Rumor Spreading on Random Regular Graphs and Expanders

R UMOR SPREADING ON RANDOM REGULAR GRAPHS AND EXP ANDERS NIKOLA OS FOUNTOULAKIS AND K ONS T ANTINOS P ANAGIOTOU Abstra ct. Broadcas ting algorithms are important building blocks of distributed systems. In this work we inv estigate the typical p erformance of the classical and w ell-studied push mo del . Assume that initially one nod e in a giv en netw ork holds some piece of informatio n. In eac h round, every one of the informed no des chooses indep endently a neighbor uniformly at random and transmits the message to it. In this pap er w e consi der random net wo rks where eac h vertex has degree d ≥ 3, i.e., th e underlying graph is drawn un iformly at random from the set of all d -regular graphs with n vertices . W e show t hat with probability 1 − o (1) the push mo del broadcasts the message to all no des within (1 + o (1)) C d ln n rounds, where C d = 1 ln(2(1 − 1 d )) − 1 d ln(1 − 1 d ) . P articularly , w e can characterize precisely the effect of the no de degree to the typical broadcast time of the push model. Moreo ver, we consider pseudo-random regular net wor ks, where w e assume that the degree of each node is very large. There w e sho w th at the broadcast time is (1 + o (1)) C ln n with probabilit y 1 − o (1), where C = lim d →∞ C d = 1 ln 2 + 1. 1. Int roduction 1.1. Rumor Spreading and the Push Mo del. In this w ork we consider the classical and w ell-studied push mo del (or push pr oto c ol ) for disseminating in formation in n et w orks. Initially , one of the no des obtains some piece of information. I n eac h succeeding roun d, eve ry no de who h as the inf ormation passes it another no d e, wh ic h it c ho oses indep enden tly and uniformly at r andom among its neighbors. The imp ortan t question is: ho w many rounds are typicall y needed until all n o des are informed? The push mo del has b een the topic of many theoretical wo rks, and its p erformance was ev aluated on seve ral typ es of netw orks . In the case wh ere the underlyin g netw ork is the complete graph, F rieze and Grimmett [16 ] p ro v ed th at with high probability (wh p.) (i.e., with pr obabilit y 1 − o (1)) the b r oadcasting is completed within (1 + o (1))(log 2 n + ln n ) rounds, where n denotes the total num b er of no des. Recen tly , this result w as extended b y the tw o authors and Hub er [14] to the classical Er d˝ os-R ´ enyi graph G n,p , whic h is obtained b y including eac h of the p ossible  n 2  edges with p robabilit y p , indep en den tly of all other edges. Among other results, th ey sho w ed that if p = ω ( ln n n ), then th e t ypical broadcast time essen tially coincides with the broadcast time on the complete graph. In other words, as long as the a ve rage d egree of the und erlying graph is significantly larger than ln n , th e num b er of r ounds needed is not affected. How ever, prior to this wo rk, ther e was n o r esult d escribing the p erformance of the pus h mo del on significant ly sparser n et w orks. The typical b r oadcast time of the push mo del was also inv estigate d for other t yp es of net w orks, alb eit not as pr ecisely . F eige et al. derive d in [13] b ounds that hold f or arbitrary graphs. Moreo ver, they pro ve d a logarithmic u pp er b ound f or the num b er of round s needed to br oadcast the inf ormation if the und erlying net w ork is a h yp ercub e. Th is result w as generalized by Els¨ asser and S auerw ald, who determined in [12] similar b oun ds for several 1 2 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU classes of Ca yley graphs. Bradonjic et al. [4] considered random geometric graphs as under- lying n etw orks, and pr o v ed that w h p. the broadcast time is essent ially prop ortional to th e diameter of these graphs. 1.2. O ur Contribution. The main con tribu tion of this pap er is the precise analysis of the push mo del on sparse random netw orks . Note that in this con text the study of the G n,p distribution is not appropriate, as w e w ould h a v e to set p = c/n for some constant c > 0. Ho w ev er, f or such p the rand om graph G n,p is t ypically not connected. In fact, if we took an y p = o  ln n n  , we would face the same pr ob lem, as suc h a p is b elo w the connectivit y threshold for G n,p (see for example [19]). A candidate class of rand om graphs that com bines th e feature of constan t a v erage d egree with that of conn ectivit y is the class of random d -regular graphs G ( n , d ) for d ≥ 3. It is well- known that a random d -regular graph on n ve rtices is connected with probabilit y 1 − o (1). Th us, a typical member of this class of graphs is suitable for the analysis of the push proto col as f ar as th e effect of d ensit y is concerned. Let T = T ( G ( n, d )) denote the broadcast time of the push mo del on G ( n, d ). Note that in this case the c hoice of the verte x where the information is placed initially d o es not matter. Theorem 1. With pr ob ability 1 − o (1) | T ( G ( n , d )) − C d ln n | = O ((ln ln n ) 2 ) , wher e C d = 1 ln(2(1 − 1 d )) − 1 d ln(1 − 1 d ) . The ab o ve theorem is in terpreted as f ollo ws: for almost all d -regular graphs on n v ertices, with probabilit y 1 − o (1) the push proto col br oadcasts the inform ation w ithin the claimed n umb er of rounds. It is easy to see that as d gro ws C d con v erges to 1 ln 2 + 1, whic h is the constan t factor of the broadcast time of the p ush pr oto col on the complete graph, as sho wn by F rieze and Grimm ett [16]. Th us our result revea ls the essential ins ensitivit y of the p erformance of the push proto col regarding the density of the un derlying net w ork and sho ws that the crucial factor is the “uniformit y” of its structure. W e explore further this asp ect and we consider regular graphs whose stru ctural charac - teristics resem ble those of a regular r andom graph. In particular, we consider expanding graphs w hose “geometry” is determined by the sp ectrum of their adjacency m atrix. In Su bsection 1.3 b elo w we giv e an int uitive description of the ev olution of the randomized proto col, th us explaining also h o w do the tw o summands in vo lv ed in C d come up. 1.2.1. R e gular exp anding gr aphs. Exp anding grap h s ha ve found n umerous a pp lications in mo dern theoretical computer science as well as in pu re mathematics. Their prop ertie s to- gether with the theory of finite Mark o v c hains ha v e led to the solution of cen tral pr oblems suc h as the appro ximation of the v olume of a con ve x b o dy , approximat e counti ng or th e ap- pro ximate uniform sampling from a class of com binatorial ob jects. The latter applications ha v e had further impact outside compu ter science such as in th e field of statistic al physics. W e refer th e reader to the excellen t surv ey of Ho ory et al. [17] for a detailed exploration of the prop erties and the numerous app lications of expanding graphs. The main feature of an expanding graph is that ev ery set of v ertices is connected to the rest of the graph by a large num b er of edges. This k ey p rop erty mak es r andom w alks on suc h graphs rapidly mixing and has led to the ab o ve men tioned applications. Moreo v er, this prop erty makes expanding graph s an attractiv e cand idate for communicatio n net wo rks. In tuitiv ely , the high expansion of a graph implies that inf orm ation th at is initially lo cated on a small part of the graph can b e sp r ead quic kly on the rest of the graph. This b ecomes p ossible as the high expansion of a graph ensu res the lac k of “b ottlenec ks”, that is, lo cal RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 3 obstructions on wh ic h a broadcast proto col would need a significan t amount of time in order to b ypass them. W e fo cu s on a sp ectral c haracterizat ion of expand ing graphs, whic h is related to the sp ectral gap of their adjacency matrix. Let G = ( V , E ) b e a connected d -regular graph and let A b e its adjacency matrix. T h e Perron-F rob eniu s Theorem im p lies (see Prop osition 2.10 in [20]) that the largest eigen v alue of A equals d and that the corresp ond ing eigen v ector is prop ortional to the all-ones v ector [1 , . . . , 1] T . Let λ 1 , . . . , λ n b e the eigen v alues of A ordered according to their v alue (note th at since A is symmetric, these are all real). Set λ := λ ( A ) := max 2 ≤ i ≤ n | λ i | . If G has n v ertices w e sa y that G is an ( n, d, λ ) gr aph . One can sh o w (see for example p. 19 in [20]) that λ = Ω( √ d ). In particular, Alon and Boppana, Nilli [23] and F riedman [15] ha v e shown that f or ev ery d -regular graph on n ve rtices w e ha v e λ 2 ≥ 2 √ d − 1(1 − o (1)). W e are in terested in the class of d -regular graphs for which λ almost attains this low er b ound . In particular, w e are concerned with the broadcast time of the randomized p ro- to col on expanding d -regular graphs on n ve rtices w ith λ = O ( √ d ). Su c h graphs can b e explicitly constructed thr ough num b er-theoretic or group theoretic metho d s (see the su r v ey of Krive levic h and S udak o v [20 ] where numerous examples are p resen ted). Informally , we sho w that if d = ω ( √ n ), then the b roadcast time is essen tially the broadcast time on th e complete grap h with n v ertices. Theorem 2. L et G b e a c onne cte d ( n , d, λ ) gr aph with λ ≤ C √ d and d ≥ 2 C p n ln 1 / 9 n . Then for any v ∈ V , with pr ob ability 1 − o (1) | T ( G, v ) − (log 2 n + ln n ) | = o (ln n ) . Again, this theorem sh o ws the insensitivit y of the b roadcast time on the densit y of the underlying net w ork. In fact, the assu mption that λ = O ( √ d ) do es not merely yield the high expansion of the graph , bu t it also implies that th e edges of the graph are distribu ted in a un iform w a y among eac h su bset of v ertices. As we shall see in the pr o of of T heorem 2, this assu mption implies that the stru ctur e of the graph is n ot v ery d ifferen t f rom th at of a random graph on n vertic es and edge probability equal to d/n . F or examp le, the num b er of edges b etw een a subset S and its co mplement is close to d n | S | ( n − | S | ), wh ic h is the exp ected v alue in th e random graph with edge p robabilit y d/n . In this sense, such graph s are pseudor andom . This notion w as in tro du ced by Thomason [26] and w as explored further b y Chung, Graham and Wilson [5 ], esp ecia lly regarding its sp ectral c haracterization. 1.3. T he evo lution of the randomized proto col in a nutshell. Roughly sp eaking, the evo lution of the pr oto col consists of thr ee phases, which ha ve differen t charac teristics regarding the rate in which the inform ation is spread. Let us consider the fi r st phase, which ends when there are at least εn informed v ertices, for some very small ε > 0. Let us denote by I t the set of informed vertic es (i.e., th ose wh o p ossess the inform ation), and by U t the set of unin formed vertice s at the b eginning of round t + 1 of the pu sh mo del. Moreo ve r, let e b e some edge that is incident to a v ertex in I t that has not b e en use d u p to now to transmit a message, and let E t b e the set of suc h edges. Then w e sh o w that th e subgraph of G ( n , d ) induced by I t is essenti ally a tr e e , and moreo v er, that E t con tains ≈ 2 t (1 − 1 d ) t edges. T o see this, note that as ev ery ve rtex informs some sp ecific neighbor w ith probabilit y 1 /d , th e exp ecte d num b er of edges fr om E t that are going to b e used is |E t | /d . This means that ≈ |E t | /d new v ertices are going to b e inform ed (as the set of informed vertice s indu ces a tree), imp lyin g that |E t +1 | ≈ |E t | − |E t | /d + ( d − 1) |E t | /d , 4 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU as for eve ry v ertex that b ecomes inform ed in th is round the n umb er of edges coun ted in E t increases by d − 1. So, |E t +1 | ≈ 2(1 − 1 d ) |E t | . Note that in this calculation we work ed only with exp ected v alues. In the actual pr o of we hav e to sh o w that all the relev ant quan tities are sharply concen trated around their exp ec tations. T o th is en d, we use a v arian t of T alagr and’s inequalit y b y McDia rmid [22] (Theorem 4), whic h has not b een used v er y frequ en tly in the analysis of distribu ted algo rithms. W e b eliev e that it could b e wid ely applicable to the analysis of existing or future randomized pr oto cols with sev eral different degrees of dep end ency . As so on as the num b er of inf ormed v ertices is ≥ εn , then after v ery few r ounds the num b er of informed vertices is already (1 − ε ) n . Here it is essen tially the expansion prop ertie s of G ( n, d ), w hic h guaran tee that ev ery large set of vertices has linearly man y neigh b ors and, th us, with high pr obabilit y a certain fraction of th ose b eco me inf ormed in eac h round . During the final ph ase, the num b er of r emaining unin formed v ertices shrinks b y a factor of (1 − 1 d ) d . Indeed, su pp ose that there are o ( n ) uninformed ve rtices. Th en w e exp ect that almost all of them hav e the pr op ert y that the num b er of their neighbors in I t is d , implying that the p robabilit y that any on e of th e remains uninformed is precisely (1 − 1 d ) d . An easy calculatio n shows that a “t ypical” subset of G ( n, d ) has this prop ert y . Ho w ev er, the set of uninformed v ertices might n ot b e t ypical at all, implyin g that we need add itional effort to guaran tee the desired pr op erties. 2. Conce ntra tion ineq ualities In this section we w ill state tw o concentrat ion inequalities th at will serv e as the b ac kb one of our pro ofs. The first one is a Chernoff-t yp e b oun d for sums of negativ ely correlated random v ariables, see e.g. [8]. Theorem 3. L et I 1 , . . . , I n b e a family of indic ator r andom variables on a c ommon pr ob- ability sp ac e, which ar e identic al ly distribute d and ne gatively c orr elate d, i.e., E ( I i I j ) ≤ E ( I i ) E ( I j ) for al l 1 ≤ i, j ≤ n . L et X := P n i =1 I i . Then, for any t > 0 P ( | X − E ( X ) | > t ) < 2 exp  − t 2 2 ( E ( X ) + t/ 3)  . The next concen tration inequalit y that we will need is du e to McDiarmid [22], and it is based on the work of T alagrand [25]. W e giv e first a few n ecessary d efinitions. Let B b e a fi nite set and let S y m ( B ) b e the set of all p er mutations on B . Assume that π is an elemen t of S y m ( B ), drawn un iformly at random. Also, let X = ( X 1 , . . . , X n ) b e a fi nite family of in dep end en t rand om v ariables, wh ere X j tak es v alues in a set Ω j . Finally , set Ω = S y m ( B ) × Q n j =1 Ω j . Theorem 4. L et c and r b e p ositive c onstants. Supp ose that h : Ω → R + satisfies the fol low ing c onditions. F or e ach ( σ, x ) ∈ Ω we have • if x ′ differs fr om x in only one c o or dinate, then | h ( σ , x ) − h ( σ, x ′ ) | ≤ 2 c ; • if σ ′ c an b e obtaine d fr om σ by swapping two elements, then | h ( σ, x ) − h ( σ ′ , x ) | ≤ c ; • if h ( σ , x ) = s , then ther e is a set of at most r s c o or dinates such that h ( σ ′ , x ′ ) ≥ s for any ( σ ′ , x ′ ) ∈ Ω that agr e es with ( σ, x ) on these c o or dinates. L et Z = h ( π , X ) and let m b e the me dian of Z . Then, for any t > 0 P ( | Z − m | > t ) ≤ 4 exp  − t 2 16 r c 2 ( m + t )  . RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 5 3. Proper ties of random r egular graphs and th e configura t ion model 3.1. T he configuration mo del. W e p erform the analysis of the randomized p roto col u s- ing the c onfigur ation mo del introdu ced by Bend er and Canfield [1] and in dep end en tly by Bollob´ a s [2]. F or n ≥ 1 let V n := { 1 , . . . , n } . Also for those n for wh ic h dn is eve n, w e let P := V n × [ d ]. W e call the elemen ts of P clones . A c onfigur ation is a p erfect matc hing on P . If we p ro ject a configuration on to V n , then w e obtain a d -regular m ultigraph on V n . Let e G ( n, d ) d enote the m ultigraph that is obtained b y c ho osing the configuration on P uni- formly at r andom. It can b e sh o wn (see e.g. [19, p . 236]) that if we condition on e G ( n, d ) b eing simple (i.e. it d o es not ha ve lo ops or multiple edges), then this is d istributed uniformly among all d -regular graphs on V n . I n other words, e G ( n, d ) conditional on b eing simple has the same distribution as G ( n, d ). Moreo ve r, Corollary 9.7 in [19] guarantee s that (3.1) lim n →∞ P ( e G ( n, d ) is simp le) > 0 . (Of course the ab ov e limit is tak en o v er those n for which dn is ev en.) Let A n b e a su bset of the set of d -regular multi graphs on V n . Altoge ther the ab o ve facts im p ly that if P ( e G ( n, d ) ∈ A n ) → 0 as n → ∞ then also P ( G ( n, d ) ∈ A n ) → 0. This allo ws u s to wo rk w ith e G ( n, d ) instead of G ( n, d ) itself. 3.2. Some useful facts. W e con tin ue by in tro d ucing some n otation. L et G b e a graph, and let S, S ′ b e subs ets of its v ertices. Then w e denote b y e G ( S ) th e num b er of edges in G joining vertice s only in S , and b y e G ( S, S ′ ) the n umb er of edges in G joining a v ertex in S to a v ertex in S ′ . Moreo v er, w e denote by Γ G ( v ) the set of neighb ors of a vertex v in G . Lemma 5. L et A , B ⊆ V n × [ d ] b e two disjoint sets of clones, and let C ⊆ V n b e a set of vertic es such that ( C × [ d ]) ∩ ( A ∪ B ) = ∅ . L et M b e a matching dr awn uniformly at r andom fr om the set of p erfe ct matchings on the union of the clones in A , B and C × [ d ] , and set N := |A | + |B| + d |C | − 1 . Then (3.2) E ( e M ( A )) =  |A| 2  1 N , E ( e M ( A , B )) = |A||B | 1 N , and E ( e M ( A , C )) = d |A||C | 1 N . Mor e over, let H ℓ denote the numb er of vertic es in C that ar e adjac ent to exactly ℓ clones in A in M , wher e 0 ≤ ℓ ≤ d . Then, if |B | ≥ |A| = ω (ln n ) (3.3) E ( H ℓ ) =  1 + o  1 ln n   · |C |  d ℓ   |A| N  ℓ  1 − |A| N  d − ℓ . Final ly, let Q = P ℓ ≥ 2 H ℓ . Then, if N ≥ 4 (3.4) E ( Q ) ≤ d 2 |A| 2 |C | N − 2 . L et X b e any of e M ( A ) , e M ( A , B ) , e M ( A , C ) or H ℓ , and let µ = E ( X ) . Then, if µ = ω (ln 2 n ) , for any ε = ω ( µ − 1 / 2 ) (3.5) P ( | X − µ | ≥ εµ ) ≤ 4 e − ε 2 64 d (1+ ε ) µ . Pr o of. Let e, e ′ b e edges w hose endp oin ts are in the union of the clones in A , B and C , and let I e , I e ′ b e the ind icator v ariables for the ev ents th at e ∈ M and e ′ ∈ M . As the num b er of matc hings with e is equal to the num b er of matc hings with e ′ w e h a v e E ( I e ) = E ( I e ′ ). Hence, as P e I e = N +1 2 alw a ys, w e infer th at E ( I e ) = 1 N . By linearit y of exp ectation this pro ve s (3.2) . T o see (3.4) let I e,e ′ b e the ev en t that b oth e and e ′ are in M . Note that if e ∩ e ′ 6 = ∅ and also e 6 = e ′ , then E ( I e,e ′ ) = 0. O therwise, let f , f ′ b e an y t wo edges satisfying f ∩ f ′ = ∅ and 6 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU f 6 = f ′ . Then, as the num b er of m atc hings w ith e, e ′ is equal to th e num b er of matc hings with f , f ′ w e infer that E ( I e,e ′ ) = E ( I f ,f ′ ). As P e 6 = e ′ I e,e ′ = N +1 2 N − 1 2 alw a ys and as there are 3  N +1 4  w a ys to choose e, e ′ suc h that e ∩ e ′ = ∅ and e 6 = e ′ w e obtain that E ( I e,e ′ ) =      0 , if e ∩ e ′ 6 = ∅ and e 6 = e ′ 1 N , if e = e ′ 2 N ( N − 2) , otherwise . Let v = { v 1 , . . . , v d } b e any vertex in C . Moreo v er, let no w e, e ′ b e distinct edges w ith one endp oint in A and the other in v , and note that there are  |A| 2  · |C |  d 2  w a ys to choose e and e ′ . If N ≥ 4, then E ( I e,e ′ ) ≤ 4 N − 2 , an d this completes the pro of of (3.4). T o see (3.3) let v ∈ C and denote by L v the even t that ther e is an ed ge in M connecting t w o clones of v . Moreo ver, let H ℓ ( v ) d enote the ev ent th at v is adjacen t to exactly ℓ clones in A . Then (3.6) P ( H ℓ ( v )) = P ( H ℓ ( v ) ∩ L v ) + P ( H ℓ ( v ) | L v ) P ( L v ) . W e estimate the ab o v e pr obabilities one by one. W e shall b egin with P ( L v ). Note that there are at most d 2 c hoices for an edge that conn ects t wo clones of v , and that the probabilit y that suc h an edge is in M is 1 N . Hence, (3.7) P ( L v ) ≤ d 2 N − 1 = o (ln − 1 n ) . Next w e estimate P ( H ℓ ( v ) ∩ L v ). Let u s f or the momen t fi x ℓ clones c 1 , . . . , c ℓ in A , and ℓ clones c ′ 1 , . . . , c ′ ℓ of v . Note that there are  |A| ℓ  c hoices for the c i ’s and  d ℓ  c hoices for the c ′ i ’s. Then the num b er of m atc hings where the c i ’s are matc hed to the c ′ i ’s, and no one of the remaining clones of v is matc hed to a clone in A , and there is no edge connecting tw o of the clones of v , is ℓ ! ·  |B | + d ( |C | − 1) d − ℓ  ( d − ℓ )! · M |A| + |B | + d ( |C |− 2) , where M n = n ! ( n/ 2)!2 n/ 2 denotes the n umb er of p erfect matc hings on n vertice s. Stirling’s form ula yields the appro ximation (3.8) M n = (1 + Θ( n − 1 )) · √ 2 n n/ 2 e − n/ 2 . Moreo v er, our assu m ption |B | ≥ |A| = ω (ln n ) implies that  |A| ℓ  =  1 + o  1 ln n   · |A| ℓ ℓ ! and  |B | + d ( |C | − 1) d − ℓ  =  1 + o  1 ln n   · ( N − |A| ) d − ℓ ( d − ℓ )! . All the ab o v e facts toget her yield that P ( H ℓ ( v ) ∩ L v ) =  1 + o  1 ln n   ·  d ℓ  |A| ℓ ( N − |A| ) d − ℓ · M N +1 − 2 d M N +1 . By app lying the estimate for M n w e infer that th e last fr action equals  1 + o  1 ln n   · e d ( N + 1 − 2 d ) N +1 − 2 d 2 ( N + 1) N +1 2 =  1 + o  1 ln n   · N − d . So, (3.9) P ( H ℓ ( v ) ∩ L v ) =  1 + o  1 ln n   ·  d ℓ   |A| N  ℓ  1 − |A| N  d − ℓ . Finally , we estimate P ( H ℓ ( v ) | L v ). Note that the ev en t H ℓ ( v ), giv en L v , im p lies that there are ℓ clones of v that are matc hed to s ome clones in A . By a similar reasoning as ab o ve w e RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 7 infer th at P ( H ℓ ( v ) | L v ) ≤  d ℓ  | A | ℓ  ℓ ! · M N +1 − 2 ℓ M N +1 ≤  1 + o  1 ln n    d ℓ   |A| N  ℓ . Note that our assump tion |B | ≥ |A| implies that |A| N ≤ |A| |A| + |B | ≤ 1 2 . So, 1 − |A| N ≥ 1 2 , and (3.7 ) together w ith (3.6) imply that P ( H ℓ ( v ) | L v ) P ( L v ) = o  P ( H ℓ ( v ) ∩ L v ) ln n  . By p lugging this in to (3.6) we thus complete the pro of of (3.3 ). W e finally pr o v e the concent ration of X by app lying Th eorem 4 as follo ws. W e will first sp ecify the families X and π . Here, X = ∅ . The random p erm utation π corresp onds to the random p erfect matc hing on the union of the v ertices in A , B and C . More precisely , assuming that this union consists of 2 k clones, which are lab eled 1 , . . . , 2 k , we consider a uniformly random p ermutation of these clones π := ( i 1 i 2 . . . i 2 k − 1 i 2 k ). Th en we matc h the clones that are in consecutiv e pairs, that is, w e c ho ose the matc hing { ( i 1 , i 2 ) , ( i 3 , i 4 ) , . . . , ( i 2 k − 1 , i 2 k ) } . This is a uniform p erfect matc hing on these clones. Not e that the p air ( X , π ) d etermines the v alue of X . Moreo v er, • if we swap tw o elemen ts of π , then X can change by at most 2; • if X = ℓ , then we need to sp ecify at most dℓ elemen ts of π in order to certify this. Th us, w e may tak e c = 2 and r = d in Theorem 4. Moreo v er, let M X b e the median of X . An easy calculation sho ws that | M X − E ( X ) | = O ( p E ( X )) (cf. Example 2.33 in [19]). Th e pro of completes by applying Theorem 4 with, sa y , t = 1 . 1 εM X .  4. Ana l ysis of the Rand omized Bro adcas ting Algorithm 4.1. T he preliminary phase. Let T 0 b e the fir st round in wh ich the num b er of informed v ertices exceeds ln 7 n . W e will show the follo wing statemen t; it is not b est p ossible, bu t it suffices for our purp oses. Lemma 6. Wi th pr ob ability 1 − o (1) we have that T 0 = O (ln ln n ) . Mor e over, for sufficiently lar ge n the sub gr aph induc e d b y the v e rtic es in I T 0 is with pr ob ability 1 − o (1) a tr e e. Pr o of. Let D i denote the num b er of v ertices at d istance i from v ertex 1. W e w ill first sho w that whp . w e h a v e |D i | = d ( d − 1) i − 1 for all 1 ≤ i ≤ √ ln n , wh ic h implies that the subgraph induced by ∪ √ ln n i =1 D i is w h p. a tree. T o see the claim, we w ork in the confi guration mo del and exp ose th e sets D i one after the other, i.e., we first exp ose the edges in the rand om matc hing that conta in the clones of v ertex 1, then the edges that con tain the (remaining) clones of the v ertices in D 1 , an d so on. Supp ose th at |D i | = d ( d − 1) i − 1 for all i ≤ j < √ ln n . This implies that all edges in the matc hing incident to the clones corresp onding to the vertic es in D 1 , . . . D j − 1 are exp osed. Moreo v er, for ev ery verte x in D j there is precisely one clone whose neighbor is exp osed, and for all other d − 1 it is not. Let us denote by F j this set of unexp ose d clones. W e ha v e |F j | = d ( d − 1) j , and let us note for futur e reference that with ro om to spare |F j | ≤ n 1 / 3 . Clearly , D j +1 consists of all v ertices in C = V n \ ( D 1 ∪ · · · ∪ D j ) for whic h at least one of their clones is connected in th e m atc hing to some clone in F j . Let Q denote the num b er of suc h vertices with the prop ert y that they are matc hed to at least t w o clones in F j , and let M b e a random p erf ect matc hing on th e union of the clones in F j and C . Th en |D j +1 | = |F j | − 2 e M ( F j ) − Q. 8 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU By app lying Lemma 5 with A = F j , B = ∅ and C as ab o ve w e obtain for large n that E ( e M ( F j )) ≤ n − 1 / 3 and E ( Q ) ≤ 2 d 2 n − 1 / 3 . So, with probabilit y at least 1 − 3 d 2 n − 1 / 3 w e ha v e that e M ( F j ) = Q = 0. The pro of of the claim completes by applying the ab o ve argument f or i = 1 . . . √ ln n . With the ab ov e fact we can pr ov e the lemma as follo ws. Let v be a vertex in D i , for some 1 ≤ i ≤ 10 ln ln n := ℓ , and denote by T v the time until v gets informed. Let v ′ b e the unique neigh b or of v in D i − 1 . T hen T v = T v ′ + X v,v ′ , where X v,v ′ is a geometricall y distributed rand om v ariable with su ccess probabilit y d − 1 . Moreo ver, X v,v ′ is indep enden t of T v ′ . I n other w ords, w e hav e that T v = P i j =1 X j , where th e X j ’s are iid. v ariables as ab o v e. So, E ( T v ) = di , and by Th eorem 3 P ( T v ≥ 20 d 2 i ) = P ( B in (20 d 2 i, d − 1 ) < i ) ≤ 2 e − (19 di ) 2 4 · 20 di ≤ 2 e − 4 di . In particular, for i = ℓ , this probabilit y is at most 2 ln − 40 d n . Moreo v er, the total n umb er of v ertices in ∪ ℓ i =1 D i is at most d d ℓ − 1 d − 2 ≤ ln 20 d n . So, b y Mark o v’s inequalit y , there is no v ertex at distance at most ℓ from v ertex 1 that will n ot b e informed in the firs t 20 d 2 ℓ = O (ln ln n ) rounds. Moreo v er, d d ℓ − 1 d − 2 = ω (ln 7 n ), and the pro of is completed.  4.2. T he Exp osure Strategy. In this section we w ill describ e our ge neral strateg y for determining the probable broadcast time of the randomized rumor spreading pr oto col. W e will denote by I t the set of in formed v ertices and by U t the set consisting of the uninf orm ed v ertices, i.e., U t = [ n ] \ I t , at the b eginnin g of round t . W e ha v e that I 1 = { 1 } . W e can sim ulate the execution of th e rumor s p reading proto col as follo ws in t wo steps. First, we c ho ose one of the clo nes of vertex 1 uniformly at ran d om, sa y c 1 . T hen, w e exp ose the edge in the random matc hing whose one endp oin t is c 1 , and pass the message to the other endp oint, say c 2 . Note that this is equiv alen t to selecting uniformly at r andom a clone c ′ differen t fr om c 1 , and j oining c 1 and c ′ b y an edge. Clearly , c 2 is a clone that corresp onds to some v ertex in the original graph, which n o w b ecomes informed . Th is completes the firs t round, and I 2 consists of v ertex 1 and the v ertex corresp ondin g to c 2 . This gradual exp osure of the graph can b e generalized to an y other round in the follo wing manner. Supp ose that we are in the b eginning of round t + 1 ≥ 0. W e will simulat e the execution of the proto col as follo ws in t w o steps. Step 1 . F or eac h v ∈ I t w e c ho ose one of its clones u niformly at random, indep enden tly for ev ery s uc h v ertex. W e shall d enote the s elected clone b y c v = c v ( t ). Step 2 . Set I t +1 = I t and let v ∈ I t . If c v b elongs to an edge in the rand om matc hing that w as exp osed in one of the previous round s , do nothing. O th erwise, c ho ose uniform ly at rand om one of the remainin g unm atc hed clones, say c , and connect it to c v b y an edge. Add the vertex corresp ond ing to c to I t +1 , if it isn ’t already con tained in I t +1 . If a clone of a verte x in U t is matc hed to c v , for some v ∈ I t , then that vertex b ec omes informe d – we d en ote b y N t +1 the set of those vertices. In sh ort, N t +1 is the set of new ly informe d ve rtices in the t + 1st r ou n d. Let us in tro du ce some further notation regarding the t w o exp osur e steps. A t th e b eg innin g of round t + 1, w e d enote by P t the s et of clones of the v ertices in I t whose n eigh b ors h a v e not b een exp osed y et (i.e., in none of the previous round s the ed ges in the matc hin g con taining those clones w ere exposed). Among those, during Step 1 w e choose a set A t +1 ⊆ P t of clones. Informally , A t +1 con tains the clones through which new RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 9 v ertices migh t get informed. Finally , w e write N t +1 = |N t +1 | , A t +1 = |A t +1 | and P t = |P t | , and n ote that P 0 consists of the d clones of vertex 1. The t w o steps of our exp osure strategy can b e also v iewed as follo ws. In the fi r st step we c ho ose according to the r u le d escrib ed ab o ve a random s ubset A t +1 of P t . Then, in Step 2, the clones in A t +1 are matc hed to the un ion of the clones in P t and the clones corresp ondin g to the v ertices in U t (as, p er defin ition, all other clones are already matc hed). I n other words, w e consider a rand om p erf ect matc hing M t +1 on the set of clones in P t and U t , and we will study its com binatorial prop erties. In particular, the follo wing claim relates the random quan tities in question. Prop osition 7. L et H i,t +1 denote the numb er of vertic es in U t that wer e informe d i times in r ound t + 1 , i.e ., a vertex v is c ounte d in H i,t +1 , if ther e ar e i clones in A t +1 that ar e matche d to the clones of v in M t +1 . Then N t +1 = d X i =1 H i,t +1 ≤ e M t +1 ( A t +1 , U t ) , (4.1) I t +1 = I t + N t +1 and U t +1 = U t − N t +1 , (4.2) P t +1 = P t − A t +1 − e M t +1 ( A t +1 , P t \ A t +1 ) + d X i =1 ( d − i ) H i,t +1 . (4.3) Pr o of. The first equalit y in Equation (4.1 ) follo ws directly from th e definition of H i,t +1 , as ev ery v ertex in U t has d unm atc hed clones, and it b ecomes inform ed as so on as at least one of th em gets matc hed in S tep 2 to a clone in A t +1 . The up p er b ound is also easy to see, as the num b er of newly inform ed vertices is at most the num b er of edges in M t +1 that hav e one endp oin t in A t +1 and the other in the set of clones corresp onding to the v ertices in U t . Equation (4.2) f ollo ws immediately from the definition of I t and N t +1 . Finally , to see (4.3 ), note first that all clones in A t +1 are excluded from P t +1 , as they are matc h ed to other clones in P t or U t ; this accoun ts f or the “ − A t +1 ” term. Moreo v er, all clones in P t \ A t +1 that are con tained in edges of M t +1 with the other endp oin t in A t +1 are excluded from P t +1 as well, as th e edge including them wa s exp ose d; this acco unts for the the “ − e M t +1 ( A t , P t \ A t +1 )” term. Finally , f or eac h newly informed vertex count ed in H i,t +1 , i.e., wh ich wa s informed i times in round t , the n umb er of clones coun ted in P t increases b y d − i .  F or futur e r eference w e pro ve already here a lemma that add resses the co ncentrat ion prop erties of A t +1 . Lemma 8. F or any t ≥ 1 and n ≥ 5 P     A t − P t d    ≥ P t d ln 2 n    P t  ≤ 2 e − P t 3 d ln 4 n . Pr o of. F or eac h clone c ∈ P t let I c b e the indicator v ariable for the ev en t that c is selected in the first step of th e t th round, i.e., “ I c = 1” iff the random decisions in Step 1 are suc h that c ∈ A t +1 . Since eac h clone h as p robabilit y 1 /d to b e selected we hav e E ( I c ) = 1 /d . Moreo v er, for t w o distinct clones c, c ′ w e hav e that E ( I c I c ′ ) = ( 0 , if c, c ′ are clones b elonging to the same v ∈ V n 1 /d 2 , otherwise ≤ 1 d 2 = E ( I c ) E ( I c ′ ) , 10 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU i.e., the I c ’s are n egativ ely correlated. W e infer that µ := E ( A t +1 | P t ) = P t d , and Th eorem 3 implies that the sought prob ab ility is at most P ( | A t +1 − µ | ≥ µ / ln 2 n | P t ) ≤ 2 exp  − µ 2 ln − 4 n 2( µ + µ/ (3 ln 2 n ))  ≤ 2 exp  − µ 3 ln 4 n  .  4.3. T he Middle Phases. Let T 1 b e the firs t r ound where the num b er of informed ve rtices is at least n − ln 7 n , or equ iv alen tly , where U T 1 ≤ ln 7 n . Th e main accomplishmen t of this section is the p ro of of the follo win g lemma, w hic h describ es the lik ely ev olution of the n umber of (u n)informed v ertices and of P t unt il t = T 1 . Lemma 9. Supp ose that P t , U t ≥ ln 7 n . Abbr eviate F t = 1 − P t d ( P t + dU t ) . Then, uniformly with pr ob ability at le ast 1 − o ( 1 ln n ) , P t +1 =  1 − o  1 ln n   ·  1 − 1 d  F t · P t + dU t ( F t − F d t )  , (4.4) U t +1 =  1 − o  1 ln n   · F d t · U t . (4.5) Pr o of. Let H i,t +1 denote the n umb er of v ertices in U t that w ere inf ormed i times in roun d t + 1, and recall that Prop osition 7 describ es the relation of the quant ities P t +1 and U t +1 to P t , U t and H i,t +1 . W e will sho w that u niformly for all t such that P t , U t ≥ ln 7 n , with probabilit y 1 − o ( 1 ln 2 n ) w e h av e (4.6) A t +1 =  1 + o  1 ln n   P t d , and (4.7) e M t +1 ( A t +1 , P t \ A t +1 ) =  1 + o  1 ln n    1 − 1 d  P t (1 − F t ) ± ln 5 n, and th at f or all 1 ≤ i ≤ d (4.8) H i,t +1 =  1 + o  1 ln n   U t ·  d i  (1 − F t ) i F d − i t ± ln 5 n. This prov es (4.4) and (4.5 ) as follo ws. First, by using (4.1) we infer th at with probabilit y 1 − o ( 1 ln n ) the n umb er of in formed v ertices in roun d t + 1 is N t +1 = d X i =1 H i,t +1 =  1 + o  1 ln n   U t · (1 − F d t ) ± d ln 5 n. So, as U t ≥ ln 7 n , with p robabilit y 1 − o ( 1 ln n ) th e num b er of uninf orm ed v ertices at the end of r ound t + 1 is U t +1 = U t − N t +1 = U t −  1 + o  1 ln n   U t · (1 − F d t ) ± d ln 5 n =  1 + o  1 ln n   F d t U t . This sho ws (4.5). T o see (4.4) recall (4.3) and note that with probabilit y 1 − o ( 1 ln n ) d X i =1 ( d − i ) H i,t +1 =  1 + o  1 ln n   U t d ( F t − F d t ) ± d ln 5 n. RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 11 Hence, b y applying (4.3) w e infer that with p robabilit y 1 − o ( 1 ln n ) P t +1 = P t − A t +1 − e M t +1 ( A t , P t \ A t ) + d X i =1 ( d − i ) H i,t +1 =  1 + o  1 ln n    P t − P t d −  1 − 1 d  P t (1 − F t ) + U t d ( F t − F d t )  , and th is sho ws (4.4). It remains to pro v e (4.6)-(4.8). W e start with (4.6). This is ea sily seen to h old, by applying Lemma 8 and using th e fact that P t ≥ ln 7 n . T o s ee (4.7) w e apply Lemma 5 with A = A t +1 , B = P t \ A t +1 and C = U t . W e in fer that µ := E ( e M t +1 ( A t +1 , P t \ A t +1 )) = A t +1 ( P t − A t +1 ) P t + dU t − 1 . Note that for sufficiently large n w e hav e with pr ob ab ility 1 − o ( 1 ln n ) that |P t \ A t +1 | ≥ |A t +1 | , and that A t +1 = ω (ln n ). By usin g (4.6 ) and the definition F t = 1 − P t d ( P t + dU t ) w e thus obtain µ =  1 + o  1 ln n   (1 − 1 d ) P 2 t d ( P t + dU t − 1) =  1 + o  1 ln n    1 − 1 d  P t (1 − F t ) . If µ ≥ ln 3 n , then b y applying (3.5) with ε = ln − 1 . 1 n w e infer that P ( | e M t +1 ( A t +1 , P t \ A t +1 ) − µ | ≥ µ ln − 1 . 1 n ) = o (ln − 1 n ) . On the other hand, if µ ≤ ln 3 n , w e obtain by Marko v’s inequalit y that P ( e M t +1 ( A t +1 , P t \ A t +1 ) ≥ ln 5 n ) = o (ln − 1 n ) . By com binin g the ab ov e statemen ts w e infer that with prob ab ility at least 1 − o ( 1 ln n ) w e ha v e that e M t +1 ( A t +1 , P t \ A t +1 ) = (1 + o ( 1 ln n )) µ ± ln 5 n i.e., (4.7) is prov ed. The pro of of (4.8) is v ery similar. By applying Lemma 5 with A = A t +1 , B = P t \ A t +1 and C = U t w e infer that µ i := E ( H i,t +1 ) =  1 + o  1 ln n   U t ·  d i   A t P t + dU t − 1  i  1 − A t P t + dU t − 1  d − i . As with probabilit y 1 − o ( 1 ln n ) w e h av e A t +1 = (1 + o ( 1 ln n )) P t d w e infer that µ i =  1 + o  1 ln n   U t ·  d i  F i t (1 − F t ) d − i . The pro of now completes with a case distinction as ab o v e, i.e., w e treat th e case µ i ≤ ln 5 n with Mark o v’s inequalit y and the case µ i ≥ ln 5 n b y using (3.5).  Lemma 9 allo ws us n o w to deriv e pr obable b ounds for T 1 . Corollary 10. With pr ob ability 1 − o (1) we have that T 1 − T 0 = C d ln n + O (ln ln n ) , wher e C d = 1 ln(2(1 − 1 d )) − 1 d ln(1 − 1 d ) . Pr o of. By applying Lemma 6 w e infer that at round T 0 with high pr obabilit y ther e are for the firs t time at least ln 7 n inf orm ed v ertices, and the set of inform ed v ertices induces a tree. Hence, w e ma y assu me that ln 7 n ≤ I T 0 ≤ 2 ln 7 n and ( d − 1) I T 0 ≤ P T 0 ≤ dI T 0 . W e will u se th ose facts in the sequel without fu rther reference. 12 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU Let p t and u t b e giv en by the r ecursions p t +1 =  1 − 1 d  f t p t + du t ( f t − f d t ) and u t +1 = f d t u t , where f t = 1 − p t d ( p t + du t ) , and p T 0 = P T 0 , u T 0 = n − I T 0 . As we are in terested in th e probab le v alues of P t and U t for t = O (ln n ) w e infer b y applying L emma 9 that p t = (1 + o (1)) P t and u t = (1 + o (1)) U t for all su c h t , pr o vided that U t , P t ≥ ln 7 n . In th e sequel w e shall therefore consider only the ev olution of p t and u t . Let q := 2  1 − 1 d  , ε = 0 . 01 and t 1 b e the minimal t such that q t − T 0 ≤ εn ln 7 n . W e will fi rst sho w that for all T 0 ≤ t ≤ t 1 (4.9) p t ≤ P T 0 · q t − T 0 and p t ≥ P T 0 · q t − T 0 − 3 P 2 T 0 · q 2( t − T 0 ) /n, and (4.10) u t = n − I T 0 − P T 0 q t − T 0 − 1 d ( q − 1) ± 9 · P 2 T 0 q 2( t − T 0 ) /n. W e pro ceed by indu ction on t . Note that for t = T 0 the statemen t trivially holds. In order to p erform the induction step ( t → t + 1) we will need some facts. Fi rst, let x = 1 − f t and note that f t − f d t = (1 − x ) − (1 − x ) d ≤ ( d − 1) x = ( d − 1) p t d ( p t + du t ) ≤ d − 1 d 2 p t u t . So, we readily obtain the up p er b ound for p t in (4.9) by usin g the th e recursion for p t as follo ws. p t +1 ≤  1 − 1 d  f t p t + du t · d − 1 d 2 p t u t ≤ 2  1 − 1 d  p t = q p t ⇒ p t +1 ≤ P T 0 · q t +1 − T 0 . T o see the lo we r b ound for p t , note first that th e indu ction hyp othesis, together with the fact th at q t − T 0 ≤ εn ln 7 n imply that p t u t < 1. Th us, 1 1+ p t du t ≥ 1 − p t du t . A similar calculation as ab o v e and by u sing the fact (1 − x ) d ≤ 1 − dx +  d 2  x 2 for x ≥ 0 reve als that f t − f d t ≥ ( d − 1) x −  d 2  x 2 ≥ d − 1 d p t p t + du t − d 2 2 p 2 t d 2 ( p t + du t ) 2 ≥ d − 1 d 2 p t u t (1 + p t du t ) − p 2 t 2 d 2 u 2 t ≥ d − 1 d 2 p t u t − 3 p 2 t 2 d 2 u 2 t . By u sing again the recursion for p t w e infer that p t +1 ≥  1 − 1 d  f t p t + du t ·  d − 1 d 2 p t u t − 3 p 2 t 2 d 2 u 2 t  ≥ q p t − 2 d p 2 t u t . Note th at the indu ction hyp othesis and the fact q t − T 0 ≤ εn ln 7 n imply that u t ≥ n/ 2. So, p t +1 ≥ q p t − 4 dn p 2 t ≥ P T 0 q t +1 − T 0 − 3 P 2 T 0 q 2( t − T 0 )+1 n − 4 dn  P T 0 q t − T 0  2 = P T 0 q t +1 − T 0 − P 2 T 0 q 2( t − T 0 +1) n  3 q + 4 dq 2  ≥ P T 0 q t +1 − T 0 − 3 P 2 T 0 q 2( t − T 0 +1) n . This pro v es the lo wer b oun d for p t in (4.9). Next w e pr o v e the b ounds for u t +1 . Note that u t +1 u t =  1 − p t d ( p t + du t )  d ≥ 1 − p t p t + du t ≥ 1 − p t du t ⇒ u t +1 ≥ u t − p t d . RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 13 A s imilar calculation using the fact (1 − x ) d ≤ 1 − dx +  d 2  x 2 for x ≥ 0 reve als that u t +1 u t ≤ 1 − p t p t + du t +  d 2  p 2 t d 2 ( p t + du t ) 2 ≤ 1 − p t du t + 3 4 p 2 t u 2 t . Recall that the indu ction hypothesis guarante es u t ≥ n/ 2. T h e ab o ve facts together with the b ounds for p t imply after a s tr aigh tforw ard but lengthy calculation (4.10). W e omit the details. The ab o ve discussion settles the growth of p t and u t up to the time t 1 . Note that t 1 = ln(2(1 − 1 d )) − 1 ln n + Θ(ln ln n ). In order to deal with t > t 1 let us first m ak e tw o imp ortant observ ations. Firs t, note that at t 1 w e ha v e that (4.11) p t 1 u t 1 = Ω(1) . Let u s n ext consider the ratio r t := p t /u t . Note that f t = 1 − p t d ( p t + du t ) = 1 − 1 d (1+ d/r t ) . The recursions for p t and u t imply that r t +1 =  1 − 1 d  f − d +1 t r t + d ( f − d +1 t − 1) ⇒ r t +1 r t =  1 − 1 d  f − d +1 t + d r t ( f − d +1 t − 1) . Consider the function g ( x ) =  1 − 1 d + d x   1 − 1 d (1 + d/x )  − d +1 − d x , and note that r t +1 r t = g ( r t ). A straigh tforwa rd calculat ion sho ws that lim x → 0 g ( x ) = 2(1 − 1 d ). In the sequel w e will argue that g is monotone increasing. This imp lies r t +1 r t ≥ g (0) ≥ 4 3 , and so we ha v e f or an y t ′ > 0 (4.12) r t + t ′ ≥ r t  4 3  t ′ ⇒ p t + t ′ ≥  4 3  t ′ u t + t ′ . This fact will b ecome ve ry useful later on. T o see why g is increasing, note that g ′ ( x ) = − T ( d 2 + x ) + dx + d 2 x 2 + xd , w here T =  1 − 1 d (1 + d/x )  − d +1 . Supp ose that there is an x 0 ≥ 0 such th at g ′ ( x 0 ) = 0. Then − T + d 2 = x 0 ( T − d ). Ho we ver, w e alw a ys h a v e 1 ≤ T < d . T h us, the right -hand side of the ab o ve equation is < 0, while the left-hand side is > 0. W e infer that there is n o suc h x 0 , and therefore the sign of g ′ ( x ) equals the sign of g ′ (0). As the latter is easily seen to b e p ositiv e, this concludes the p ro of of th e monotonicit y of g . Let t 2 b e the m inimal t suc h that p t 2 ≥ u t 2 ln 2 n . The Equations (4.11) and (4.12) guaran tee that t 2 = t 1 + O (ln ln n ), and moreo v er that for an y t > t 2 suc h th at u t > 0 w e ha v e p t ≥ u t ln 2 n ≥ 1. Und er these conditions n ote that f d t =  1 − p t d ( p t + du t )  d =  1 + O (ln − 2 n )   1 − 1 d  d . Th us, for an y t su c h that t = t 2 + O (ln n ) we ha v e that u t = (1 + o (1)) ·  1 − 1 d  d ( t − t 2 ) u t 2 . 14 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU Recall that T 1 is the first t suc h th at U T 1 ≤ ln 7 n . As u t 2 ≤ n , we readily obtain that T 1 ≤ t 1 + O (ln ln n ) − 1 d ln((1 − 1 d )) ln n = C d ln n + O (ln ln n ). T o see the corresp onding low er b oun d for T 1 , note that as long as p t ≥ 1 w e alw a ys ha v e u t +1 ≥  1 − 1 d  d u t . The pro of completes with the fact u t 1 = Θ( n ).  4.4. T he Final Phase. Let T 1 b e the first time su c h that the n umber of uninformed v er tices drops b elo w ln 7 n . In the previous section w e argued that T 1 = C d ln n + O (ln ln n ), wh ere C d is give n in Corollary 10. The m ain aim of th is section is to pr o v e that the br oadcasting of th e message completes after additional O ((ln ln n ) 2 ) r ounds. Th is is s ho wn in the next lemma. Lemma 11. With pr ob ability 1 − o (1) we have T − T 1 = O ((ln ln n ) 2 ) . Pr o of. Be fore w e sho w the claim let us pro ve a au x iliary fact. Let S b e an y subset of the v ertices of e G ( n, d ) of size at most ln 7 n . W e will sho w that with pr ob ab ility 1 − o (1) e ( S ) < 1 . 1 | S | . T o see the claim, s u pp ose that th ere is an S suc h that e ( S ) ≥ 1 . 1 s , where we set s = | S | . There are  n s  ≤ ( en s ) s c hoices for the set S . Moreov er, there are at m ost s 2 . 2 s w a ys to c ho ose 1 . 1 s ed ges in S . Finally , the p r obabilit y that the chosen edges are in e G ( n, d ) is M dn − 2 . 2 s M dn , where M x denotes th e num b er of p erfect matc hings on x vertic es. Using (3.8 ) we inf er that P ( ∃ S : e ( S ) ≥ 1 . 1 | S | ) ≤ (1 + o (1))  en s  s · s 2 . 2 s · e dn/ 2 ( dn ) dn/ 2 ( dn − 2 . 2 s ) dn/ 2 − 1 . 1 s e dn/ 2 − 1 . 1 s ≤ (1 + o (1))( e 2 . 1 s 1 . 2 n ) s · ( dn ) − 1 . 1 s . This expr ession is n − Ω(1) for any s ≤ ln 7 n ; this conclud es the pr o of of the auxiliary claim. In particular, e G ( n, d ) is suc h th at any set S of at most ln 7 n ve rtices satisfies with ro om to spare e ( S, V n \ S ) ≥ ( d − 2 . 2) s ≥ ds/ 4 . With this fact at han d it is routine to complete the pro of of the lemma. Ind eed, let S b e the s et of uninf orm ed v ertices at some p oin t in time after T 1 . So, | S | ≤ ln 7 n . As e ( S, V n \ S ) ≥ ds/ 4, we know that at least s/ 4 vertice s in S ha v e at least one n eigh b or in V n \ S . More precisely , there is a set S ′ ⊆ S suc h that | S ′ | ≥ s/ 4 and f or all v ∈ S ′ there is at least one v ′ ∈ V n \ S su c h that v and v ′ are joined by an edge. Denote b y B the ev ent that after 10 ln ln n rounds there is a v ∈ S ′ that w as not informed b y v ′ . The probabilit y for this even t is at most | S ′ | ·  1 − 1 d  10 ln l n n ( | S ′ |≤ ln 7 n ) = o (ln − 1 n ) . So, after 10 ln ln n round s the new s et of un informed v ertices has size at most | S \ S ′ | ≤ 3 4 | S | . Iterating the ab ov e argum en t O (ln ln n ) times fi nally completes the pro of.  RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 15 5. Ran domized bro adcast ing on exp a nding graphs: proof of Theorem 2 In this sectio n we p ro v e Theorem 2, th us b oun ding the broadcast time on conn ected ( n, d, λ ) graphs with λ = O ( √ d ). In order to a void an y confusion w e s tr ess that this condition is in terpreted as follo ws: there is a C > 0 such that for any n su fficien tly large λ ≤ C √ d . W e will use the main result f r om [14]. Before w e state it, let us first introdu ce the n otion of a ( p, ε )- typic al grap h . A grap h G = G ( V , E ) on n vertic es is called ( p, ε )- typic al , if the follo wing three conditions are satisfied: • F or an y S ⊆ V with | S | ≥ ε 2 n , there is a set X S ⊆ V \ S with | X S | ≤ 8 n ln n suc h that ∀ v ∈ ( V \ S ) \ X S : d S ( v ) = (1 ± ε ) p | S | . • F or any S ⊆ V with | S | ≤ ε 2 n , there is a set X S ⊆ V \ S with | X S | ≤ ε | S | such that ∀ v ∈ ( V \ S ) \ X S : d S ( v ) ≤ εpn. • F or all S ⊆ V we ha ve e ( S, V \ S ) = | S | ( n − | S | ) p  1 ± 8 √ ε  . The follo wing app ears in [14]. Lemma 12. L et ε = ε ( n ) b e a p ositive r e al-value d function such that ε ( n ) → 0 , as n → ∞ , but ε ≥ ln − 1 / 9 n . L et p ≥ 1 ε 2 ln n n . If G is a ( p, ε ) -typic al gr aph and v ∈ V , then with pr ob ability 1 − o (1) | T ( G, v ) − (log 2 n + ln n ) | ≤ 3 ε 1 / 3 ln n. W e will sho w that an ( n, d, λ ) graph is ( p, ε )-t ypical with p = d/n and ε ≥ ln 1 / 9 n . In particular, w e will p ro v e the first t w o conditions b y sampling uniformly at r an d om a v ertex in V , a nd then sh o wing with Chebysc hev’s in equalit y that its degree in a giv en set S is concen trated around its exp ecte d v alue which, as we shall see, equals d | S | /n . Let A b e the adjacency matrix of G and let e 1 , . . . , e n b e an orthonorm al basis of R n consisting of the eigenv ectors of A , ordered according to the m o duli of the corresp ond ing eigen v alues λ 1 , . . . , λ n . S ince G is d -regular and connected, we h a v e e 1 := 1 √ n [1 , . . . , 1] T (cf. Prop osition 2.10 in [20]) an d the corresp onding eigen v alue is d . F or the sak e of notational con v enience, we w ill fix an ordering on V , n amely v 1 , . . . , v n and w e will assume that the i th en try of eac h v ector corresp ond s to v i . Let S b e an arbitrary s ubset of V and let χ S b e the c haracteristic v ector of S , that is, the v ector indexed b y V where the elemen ts corresp onding to the vertic es of S are equal to 1 and the r emaining ones are equ al to 0. W e set d S := Aχ S and n ote that d S = [ d S ( v 1 ) , . . . , d S ( v n )] T . Let v b e a vertex in V c hosen u niformly at rand om. T h us E ( d S ( v )) = 1 n P u ∈ V d S ( u ). Note that this sum is just h d S ,e 1 i √ n , where h· , ·i denotes the usual dot pro duct in R n . On the other hand, we can express d S = Aχ S also by taking the exp ansion of χ S with resp ect to the basis e 1 , . . . , e n and then m ultiplying b y A . Note that h χ S , e 1 i e 1 = | S | n [1 , . . . , 1] T . Thus χ S = | S | n [1 , . . . , 1] T + X i ≥ 2 h χ S , e i i e i . Therefore Aχ S = d | S | n [1 , . . . , 1] T + X i ≥ 2 λ i h χ S , e i i e i . 16 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU Since e 1 is orthogonal to the ve ctors e 2 , . . . , e n , we ha v e h d S , e 1 i = h Aχ S , e 1 i = d | S | n √ n, implying that ¯ d := E ( d S ( v )) = d | S | n . In the follo wing we will b ound the v ariance of d S ( v ). W rite V ar ( d S ( v )) = D /n , where D := P u ∈ V d 2 S ( u ) − n ¯ d 2 . But P u ∈ V d 2 S ( u ) = k d S k 2 . By Pythagoras’ Theorem k d S k 2 = n X i =1 h d S , e i i 2 = n ¯ d 2 + n X i =2 h d S , e i i 2 . Therefore, D = P n i =2 h d S , e i i 2 . T o b ound the latter sum note that n X i =2 h d S , e i i 2 = n X i =2 h Aχ S , e i i 2 = n X i =2 h χ S , Ae i i 2 = n X i =2 λ 2 i h χ S , e i i 2 ≤ λ 2 n X i =2 h χ S , e i i 2 = λ 2  k χ S k 2 − h χ S , e 1 i 2  = λ 2  | S | − | S | 2 n  = λ 2 | S |  1 − | S | n  . Th us V ar( d S ( v )) = D n ≤ λ 2 | S | n  1 − | S | n  . No w we are ready to d eriv e the first tw o conditions of the definition of ( d/n, ε )-t ypicalit y , when λ ≤ C √ d . • Let S b e such th at | S | ≥ ε 2 n . T hen th e size of X S is b ounded f rom abov e by n P ( | d S ( v ) − ¯ d | > ε ¯ d ). W e b ound this probabilit y with Chebysc hev’s inequalit y . Indeed, P ( | d S ( v ) − ¯ d | > ε ¯ d ) ≤ V ar( d S ( v )) ε 2 ¯ d 2 ≤ λ 2 | S | /n ε 2 ¯ d 2 = λ 2 dε 2 ¯ d ≤ nC 2 ε 2 d | S | ≤ C 2 ε 4 d . By the c hoice of ε , the ab o v e b ound is at most 8 / ln n and therefore | X S | ≤ 8 n/ ln n . • No w let | S | ≤ ε 2 n . Thus ¯ d ≤ dε 2 . Here the size of X S is b oun ded from ab ov e b y n P ( d S ( v ) > dε ). Since for n large enough dε − dε 2 > dε/ 2, this p robabilit y is at most P ( d S ( v ) − ¯ d > dε/ 2). Again, Chebysc hev’s inequalit y implies P ( d S ( v ) − ¯ d > dε/ 2) ≤ 4V ar( d S ( v )) ε 2 d 2 ≤ 4 λ 2 | S | /n ε 2 d 2 ≤ 4 λ 2 ε 2 ε 2 d 2 = 4 λ 2 d 2 ≤ 4 C 2 d . Th us | X S | ≤ 4 C 2 n d . W e w an t to d educe that this is at most ε | S | . W e ma y assum e that | S | ≥ dε , as otherwise what w e are aiming at holds trivially . S o, it suffices to deduce that 4 C 2 n d ≤ dε . But this holds by our assumption d ≥ p 4 C 2 n ln 1 / 9 n ≥ √ 4 C 2 ε − 1 n . The third condition in the definition of ( d/n, ε )-t ypicalit y is a standard p rop erty of ( n, d, λ ) graphs. Theorem 13 (Theorem 2.11 in [20]) . L et G = G ( V , E ) b e an ( n, d, λ ) gr aph. Then for any two sub se ts U, W ⊂ V we have     e ( U, W ) − d | U || W | n     ≤ λ s | U || W |  1 − | U | n   1 − | W | n  . RUMOR SPREADING ON RANDOM REGULAR GRAPHS AN D EXP ANDERS 17 W e set U = S and W = V \ S . Th en the ab o v e implies     e ( S, V \ S ) − d | S | ( n − | S | ) n     ≤ λ s | S | ( n − | S | )  1 − | S | n   1 − n − | S | n  = λ | S | ( n − | S | ) n = λ d d | S | ( n − | S | ) n ≤ C √ d d | S | ( n − | S | ) n . Since d ≥ √ 4 C 2 ε − 1 n , we ha ve C √ d ≤ ε 1 / 4 2 √ n . But ε ≥ ln − 1 / 9 n and , therefore, the latter b ound is at most 8 ε 1 / 2 , as required to satisfy th e third condition. Ac kno wledgmen t W e wo uld like to thank Colin McDiarmid for suggesting the u se of his concen tration inequalit y (Theorem 4), whic h greatly f acilitat ed our pro ofs. Referen ces [1] E.A. Bender and E.R. Canfield, The asymptotic number of lab elled graphs with given degree sequences, J. Combin. The ory Ser. A 24 (1978), 296–307. [2] B. Bollob´ as, A probabilistic pro of of an asymp totic form ula for the num b er of lab elled regular graphs, Eur op. J. Combin. 1 (1980), 311–3 16. [3] S. Botros and S. W aterhouse, Search in jxta and oth er distributed netw orks, In Pr o c e e dings of the 1st IEEE International Conf er enc e on Pe er-to-Pe er Computing (P2P ’01) , 2001, pp. 30–35. [4] M. Bradonjic, R Els¨ asser, T. F riedrich, T. Sauerw ald and A. S tauffer, Efficien t broadcast on rand om geometric graphs, I n Pr o c e e dings of the 21st Annual ACM-SIAM Symp osium on Discr ete Algorith ms (SOD A ’10) , 2010, pp. 1412–142 1. [5] F.R.K. Chung, R. Graham and R.M. Wilson, Qu asi-random graphs, Combi natoric a 9 (1989), 345–362. [6] Clip2.com I nc. Gnutella: T o the band width barrier and b eyond. Published online, 2000. [7] A. Demers, D. Greene, C. Hauser, W. Irish, J. Lar son, S. Shenker, H. S turgis, D. Swi nehart and D. T erry , Epidemic algorithms for replicated database main tenance, I n Pr o c e e dings of the 6th Annual ACM Sym- p osium on Principl es of D i stribute d Computing (PODC ’ 87) [8] D. D ubhashi and A. Panconesi, Conc entr ation of M e asur e for the Analysis of R andomize d A lgorithms , Cam bridge U niversi ty Press, 2009. [9] R. D urrett, R andom Gr aph Dynamics , Cam bridge Universit y Press, New Y ork, 2007. [10] R. Els¨ asser, On randomized broadcasting in pow er la w netw orks, In Pr o c e e di ngs of the 20th I nternational Symp osium on Di stribute d Computing (DISC ’06) , 2006, pp. 370–384. [11] R. Els¨ asser, L. Gasieniec and T. Sauerwald, On radio broadcasting in random geometric graphs, In Pr o c e e di ngs of the 22nd International Symp osium on Distribute d Computing (DISC ’08) , 2008, pp. 212– 226. [12] R. Els¨ asser and T. Sauerwald, On Broadcasting vs. mix ing and information dissemination on Caley graphs, I n Pr o c e e dings of the 24th Symp osium on The or etic al Asp e cts of Computing (ST ACS ’07) , 2007, pp. 163–174. [13] U. F eige, D. P eleg, P . Ragha v an, and E. Upfal, R andomized broadcast in n etw ork s, R andom Structur es and Algorithms 1 (4) (1990), 447–460. [14] N. F ountoulakis, A. H ub er and K. Panagi otou, Reliable broadcasting an d the effect of den sit y , to app ea r in Pr o c e e di ngs of INFOCOM ’10 . [15] J. F riedman, A pro of of Alon’s second eigenv alue conjecture, In Pr o c e e dings of the 35th annual ACM Symp osium on The ory of Computing (STOC ’03) , 2003 ,pp. 720–724. [16] A.M. F rieze and G.R. Grimmett, The shortest-path prob lem fo r graphs with rand om arc-lengths, Discr ete Appl. Math. 10 (1985), 57–77. [17] S. Ho ory , N . Linial, and A. Wigderson, Expander graphs an d their applications, B ul l. AMS 43 (2006), 439–561 . [18] S. Jagannathan, G. Pandrurangan an d S. Sriniva san, Qu ery proto cols for highly resilien t p eer-to-p eer netw orks. I n Pr o c e e dings of the I SCA 19th International Confer enc e on Par al lel and Distribute d Com- puting Systems (ISCA PDCS’ 06) , 2006, p p. 247–252 . [19] S. Janson, T. Luczak and A. Ru ci´ nski, R andom Gr aphs , Wiley , 2000. [20] M. K rivel evich and B. S udako v, Pseudo- random graphs, In Pr o c e e di ngs of the Confer enc e on Fi ni te and Infinite Sets, Bolyai So ci ety Mathematic al Studies 15 , Springer,2006, pp. 199–262 . 18 NIKOLA OS FOUNTOULAKIS AND KONST ANTINOS P ANA GIOTOU [21] C. Law and K.-Y. S iu, D istributed construction of random ex pander netw oks, In Pr o c e e dings of IEEE INFOCOM’ 03 , 2003, pp . 2133–2143. [22] C. McDiarmid, Concentration for indepen dent permutations, Combinatorics, Pr ob ability and Computing (2002) 11 , 163–178. [23] A. Nilli, On the second eigenv alue of a graph, Discr ete Math. 91 (1991), 207–2 10. [24] G. P andurangan, P . Ragha v an and E. Upfal, Building lo w-diameter p eer-to-p eer net w orks, IEEE Journal on Sele cte d Ar e as in Communic ations 21 (2003), 995–1002. [25] M. T alagrand, Concentration of measure and isoperimetric inequalities in p rodu ct spaces, Inst. Hautes ´ Etudes Sci. Publ. Math. 81 (1995), 73–205. [26] A. Thomason, Pseudo-random graphs, In Pr o c e e dings of R andom Gr aphs (M. Karonski, ed.), 1987, pp . 307–331 .