Vertex Percolation on Expander Graphs

V ertex P ercolation on Expander Graphs Sonn y Ben-Shimon ∗ Mic hael Kriv elevic h † No v ember 6, 2018 Abstract W e sa y that a graph G = ( V , E ) on n vertices is a β -exp an der for some constant β > 0 if every U ⊆ V of cardinali ty | U | ≤ n 2 satisfies | N G ( U ) | ≥ β | U | where N G ( U ) denotes the n eig hb orhoo d of U . In this w ork we explore the process of deleting vertices of a β - expander indep endently at random with probability n − α for some constan t α > 0, and study the prop erties of the resulting graph. Our main result states that as n tends to infinity , the d eletio n pro cess p erfor med on a β -expander graph of bound ed degree will result with high probability in a graph comp osed of a giant comp onen t con taining n − o ( n ) vertices that is in itself an exp ander graph, and constan t si ze components. W e pro ceed by applying the main result to ex pander graphs with a p ositiv e sp ectral gap. In the p articular case of ( n, d, λ )-graphs, that are such expanders, we compute the values of α , un der add itio n al constraints on the graph, for which with high probabilit y the resulting graph will stay connected, or will be comp osed of a gian t comp onent and isolated vertice s. As a graph sampled from th e un iform probability space of d - regular graphs with high probabilit y is an expander and meets the add itio n al constraints, t h is result strength ens a recent result due to Greenhill, Holt and W ormald ab out vertex p ercolatio n on random d -regular graphs. W e conclude by sh owing that performing the ab o ve describ ed deletion process on graphs that expand sub -linear sets by an unbound ed expansion ratio, with high probability results in a conn ected expander graph. 1 In tro duction In this pap er we analyze the pro cess o f deleting vertices indep enden tly at random from an expander graph and describe t ypical prop erties and the structure of the resulting gra ph. W e focus on the case where the initial graph, G , is of b ounded degree and the deletion pro ba bilit y equals n − α , for any fix e d α > 0, wher e n denotes the n umber of v ertices in G . W e are mainly in ter ested in in vestigating when the resulting gr aph with high proba bilit y will p ossess some expansion pro p erties as will b e discussed in Section 1 .3. In a r e cen t pap er of Greenhill, Holt and W ormald [9], the a uthors p erform a very simila r a nalysis wher e the initial g raph is sampled from the uniform probability space of all d -regula r graphs for some fixed d ≥ 3. Our current result, genera lizing and impro ving [9], can be in terpreted a s pr o viding sufficient deterministic conditions o n ∗ Sc ho ol of Computer Science, Ra ymond and Bev erl y Sac kler F aculty of Exact Sciences, T el Aviv University , T el Aviv 69978, Israel. E-mail: s onn y@post.tau.ac.il. Research conducted as part of the author’s Ph.D. thesis under the supervi sion of Prof. Michae l Krive l ev i c h. † Sc ho ol of Mathematical Sciences, Raymond and Bev erly Sackler F acult y of Exact Sciences, T el Aviv Universit y , T el Aviv 69978, Israel. E-mail: krivelev@post.tau.ac.il. Researc h supp orted in part by USA- Israel BSF Grant 2006-322, and by gr an t 526/05 fr om the Israel Science F oundation. 1 the initial graph that imply the result of [9]. W e a re also able to prove s ome results when the initial graph has an unbounded ex pansion ratio, and apply it to the case of ra ndom d - r egular gr aphs when d = o ( √ n ). 1.1 Notation Given a graph G = ( V , E ), the neighb orho o d N G ( U ) of a subset U ⊆ V of vertices is the set of vertices defined by N G ( U ) = { v / ∈ U : v has a neighbor in U } . F or any f :  ⌊ n 2 ⌋  → R + , we say that a gr aph G = ( V , E ) on n vertices is an f - exp ander if ev er y U ⊆ V of cardinality | U | ≤ n 2 satisfies: | N G ( U ) | ≥ f ( | U | ) · | U | . When f is a consta n t function equal to some β > 0 we say that G is a β - exp ander . When a function f : A → R + satisfies: f ( a ) ≥ c for a n y a ∈ A , where c ≥ 0 is a constant, we simply write f ≥ c . Expanders in g eneral are highly-co nnected spa r se graphs. T he r e are many other notions a nd definitions of expander graphs different from the one descr ib ed above, s o me of which w ill be addressed in the coming sections. Expander gr a phs is a sub ject of utmost importance to the fields of both applied and theo retical Computer Science, Co m binatorics , Pr obabilit y Theory etc. Mo nograph [11] pro vides an excelle n t survey on expander gr aphs and their applications. In our setting, w e start with a graph G = ( V , E ) on n vertices. W e delete every vertex o f V with probability p = n − α for some fixed α > 0 indep endent ly at random. T o simplify no tation, from here on, we will denote the resulting g r aph of this pro cess b y b G = ( b V , b E ), and for e very X ⊆ V , we denote by b X = X ∩ b V the subset o f X that was not deleted by the deletion pro cess. W e denote by b n the ca rdinalit y of b V , by R the set of dele ted vertices, i.e. b G = G [ V \ R ], and b y r its car dinalit y , i.e. b n = n − r . When consider ing the neighborho o d in the graph b G of a subset of vertices U ⊆ b V w e denote it b y N b G ( U ). The main re s earc h interest o f this pa p er is the asy mpto tic b eha vio r of prop erties of the graph b G a s we let the n umber of vertices, n , gro w to infinit y . In this context, o ne needs to be prec ise when formulating such claims. When stating an asymptotic claim for every graph G on n vertices that sa tisfies a set of prop erties P n (the prop erties may depend on n ), one actually means that for every family o f gr a phs G = { G n } , such that G n is a gra ph on n v ertice s sa tis fying P n , there exis ts a v a lue n 0 such that the claim is co rrect for every G n where n > n 0 . W e say that an even t A in o ur probability space o ccurs with high probability (or w.h.p. for brevity) if Pr [ A ] → 1 a s n goes to infinity . Therefore, from now on and throughout the rest of this work, w e will alwa y s a ssume n to b e la rge enough. W e use the usual as ymptotic notation. That is, for tw o functions of n , f ( n ) a nd g ( n ), we denote f = O ( g ) if there exists a cons ta n t C > 0 such that f ( n ) ≤ C · g ( n ) for la rge enoug h v a lues of n ; f = o ( g ) or f ≪ g if f /g → 0 as n go es to infinit y; f = Ω( g ) if g = O ( f ); f = ω ( g ) or f ≫ g if g = o ( f ); f = Θ( g ) if b oth f = O ( g ) a nd f = Ω( g ). 1.2 Motiv ation Let G n,d denote the random gr aph model consisting o f the unifor m distribution o f all d -reg ular graphs on n labeled vertices (where dn is even). One of the motiv ations of this pa per is the fo llowing r esult, r ecen tly prov ed by Gree nhill, Holt and W ormald in [9 ]. Theorem 1.1 (Greenhill, Holt and W or mald [9]) . F or every fi x e d α > 0 and fixe d d ≥ 3 ther e exists a c onstant β > 0 , such t hat if p = n − α and G is a gr aph sample d fr om G n,d , t hen w.h.p. b G h as a c onne cte d c omp onent of size n − o ( n ) that is a β -ex p ander and al l other c omp onents ar e of b oun de d size. Mor e over, 2 1. if α > 1 2( d − 1) , w.h. p. all smal l c onne cte d c omp onents of b G ar e isolate d vertic es. 2. if α ≥ 1 d − 1 , w.h. p. b G is c onne cte d. Theorem 1.1 sugges ts a few questions that may b e of interest to a ddress. First, o ne might consider the ques tion whether the deletion probability p for which the desired pro perties hold is best po ssible. This question has be e n answered in [9]. Simple pro babilistic arguments show that the ab o ve result is indeed optimal in the sense that if we le t α = o (1), the largest comp onent of b G will c on tain w.h.p. man y induced paths of length O (1 /α ), and hence cannot b e an expander . N e x t, one may ask what ar e the proper ties o f random d -regular graphs that make the ab ov e claim true. One of the research motiv a tions of this pap er is precisely that, as will be described below. Mor eo ver, Item 2 of Theor e m 1.1 do es not seem to b e optimal due to the following argument. As random d - regular graphs (for co nstan t v alues of d ) w.h.p. lo cally lo ok like trees (i.e. there are very few cycles of cons tan t leng th) it would seem na tur al to think that to disco nnect such a graph one w ould need to find the deletion probability that is “just enough” to disconnect a single vertex. A simple first mo men t argument would imply that α > 1 d should suffice. In Section 3.2 we confirm this hyp o thesis in the mo r e g eneral s etting of pseudo-rando m ( n, d, λ )-graphs. Lastly , Theorem 1.1 do es not consider the case of sampling a random d -reg ular graph when d = ω (1), i.e. d goes to infinity with n . This setting is addr essed in Section 4. 1.3 Main result The main result o f this paper states that the deletion of vertices of an expander gr aph G independently at random with probabilit y n − α w.h.p. results in b G con taining one giant comp onen ts that is in itself an expander gr aph. Moreov er, the expa nsion pro perties of G imply a b ound o n the sizes of the small co nnected comp onen ts of b G . Theorem 1.2. F or every fix e d α, c > 0 and fixe d ∆ > 0 , ther e exists a c onstant β > 0 , such that if G is an f -exp ander gr aph on n vertic es of b ounde d maximum de gr e e ∆ , and f ≥ c , then w. h.p. b G h as a c onne cte d c omp onent of size n − o ( n ) that is a β - ex p ander, and the r est of its c onne cte d c omp onents have at most K − 1 vertic es, wher e K = min  u : ∀ k ≥ u k f ( k ) > 1 α  . (1) W e note that K is w ell defined as f ≥ c implies that K < 1 cα . As mentioned in Section 1 .2, Theorem 1.2 is optimal with r espect to the deletion pr obabilit y if we require the giant compo nen t of the r esulting graph to p ossess expansion prop erties. It is well known that for fixed d ≥ 3 random d -regular gr aphs are w.h.p. expander graphs. Th us our result strengthens Theor em 1.1, as will b e formalized in Section 3.3. It should b e stress e d that the techniques used in the present pa p er and in [9] are quite different. Whereas in [9] the analysis is done directly in the so called Configuratio n Mo del in a pr obabilistic setting, we r ely up on a deterministic prop erty of a gra ph, namely , being an expander. The approach of first proving so me claim under deter ministic as sumptions, a nd then showing tha t these c o nditions app ear w.h.p. in so me pro babilit y space, allows us to, a rguably , simplify the pr oof, a nd to get a strengthened r esult for families of pseudo-r andom graphs and the random d -r e gular graph. 3 1.4 Related w or k The pro cess of random deletio n of vertices of a g raph received r a ther limited attention, mainly in the co n text of fault y storag e (see e.g. [2]), communication netw orks , and distributed computing. F or instance, the main motiv ation of [9] is the SW AN p eer-to-p eer [10] netw or k who s e topo logy p ossess some prop erties o f d -regula r graphs, a nd may ha ve fault y nodes. O ther w or ks a re mainly interested in connectivity and r outing in the resulting graph after pe r forming (p ossibly adversarial) vertex deletions on some presc r ibed g raph topo logies. The pr ocess of deleting edges, sometimes referred to by e dge-p er c olation (or b ond-p er c olation ) has b een more extensively studied. The main in teres t o f edge- percola tion is the ex istence o f a “ gian t comp onent”, i.e. a connected co mponent consisting of a linear size of the v ertice s , in the res ulting graph. When the initial graph is taken to b e K n , edge-p ercolatio n b ecomes the famous G ( n, p ) random gra ph model. In [1, 8, 14] the edge p ercolatio n on an expander graph is considered, the authors determine the thr e shold of the deletion probability at whic h the giant co mp onent emerges w.h.p.. It s ho uld b e no ted that in the context of this pap er the ex p ected n umber of deleted vertices is far low er than p ermissable in order to retain a giant co mponent in the gr aph, as is clearly seen in Lemma 2.2. 1.5 Organization of pap er The rest of the paper is orga niz e d as follows. In Section 2 w e give a pr oof of Theorem 1.2. W e pro ceed in Section 3 to a straightforw ar d applica tion of our result to expa nder graphs arising fro m constraints on the sp ectrum of the graph. W e c o n tinue in Section 3.2 to the particular case of ( n, d, λ )-gr aphs, and under additional constraints on the graph compute the v a lues o f α for whic h the resulting graph will w.h.p. stay connected o r will b e comp osed of a giant component and iso lated v er tice s . In Section 3.3 we sho w tha t a graph sampled fro m the uniform pro babilit y space o f d -regular graphs s atisfies all constraints, providing an alternative pro of o f the main result of [9] and ev e n improving it. As a fina l res ult, in Section 4 we a nalyze the ca se o f graphs of unbo unded expansion ra tio fo r sub-linear sets with the sa me deletion proba bilit y , and extend our result to ra ndom d -r egular graphs where 1 ≪ d ≪ √ n . W e c o nclude in Sec tion 5 with a short summary and op en problems fo r further research. 2 Pro of of Theorem 1.2 Let G b e an f -expander, where f ≥ c for some constant c > 0. The num b er of deleted v ertice s , r , is clearly distributed b y r ∼ B ( n, p ), and hence by the Chernoff bound (see e.g. [3]) r is hig hly concentrated aro und its exp ectation. Claim 2. 1. W .h .p. (1 − o (1)) n 1 − α ≤ r ≤ (1 + o (1)) n 1 − α . As for α > 1 w.h.p. no vertices are deleted fro m the g raph G and the pro of of Theorem 1.2 beco mes trivial, w e will assume from now on that α ≤ 1. Denote b y b V 1 , . . . , b V s the par titio n of b V to its connected comp onen ts ordered in desce nding order of ca rdinalit y . W e call b V 1 the big co mponent of b G , and b V 2 , . . . , b V s the smal l comp onen ts of b G . Lemma 2. 2. W. h.p. | b V 1 | ≥ (1 − C n − α ) n for any C > 1+ c c . 4 Pr o of. Fir st, w e show that | b V 1 | > n 2 . Assume other wise, a nd tak e c W = S j i =1 b V i for some j ∈ [ s ] suc h that n 4 ≤ | c W | ≤ n 2 . Suc h a j surely exists. By our condition on f , w e hav e that | N G ( c W ) | ≥ c | c W | = Θ( n ). But surely , N G ( c W ) ⊆ R , and hence, by Claim 2.1 | N G ( c W ) | = o ( n ), a contradiction. Now, set b U = b V \ b V 1 . F r om the ab o ve, it follows that | b U | < n 2 . Clearly , N G ( b U ) ⊆ R , and | N G ( b U ) | ≥ c | b U | . P utting these together yields that | b U | ≤ | R | c , and hence, b y Claim 2.1, w.h.p. | V \ b V 1 | = | R ∪ b U | ≤ (1 + o (1)) 1+ c c n 1 − α , completing the pro of. In a graph H , w e call a s ubs et of vertices U ⊆ V ( H ) c onne cte d if the cor responding spanning subgraph H [ U ] is connected. The following well known lemma (see e.g. [13, Exercise 11, p.396]) helps us to b ound the num b er of connected subsets of vertices in a graph of b o unded maxim um deg ree. Lemma 2.3. If H = ( V , E ) is a gr aph of maximum de gr e e D , then V c ontains at most | V | ( De ) k k c onne cte d subsets of c ar dinality k . Keeping in mind tha t ∆ is a constant, L e mma 2.3 turns out to b e quite cr ucial to our forthcoming calculations, for it allows us to bound probability of ev ents using union b ound arguments b y summing ov er connected s ubgraphs of a prescr ibed c ardinalit y instead of summing ov er all subgra phs of the resp ectiv e cardinality . W e cont inue by showing that w.h.p. all small connected co mponents of b G must span less than K v ertices . Lemma 2. 4. W. h.p. every smal l c onn e cte d c omp onent of b G is of c ar dinality at most K − 1 . Pr o of. B y Lemma 2.2, every s mall connected comp onen t of b G will be of cardinality at most ¯ u = O ( n 1 − α ). Let U ⊆ V b e a subset of vertices of cardinality u ≤ ¯ u , a nd let W = N G ( U ) b e its neighbor hoo d in G , where | W | = w . W e first b ound the probabilit y tha t b U is a small connected compo nen t o f b G by the pr obabilit y that all of W was de le ted, Pr h ∃ j > 1 s.t. b U = b V j i ≤ p w ≤ n − αf ( u ) u . By Lemma 2.3 w e know that there a re at most n ( e ∆) u u connected s panning subgraphs of car dinalit y u , th us w e ca n b ound the pro babilit y of app earance of a sma ll co nnected compo ne nt of cardina lit y u . Pr h ∃ j > 1 s.t. | b V j | = u i ≤ n ( e ∆) u u · n − αf ( u ) u . (2) Setting ε = min { k f ( k ) − 1 α : k ≥ K } , the definition of K implies ε is a p ositive constant . Applying (2) and summing over all p ossible v a lues of u , we can b ound the probability there will be in b G a small connected comp onen t of ca rdinalit y at lea st K . First assume K < ⌈ 2 cα ⌉ . Pr h ∃ j > 1 s.t. | b V j | ≥ K i ≤ ¯ u X u = K n ( e ∆) u u · n − αf ( u ) u ≤ ⌈ 2 cα ⌉− 1 X u = K n 1 − αuf ( u )+ o (1) + ¯ u X u = ⌈ 2 cα ⌉ n 1 − u ( cα − o (1)) ≤  2 cα  − K  n − αε + o (1) + O  n 2 − α − ⌈ 2 cα ⌉ ( cα − o (1))  = o (1) . Finally , if K ≥ ⌈ 2 cα ⌉ the above co mputatio n b ecomes simpler a s we are left with only the second summand in the sec o nd line and the statement holds in that cas e as well. 5 Having s hown that the small connected comp onen ts of b G a re w.h.p. of b ounded size, we mov e on to show that larg er connected subs ets o f b G w.h.p. expand. Lemma 2.5. W.h.p. every c onne ct e d subset U ⊆ b V of b G of c ar dinality u s.t. K ≤ u ≤ b n 2 satisfies | N b G ( U ) | ≥ αc 4 u . Pr o of. Simila rly to the proof of Lemma 2.4 let W = N G ( U ) and c W = N b G ( U ) b e the neigh b orho ods of U in G and b G , resp ectively , and let w and b w denote their resp ectiv e cardinalities . By Lemma 2.4 w e hav e that w.h.p. every connected subset of v er tices U of cardinality K ≤ u ≤ 4 αc is not disconnected from the graph, and th us has a t leas t one edge leaving it. Setting η = αc 4 this implies that for every such connected subset U , b w ≥ η u . Assuming 4 αc < u ≤ b n 2 , relying on b w ∼ B ( w, 1 − p ) we ha ve Pr [ b w < η u ] ≤  w ⌊ η u ⌋  · p w − ⌊ ηu ⌋ ≤  ew η u  ηu p uf ( u ) − η u ≤  e ∆ η  ηu n − αu ( f ( u ) − η ) . T o b ound the pr obabilit y there exis ts a c onnected subset in b G of cardinality u whose neighborho o d contains less than η u vertices, we a pply the ab o ve with the union bound on all connected s ubs e ts of G from Lemma 2.3 as follows. n ( e ∆) u u ·  e ∆ η  ηu · n − αu ( f ( u ) − η ) ≤ η − ηu · ( e ∆) u ( η +1) · n 1 − αu ( f ( u ) − η ) ≤  ( e ∆) η +1 n αc/ 4 η η  u · n − 1 = o ( n − 1 ) . The inequality from the fir st to the seco nd line follo ws fr om the fact that 1 − αu ( f ( u ) − αc 4 ) ≤ − (1 + αuc 4 ), or equiv alently αuf ( u ) − αuc 4 (1 + α ) ≥ αuc 2 ≥ 2 using that α ≤ 1 and u > 4 αc . Summing over all p ossible v a lues of u implies that w.h.p. there is no connected subset U in b G o f car dinalit y a t leas t 4 αc that satisfies | N b G ( U ) | < η | U | , completing the pro of. Lemma 2.5 states that w.h.p. all co nnected subsets of G [ b V 1 ] expa nd. As G is of b ounded maximum degree, this is sufficien t to imply that w.h.p. all subsets o f G [ b V 1 ] expand. Lemma 2. 6. W. h.p. G [ b V 1 ] is a β -exp ander, wher e β = 1 ∆ · min { 1 K , αc 4 } . Pr o of. Set η = αc 4 as defined in Lemma 2 .5 , a nd γ = min { 1 K , η } . F or every U ⊆ b V 1 of cardinality | U | = u ≤ K ≤ 1 /γ , tr ivially | N b G ( U ) | ≥ 1 ≥ γ u , as U ha s at least o ne edge emitting out of it. Assume u > K and and denote b y U 1 , . . . , U t the deco mposition of U to its connected subsets, and by u 1 , . . . , u t their res pective cardinalities. As every connected subset U i satisfies w.h.p. | N b G ( U i ) | ≥ γ u i by L emma 2.5, it follo ws tha t w.h.p. | N b G ( U ) | ≥ γ ∆ u completing the pro of. Combining Lemmata 2.2, 2.4, and 2.6 co mpletes the proo f of Theor em 1.2 3 Applications to differen t expander graph famili es 3.1 Expansion via the sp ectrum of a graph The adjac ency matrix of a g raph G on n vertices labeled by { 1 , . . . , n } , is the n × n binary matr ix , A ( G ), where A ( G ) ij = 1 iff i ∼ j . The c ombinatorial L aplacian of G is the n × n ma trix L ( G ) = D − A ( G ) wher e D 6 is the dia gonal ma trix de fined b y D i,i = d G ( i ). It is well known that for every graph G , the ma tr ix L ( G ) is po sitiv e semi-definite (see e.g. [6]), and hence has an o rthonormal bas is o f eigenvectors and a ll its eig en v alues are non-negative. W e denote the eigenv a lues of L ( G ) in the ascending order b y 0 = σ 0 ≤ σ 1 . . . ≤ σ n − 1 , where σ 0 corres p onds to the eigenvector of all ones. W e deno te by e d = e d ( G ) the av erag e degr ee of G , a nd let θ = θ ( G ) = max { | e d − σ i | : i > 0 } . The celebrated expander mixing lemma (see e.g. [3]) and its generaliza tion to the non-regula r case (see e.g. [6 ]) state roughly that the smaller θ is, the more ra ndom-lik e is the graph. This easily implies several corollarie s on the distribution of edges in the graph. In particular, one can deduce the following e xpansion prop erty of G in terms of e d a nd θ . Its full pro of can b e found in [6]. Prop osition 3.1. L et G b e an gr aph on n vertic es. Then G is an h n, e d,θ -exp ander, wher e h n, e d,θ ( i ) = e d 2 − θ 2 θ 2 + e d 2 i n − i for 1 ≤ i ≤ j n 2 k . (3) Assume G is a gr aph of b ounded maximum degree, implying e d = O (1), and let H ( i ) = i · h n, e d,θ ( i ). Straightforw a rd ana lysis implies H ( i ) is monotonically increasing for 1 ≤ i ≤ j nθ e d + θ k , and mono tonically decreasing fo r l nθ e d + θ m ≤ i ≤  n 2  . When e d − θ > ε for s ome ε > 0, w e ha ve that h n,d,θ ≥ c , where c = e d 2 − θ 2 e d 2 + θ 2 is a constant depending on e d and θ . W e note that H (  n 2  ) = O ( n ) ≫ 1 / α . Setting k = θ 2 ( e d 2 − θ 2 ) α + 1, we hav e that k ≤ j nθ e d + θ k and H ( k ) = θ 2 ( e d 2 − θ 2 ) α + 1 ! · e d 2 − θ 2 θ 2 + o (1 ) ! > 1 α . Our analysis of H ( i ) implies that the v alue K defined in (1) sa tis fie s K ≤ k . Prop osition 3.1 th us enables us to apply Theo rem 1.2 to such graphs. Theorem 3.2. F or every fixe d α, ε > 0 and fixe d ∆ ≥ 0 ther e exists a c onstant β > 0 , such tha t if G is a gr aph on n vertic es of maximum de gr e e ∆ , and e d − θ > ε , then w.h. p. b G has a c onne cte d c omp onent of size n − o ( n ) that is a β -exp ander, and all o t he r c omp onents ar e of c ar dinality at most θ 2 ( e d 2 − θ 2 ) α . 3.2 ( n, d, λ ) -graphs When the gr aph G is d -r egular, L ( G ) = dI − A ( G ), and hence if λ 0 ≥ λ 1 ≥ . . . ≥ λ n − 1 is the sp ectrum of A ( G ) we hav e that λ i = d − σ i . In the c a se of d -r egular g raphs it is customary and, ar guably , more natural to use the sp ectrum of A ( G ) r ather than o f L ( G ) to a ddress expansion prop erties of the gr aph. As the la rgest eigenv alue of A is clearly λ 0 = d and it is maxima l in absolute v alue, w e hav e that θ ( G ) = max { | λ 1 ( G ) | , | λ n − 1 ( G ) |} . In the case o f d -regular graphs it is cus tomary to deno te θ ( G ) by λ ( G ) = λ , and to call such a graph G an ( n, d, λ )-graph. F o r an extensive survey of fascinating prop erties of ( n, d, λ )-graphs the reader is referr ed to [12]. In the case of ( n, d, λ )-gr aphs Pr o position 3.1 and Theo rem 3 .2 translate to the following. Prop osition 3.3. L et G b e an ( n, d, λ ) - gr aph, then G is an h n,d,λ -exp ander, wher e h n,d,λ (1) = d ; and h n,d,λ ( i ) = d 2 − λ 2 λ 2 + d 2 i n − i for 2 ≤ i ≤ j n 2 k . (4) 7 Theorem 3. 4. F or every fixe d α, ε > 0 and fixe d d ≥ 3 ther e exists a c onst ant β > 0 , su ch that if G is an ( n, d, λ ) - gr aph wher e d − λ > ε , then w.h.p . b G has a c onne ct e d c omp onent of size n − o ( n ) t ha t is a β -exp ander, and al l other c omp onents ar e of c ar dinality at most λ 2 ( d 2 − λ 2 ) α . The next tw o prop ositions allow us to g et improved bounds on the sizes of the sma ll connected comp onen ts of b G . In Prop osition 3 .5 w e a re interested in the v alues of α for which b G is w.h.p. co nnected, and in Prop osition 3.6 in the v alues for which w.h.p. the small connected co mponents of b G are all isola ted vertices. W e compute these v alues of α under so me additional assumptions on the ( n, d, λ )-graph. Sp ecifically , we require the graph to b e lo cally “spa rse” and the sp e ctr al gap , i.e. d − λ , to b e relatively large. Although these constraints ma y seem somewhat a rtificial, they ar ise na turally in the s etting of random d -regular graphs a s will b e exp osed in Section 3.3. F or any graph G = ( V , E ) we deno te b y ρ ( G, M ) = max  e ( U ) | U | : U ⊆ V s.t. | U | ≤ M  , where e ( U ) denotes the num b e r o f edges of G that have b oth endp oin ts in U . Prop osition 3.5. F or every α > 1 d and fixe d d ≥ 3 , if G is an ( n, d, λ ) -gr aph satisf ying λ ≤ 2 √ d − 1 + 1 40 and ρ ( G, d + 29) ≤ 1 , then w.h.p. b G is c onne cte d. Pr o of. W e prov e that suc h a graph G is a n f -ex pa nder wher e if ( i ) ≥ d for ev e r y 1 ≤ i ≤ ⌊ n 2 ⌋ . Lemma 2 .4 will then imply tha t K = 1, and hence b G is w.h.p. connected. Prop osition 3.3 guarantees that f ( i ) ≥ h n,d,λ ( i ) (with h n,d,λ as defined in (4)). T aking i ≥ 30 and plugging our assumption on λ in the definition of h n,d,λ , we hav e tha t for every d ≥ 3 f (30) ≥ d 2 − 4 ( d − 1 ) − √ d − 1 10 − 1 1600 4( d − 1) + √ d − 1 10 + 1 1600 + d 2 30 n − 30 > d 30 . The analysis of H therefore guarantees that if ( i ) ≥ d for a ll i ≥ 30. Now, let U b e a subset o f vertices of ca r dinalit y u ≤ 29, and set s = | N G ( U ) | and w = | U ∪ N G ( U ) | . It now suffices to show that s ≥ d for suc h a set U . If u = 1, trivially s = d , as ev ery v er tex has d neighbors . Now, our assumption on G implies that all triangles in the g r aph must b e edge disjo in t. T aking u = 2 , if the t wo v er tices in U ar e non- adjacent trivially s ≥ d , and if they ar e adjacent, they must have at most one common neigh b or, implying s ≥ 2 d − 3 ≥ d . T a king 3 ≤ u ≤ 29, if w ≥ u + d w e are done. Otherwis e, the assumption on G implies e ( U ) ≤ u and e ( U ∪ N G ( U )) ≤ w , a nd hence du − u ≤ du − e ( U ) ≤ e ( U ∪ N G ( U )) ≤ w = u + s . This implies s ≥ u ( d − 2 ) ≥ d which completes the pro of. F o r any graph G w e denote by t ( G ) the num b er of triangles in G . T o analyze the v a lues of α for which w.h.p. all small connected components of b G are isolated vertices, w e additiona lly r equire that the num b er of triangles in G is b ounded b y a cer tain p ositive p ow er of n . This requirement as w ell is quite na tural in the case o f ra ndo m d -r e gular g raphs. Prop osition 3. 6. F or every α > 1 2( d − 1) and fixe d d ≥ 3 , if G is an ( n, d, λ ) -gr aph satisfying λ ≤ 2 √ d − 1 + 1 40 , ρ ( G, 39 + 2( d − 1)) ≤ 1 , and t ( G ) = O ( n 2 d − 3 2( d − 1) ) , then w. h.p. al l smal l c onne cte d c omp onents of b G ar e isolate d vertic es. Pr o of. F ollowing the spirit of the pro of of P ropo s ition 3.5, we would like to show that G is an f -expander where if ( i ) ≥ 2 ( d − 1) for every 2 ≤ i ≤ ⌊ n 2 ⌋ , which completes the pro of b y using Lemma 2.4. B y P ropos itio n 8 3.3 we c an a ssume tha t f ( i ) ≥ h n,d,λ ( i ). Our assumption on λ giv es f (40) ≥ d 2 − 4 ( d − 1) − √ d − 1 10 − 1 1600 4( d − 1) + √ d − 1 10 + 1 1600 + d 2 40 n − 40 > d − 1 20 , which in turn, using our ana lysis of H , implies if ( i ) ≥ 2 ( d − 1) for i ≥ 40 . When trying to complete the pro of b y sho wing tha t f ( i ) ≥ 2 d − 1 i for 2 ≤ i ≤ 39, it turns out that in this setting this will no t be the case, as there can be small subse ts tha t viola te this strict expansio n require ment. F ortunately , we ca n prove tha t there cannot be to o many s uch subsets, which allows us to pr o ve the above probabilistic sta temen t. Let U b e a subs e t of vertices of cardinality 2 ≤ u ≤ 39, a nd set s = | N G ( U ) | and w = | U ∪ N G ( U ) | . W e call U exc eptional if s < 2 ( d − 1 ). Let x i denote the num b er of exc e ptional s ets o f cardinality i . If 4 ≤ u ≤ 39, our assumption o n G implies e ( U ) ≤ u and e ( U ∪ N G ( U )) ≤ w . It follows that du − u ≤ e ( U ∪ N G ( U )) ≤ w = u + s , implying s ≥ u ( d − 2 ) ≥ 2( d − 1). If u = 2 and the tw o v e r tices of U are non-a dja c e n t then 2 d ≤ e ( U ∪ N G ( U )) ≤ w = 2 + s , hence s ≥ 2( d − 1). If the tw o vertices a r e adjacent, but are no t part of a tria ngle, then again s ≥ 2( d − 1). F or u = 3 , if U spans at most one edg e, then 3 d − 1 ≤ e ( U ∪ N G ( U )) ≤ w = 3 + s , yielding s ≥ 2( d − 1). If U spa ns a triang le and d ≥ 4, as all triangles of G must b e edge disjoin t w e get that s ≥ 3( d − 2) ≥ 2 ( d − 1 ). If U spans ex a ctly tw o edges, easy ca s e analysis, relying on the fact that no small subgr aph spans more edg es then vertices, sho ws that s ≥ 3 d − 5 ≥ 2( d − 1). Lemma 2.4 a nd the previous co mputation a ssure that for d ≥ 4 w.h.p. all s ma ll connected comp onen ts hav e a t most tw o vertices, and for d = 3, w.h.p. all small connected co mponents hav e at most three v er tice s . W e conclude by showing that since in b oth cas es ther e ar e o nly a small num b er of exceptional sets, w.h.p. all sma ll connected comp onen ts will b e isola ted vertices. Similarly to the pro of of Lemma 2.4, we b ound the probability of app earance o f a connected comp onen t of cardinality 2. The exceptional sets o f car dinalit y 2 are edges that participate in a triang le and a ll triang les in G are edge disjoint, therefore ther e are x 2 = 3 t ( G ) such exceptio na l sets, and each has e x actly 2 d − 3 neighbors . Go ing over all connected sets o f G of ca r dinalit y 2, i.e. the edges o f G , we b ound the pro ba bilit y that one of thes e sets bec o mes dis connected. Pr h ∃ j > 1 s.t. | b V j | = 2 i ≤ x 2 p 2 d − 3 +  dn 2 − x 2  p 2( d − 1) ≤ O  n 2 d − 3 2( d − 1) − α (2 d − 3)  + O  n 1 − 2 α ( d − 1)  = o (1) . The a bov e completes the proo f when d ≥ 4. W e are left with the case of exceptiona l triples tha t may exist when d = 3. Since the exceptiona l sets of cardinality 3 ar e the triangles in G , there are exactly x 3 = t ( G ) such exceptional sets ea ch having exactly 3 neighbo rs. V ery similarly to the prec e ding computation, we go o ver all connected sets of G of cardina lit y 3, i.e. sets that spa n tw o or three edges, and compute the probability that one of these s e ts b ecome disconnec ted. Recall that for d = 3 we hav e that α > 1 4 . Pr h ∃ j > 1 s.t. | b V j | = 3 i ≤ x 3 p 3 + (3 n − 3 t ( G )) p 4 ≤ O  n 3 4 − 3 α  + O  n 1 − 4 α  = o (1) . Prop ositions 3 .5 and 3.6 are easily se e n to b e optimal in some sense, for if α ≤ 1 d or α ≤ 1 2( d − 1) , then the exp ected n umber o f isolated vertices or edges resp ectiv ely is g r eater tha n 1 . 9 3.3 Random d -regular graphs Consider the random gra ph mo del co nsisting of the uniform dis tribution on all d -reg ular gr aphs on n vertices (where dn is even), a nd denote this probabilit y space by G n,d . Assume thr oughout this s e c tion that d ≥ 3 is a constant. Let G be a gr aph sampled from G n,d . Note that the multiplicit y of the eigen v alue d of the graph G is w.h.p. 1 as G is w.h.p. connected and non-bipartite (see e.g. [15]), hence w.h.p. λ ( G ) < d . F r iedman, c onfirming a conjecture of Alon, g iv es an ac c ur ate ev aluation o f λ ( G ) for most random d -reg ular graphs when d is a constant. Theorem 3.7 (F riedman [7]) . F or any ε > 0 and fixe d d ≥ 3 , if G is sample d fr om G n,d then w.h.p . λ ( G ) ≤ 2 √ d − 1 + ε. (5) Combining Theo rems 3.4 and 3 .7, implies explicitly the first part of Theore m 1.1. Corollary 3.8. F or every fix e d α > 0 and fixe d d ≥ 3 ther e exists a c onstant β > 0 , such that if G is a gr aph sample d fr om G n,d , then w.h.p. b G has a c onne cte d c omp onent of size n − o ( n ) that is a β -exp ander and al l other c omp onents ar e of c ar dinality at most 4( d − 1) α ( d − 2) 2 + 1 . The seco nd par t of Theorem 1.1 analyzes the v a lues o f α for w hich w.h.p. the gr aph b G is connected, and the v alues of α for which the w.h.p. small connected co mponents ar e all is olated vertices. Plugging The o rem 3.7 into T he o rem 3.4, as ab o ve, implies a similar r esult, but not as strong. T o get Theorem 1.1 in full, and even to impr o ve it, we use Prop ositions 3.5 and 3.6. T o do s o, we s tate the following well known a symptotic pr operties of G n,d (see e.g. [15]). Let G b e a graph sampled from G n,d , for any fixed d ≥ 3, then w.h.p. the minimal distance b et ween t wo cycles of constant length in G is ω (1 ). This statemen t is equiv alent to saying that for ev ery constant M > 1 w.h.p. ρ ( G n,d , M ) ≤ 1. Moreover, a s n tends to infinit y t ( G n,d ) ∼ P oisson  ( d − 1) 3 6  , and by so Mar k ov’s inequality w.h.p. t ( G n,d ) = O ( n 2 d − 3 2( d − 1) ) (with roo m to spare). Now, using Prop ositions 3.5 and 3.6 com bined with Cor ollary 3.8 we get the des ired result for G n,d . Theorem 3 .9. F or every fixe d α > 0 and d ≥ 3 ther e exists a c onstant β > 0 , such that if p = n − α and G is a gr aph sample d fr om G n,d , then w.h.p . b G has a c onne cte d c omp onent of s ize n − o ( n ) that is a β -exp ander and al l other c omp onents ar e of c ar dinality at most 4( d − 1) α ( d − 2) 2 + 1 . Mor e over, 1. if α > 1 2( d − 1) , w.h. p. all smal l c onne cte d c omp onents of b G ar e isolate d vertic es. 2. if α > 1 d , w.h. p. b G is c onne cte d. It s hould b e noted that Theor em 3.9 improv es up on Theo rem 1.1 for the v a lue s of α guaranteeing that b G stays connected w.h.p.. As mentioned in the Section 3.2, this improv ement is b est poss ible, for if α ≤ 1 d , then the ex pected num b er of iso lated vertices will b e at least one, and by some standard concentration arg umen ts it can a lso b e shown that the num b er of isolated vertices is hig hly concentrated around this exp ectation. Hence, for α ≤ 1 d the gra ph b G has isolated vertices, and is thus disconnected, with some pr obabilit y b ounded aw ay from 0. As a final note, it sho uld be men tioned that in the original s tatemen t o f the main result of [9], it is prov ed that w.h.p. all s ma ll connected comp onen ts ar e tre es, and that f o r α > 1 2( d − 1) w.h.p. the nu mber of iso lated vertices is o ( n ( d − 2) / 2( d − 1) ). These res ults as well ca n b e derived fro m simple proba bilistic arguments bas e d on proper ties of G n,d , but we omit these technical details. 10 4 Un b ounded expansion of small sets So far w e ha ve consider e d gr a phs of bounded maxim um degr e e (and in particular d -r egular gra phs for d = O (1)) that expand by a co ns tan t factor. When consider ing graphs tha t expand sets of sub-linear cardinality by an ω (1 ) factor (in pa r ticular in such g raphs δ ( G ) = ω (1), i.e. the minimal degree of G go es to infinity with n ) a simple union bound argument implies the following r esult. The pro of is quite s imila r to those w e hav e previously pre s en ted, only in this case w e can use a union b ound over all subsets of vertices with no need to go ov er all connected s ubs ets firs t, i.e. we do not make use of Lemma 2.3 . Theorem 4 .1. F or every fixe d α, c, ε > 0 if G is an f -exp ander gr aph o n n vert ic es wher e f ( u ) = ω (1) for every u = o ( n ) , and f ≥ c , then w.h. p. b G is a ( c − ε ) -exp ander. Pr o of. Le t U ⊆ V be a subset of vertices of cardinality u ≤ n 2 , a nd let W = N G ( U ) be its neighborho o d in G , wher e | W | = w . Set β = c − ε , and let us denote a s ubset of vertices U as b ad if b U = U and b w < β u . If b G is not a β -expa nder then it must contain such a bad set. W e b ound the pro babilit y of a subset U to b e bad by Pr [ U is ba d] ≤  w ⌊ β u ⌋  · p w − ⌊ βu ⌋ ≤  ew β u  β u p u ( f ( u ) − β ) . Assuming u = o ( n ), we hav e Pr [ ∃ U ⊆ V s.t. | U | = u and U is bad] ≤  n u   ew β u  β u p u ( f ( u ) − β ) ≤ n u (1+ β (1+ α )+ o (1) − αω ( 1)) = o ( n − 1 ) . In the ca se that Θ ( n ) = u ≤ n 2 , we ha ve Pr [ ∃ U ⊆ V s.t. | U | = u and U is bad] ≤  n u   ew β u  β u p u ( f ( u ) − β ) ≤ n u ( o (1)+ o (1) − α ( c − β )) = o ( n − 1 ) . Applying the unio n b ound ov er all p ossible v alues o f u completes the pro of. It should be noted that Theorem 4.1 implies that when p = n − α for a n y fixed α > 0, b G is w.h.p. an expander, and in particular stays co nnected as o pposed to the case of b ounded maximum degr ee. When d = o ( √ n ), B roder e t al. [5, Lemma 18 ] pr ovide an upper bound on the second eig en v alue of mos t of the d -r egular gr aphs. Theorem 4.2 (Broder et al. [5]) . F or d = o ( √ n ) , if G is sample d fr om G n,d then w.h.p. λ ( G ) = O ( √ d ) . (6) Plugging Theorem 4.2 into Pro position 3.3 assures that w.h.p. all conditions needed in The o rem 4.1 ar e met when the graph sampled from G n,d for 1 ≪ d ≪ √ n , and hence we get the following result. Theorem 4.3. F or every fixe d α > 0 and 1 ≪ d ≪ √ n ther e exists a c onstant β > 0 , such t ha t if G is a gr aph sample d fr om G n,d , then w.h.p. b G is a β - exp ander. When sampling a gr a ph fr o m the binomial ra ndo m graph mo del G n,p (i.e. the probability space of all g raphs on n lab eled vertices, where e a c h pair of vertices is chosen to b e an edg e independently with probability p ) with p = d n for d = Ω( √ n ), the graph is ea sily seen to b e “ a lmost d -regular” as a ll degrees 11 of the v er tices a re highly concentrated around d . F urthermore, it can b e easily s hown that when the initial graph is sa mpled from G n,p with the pre scribed v alues of p , a similar cla im to Theorem 4.1 holds. Therefore, one s hould expect Theorem 4.1 to extend to v alues of d = Ω( √ n ), but unfortunately , the techniques that are commonly us e d to deal with random reg ular graphs s eem to fail for these higher v alues of d . W e no te that in [4] the authors prove a result on the distribution o f edg es in G n,d for d = o ( √ n ), that can be eas ily used to der ive v ertex- expansion prop erties of G n,d for 1 ≪ d ≪ √ n , and co m bined with Theo r em 4.1 provides an alternative proo f of Theo r em 4 .3. 5 Concluding remarks and op en problems In this pa per w e analyzed the pro cess of deleting uniformly at r andom vertices from an expander g raph. W e hav e shown that for small enough dele tion pro babilities the res ulting gr aph w.h.p. re ta ins so me expansion prop erties (if not in the graph itself then in its larg est connected comp onen t). W e have also prov ed tha t for these deletion probabilities w.h.p. all small co nnected co mponents must b e of b ounded siz e. Lastly , we hav e shown how this result ca n b e applied to the random d -reg ular gr aph mo del for d = o ( √ n ). In Sec tio n 3.3, in order to apply our res ults from pr evious sections to the case of ra ndom d -reg ula r g raphs, we made use o f several theor ems that describ e some prop erties that o ccur w.h.p. in graphs that are sampled from G n,d , suc h as Theorem 3.7 of F riedma n [7]. This very strong result, whose proo f is far from simple, seems to b e an o verkill to prov e our claims. One could go about by showing that gra phs fro m G n,d w.h.p. po ssess s o me expansio n prop ert y (by ana ly zing the mo del directly using, e.g., the Configur a tion Mo del or the Switc hing T echnique) and then by applying Theorem 1.2 directly . This metho d would undoubtedly provide a pro of that do es not require any “heavy duty machinery”, but do es requir e more meticulous computations. Nonetheless, we hope that the reader finds the use of the connectio n betw een sp ectral graph theory and expansion pro perties (or pseudo-r andomness o f a g raph) to b e both elegant and concis e. In light of Theor em 4.3 it would b e interesting to analyze the e x pansion prop erties of ra ndom d -r egular graphs for d = ω (1) for higher v alues of p , i.e. taking p = n − o (1) , as for d = ω (1) it is no longer true trivially that for these v alues o f p w.h.p. there will be lo ng induced paths in b G . Ac knowledge ments The author s w o uld lik e to thank the ano n ymous r eferees for their helpful corrections and commen ts. The first author would like to thank Itai Benjamini for dis cussing this problem with him. References [1] N . Alon, I. Benjamini and A. Stacey , Percolation on finite g raphs and iso perimetric inequalities, Annals of Pr ob ability , V ol. 32(3):1727– 1745, 200 4. [2] N . Alon, H. Kaplan, M. Krivelevich , D. Malkhi and J. Stern, Scalable secure storag e when half the system is faulty , Information and Computation , V ol. 17 4(2):203–21 3 , 2002. 12 [3] N . Alon and J. 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