − αγk ≤ γn log n ∞ X k = αγ log n e − αγk = o (1) . (4)Now assume that γd ≤ k ≤ γn and observe that (1) implies ξ k ≤ γ + o (1) in thisrange. Then γn X k = αγ log n E C k ≤ nα γn X k = αγ log n k − ( αe − α + αγ + o (1) ) k ≤ nα γn X k = αγ log n k − e − ( αγ − o (1)) k = o (1) . (5) ✷ ow let us show that there are many vertices on small isolated trees. Let f ( α ) = ∞ X k =1 k k − k ! α k − e − αk . It is known, see for example Erd˝os and R´enyi [8] that f ( α ) = ( α ≤ . ¯ αα α > . Lemma 5.
Let k = d / . Then Pr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X k =1 C k − nf ( α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ n / log n ! = o (1) . Proof
Note that kξ k = O ( d − / ) and kη k = O ( d − / ) for k ≤ k . Thus fromLemma 3(a) we have E k X k =1 C k ≤ (1 + O ( d − / )) n k X k =1 k k − k ! α k − e − αk = (1 + O ( d − / )) nf ( α ) . (6)On the other hand, Lemma 3(b) implies, E k X k =1 C k ≥ E k X k =1 T k ≥ (1 − O ( d − / )) n k X k =1 k k − k ! α k − e − αk = (1 − O ( d − / )) nf ( α ) . (7)We now use the Azuma-Hoeffding martingale tail inequality [3] to show that therandom variable Z = P k k =1 C k is sharply concentrated. We switch to the model G m , m = αn . Changing one edge can only change Z by at most 2 k and so forany t > Pr ( | Z − E Z | ≥ t ) ≤ (cid:26) − t mk (cid:27) . Putting t = n / k log n yields the lemma, in conjunction with (6), (7) and d → ∞ . ✷ The first part of Theorem 1 now follows easily. Since α < f ( α ) = 1and so by Lemma 4 and Lemma 5 whp there are at least n − n / log n vertices incomponents of size at most αγ log n . Applying Lemma 4 again, we see that whp the remaining vertices X must be in components of size at least γn . So if X = ∅ then | X | ≥ γn . But we know that whp | X | ≤ n / log n and so X = ∅ whp .For the second part of the theorem where α > whp there are ¯ αα n + O ( n / log n ) vertices on components of size ≤ αγ log n and the remainingvertices lie in large components of size at least γn . This statement remains true if e consider G m − log n . Let S , S , . . . , S s be the large components of G m − log n , where s ≤ /γ . We now show that whp adding the remaining log n random edges Y puts S , S , . . . , S s together in one giant component of size (cid:0) − ¯ αα (cid:1) n + O ( n / log n ).We also whp have ¯ αα n + O ( n / log n ) vertices on components of size ≤ αγ log n and Lemma 4 shows that this accounts for all the vertices.So let us show thatΠ = Pr ( ∃ ≤ i < j ≤ s : Y contains no edge joining S i and S j ) = o (1) , (8)completing the proof of Theorem 1. Now by Lemma 1, G contains at least (1 − o (1)) γ dn edges between S i and S j , and the probability that Y contains none ofthese is at most (cid:16) − (1 − o (1)) γ dn dn (cid:17) log n ≤ n − γ + o (1) . So Π ≤ γ − n − γ + o (1) = o (1),proving (8). ✷ Let k ± = ζ (log n − log log n ) ± ω . Let N k denote the number of tree componentsof size k in G p .Assume that k − ≤ k ≤ αγ log n . Then from the proof of Lemma 3, (notice k k − inplace of k k − , we are counting trees, not vertices on trees), E N k ≤ n k k − k ! α k − e − αk (1 − ξ k ) = (1 + o (1)) nαk / √ π e − ζ − k (9)and when k = o ( d / ) E N k = (1 + o (1)) nαk / √ π e − ζ − k (10)(a) Let γ be as in Lemma 4. Using (9), αγ log n X k = k + E N k = O αγ log n X k = k + e − ζ − ( ω + k − k + ) = o (1)and part (a) will follow once we verify that when α >
1, the giant component isnot a tree. However, the number of edges in the giant is asymptotically αn − nα ∞ X k =1 ( k − k k − k ! ( αe − α ) k = αn − α ∞ X k =1 ( k − k k − k ! ( ¯ αe − ¯ α ) k ! = αn (cid:18) − ¯ α α (cid:19) . ote that n ¯ α ∞ X k =1 ( k − k k − k ! ( ¯ αe − ¯ α ) k = ¯ α n which can be seen from the fact that the LHS is asymptotically equal to theexpected number of edges of G n, ¯ αn which lie on trees. So, the ratio of edges tovertices for the giant is asymptotically equal to α +¯ a > k = k − . Then from (10), E N k = Ω( e ζ − ω ) → ∞ . Bounding the number of G -edges inside and between two disjoint subtrees by 3 k we estimate E N k ≤ t k p k − (1 − p ) dk − k = (1 + o (1))( E N k ) and (b) follows from the Chebychev inequality.(c) Let U k denote the number of isolated unicyclic components in G p of size k .Then n X k =3 E ( kU k ) ≤ n X k =3 kt k (cid:18) k (cid:19) p k (1 − p ) dk − k ≤ (1 + o (1)) n d n X k =3 k k +1 k ! ( αe − α + o (1) ) k ≤ (1 + o (1)) n d n X k =3 k / √ π ( αe − α + o (1) ) k = O (1)since we are assuming that d = Ω( n ) here. Part (c) follows from the Markovinequality.(d), (e) Let COM P k denote the number of components with k vertices and atleast k + 1 edges. We can restrict our attention to 4 ≤ k ≤ γn since if α < whp . Then, as in (3), E γn X k =4 COM P k ≤ γn X k =4 t k (cid:18) k (cid:19) p k +1 (1 − p ) max { a k ,b k } ≤ (1 + o (1)) nα √ πd γn X k =4 k / ( αe − α + αγ + o (1) ) k . ≤ (1 + o (1)) nα √ πd γn X k =4 k / e − ( αγ − o (1)) k = o (1) ince n/d → ✷ References [1] N. Alon and F. R. K. Chung, Explicit construction of linear sized tolerantnetworks,
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