Convergence Speed of Binary Interval Consensus

Con v ergence Sp eed of Binary In terv al Consensus ∗ Mo ez Draief † Milan V o jno vi´ c ‡ No vem b er 27, 2024 Abstract W e consider the conv ergence time for solving the binary consensus problem using the in terv al consensus algorithm proposed by B ´ en ´ ezit, Thiran and V etterli (2009). In the binary consensus problem, each no de initially holds one of t wo states and the goal for eac h no de is to correctly decide which one of these t w o states was initially held by a ma jorit y of no des. W e deriv e an upp er bound on the expected con vergence time that holds for arbi- trary connected graphs, which is based on the lo cation of eigenv alues of some contact rate matrices. W e instan tiate our bound for particular netw orks of interest, including complete graphs, paths, cycles, star-shap ed netw orks, and Erd¨ os-R ´ en yi random graphs; for these graphs, we compare our b ound with alternativ e computations. W e find that for all these examples our b ound is tight, yielding the exact order with resp ect to the n umber of no des. W e pinp oint the fact that the exp ected conv ergence time critically dep ends on the v oting margin defined as the difference b e t ween the fraction of nodes that initially held the ma jority and the minorit y states, resp ectiv ely . The c haracterization of the expected con vergence time yields exact relation b et ween the expected con vergence time and the v oting margin, for some of these graphs, which reveals ho w the exp ected conv ergence time go es to infinit y as the voting margin approac hes zero. Our results pro vide insigh ts into ho w the exp ected conv ergence time dep ends on the net work top ology whic h can b e used for performance ev aluation and netw ork design. The results are of in terest in the con text of netw ork ed systems, in particular, p eer-to- p eer netw orks, sensor net works and distributed databases. 1 In tro duction Algorithms for distributed computation in netw orks ha ve recen tly attracted considerable in terest b ecause of their wide-range of applications in netw ork ed systems such as p eer-to- p eer net w orks, sensor net works, distributed databases, and on-line social net w orks. A specific algorithmic problem of in terest is the so called binary c onsensus [2, 3, 4, 5] where, initially , eac h node in the netw ork holds one of tw o states and the goal for eac h no de is to correctly ∗ A preliminary v ersion without proofs of this w ork first app eared in [1] † Imp erial College London, SW7 2AZ London, UK (m.draief@imperial.ac.uk). ‡ Microsoft Researc h, J.J. Thomson Av enue, CB3 0FB Cambridge, UK, ( milan v@microsoft.com) 1 decide which one of the t w o states w as initially held b y a ma jority of no des. This is to b e achiev ed b y a decen tralized algorithm where each no de main tains its state based on the information exchanged at con tacts with other no des, where the con tacts are restricted b y the net work top ology . It is desired to reach a final decision by all nodes that is correct and within small con vergence time. A typical application scenario of the binary consensus corresp onds to a set of agen ts who w ant to reach consensus on whether a given ev ent has o ccurred based on their individual, one-off collected, information. Such co op erativ e decision-making settings arise in a n umber of applications suc h as en vironmental monitoring, surv eillance and securit y , and target track- ing [6], as well as voting in distributed systems [7]. F urthermore, it has b een noted that one can use multiple binary consensus instances to solv e multiv alued consensuses; we refer to [8, 9] for an account on such algorithms. W e consider a decen tralized algorithm kno wn as interval or quantize d c onsensus prop osed b y B ´ en ´ ezit, Thiran, and V etterli [4]. The aim of this algorithm is to decide whic h one of k ≥ 2 partitions of an in terv al con tains the a v erage of the initial v alues held by individual no des. In this paper, we fo cus on binary interv al consensus, i.e. the case k = 2. An attractiv e feature of the interv al consensus is its accuracy; it was shown in [4] that for an y finite connected graph that describ es the net work top ology , the in terv al consensus is guaran teed to conv erge to the correct state with probability 1. How ever, the follo wing imp ortant question remained op en: How fast do es the interval c onsensus c onver ge to the final state? W e answer this question for the case of binary in terv al consensus. The in terv al consensus could b e considered a state-of-the-art algorithm for solving the binary consensus problem as it guarantees conv ergence to the correct consensus (i.e. has zero probability of error) for arbitrary finite connected graphs. Besides, it only requires a limited amount of memory and communication by individual no des (only four states). Some alternativ e decen tralized algorithms require fewer states of memory or comm unication but fail to reach the correct consensus with strictly p ositive probabilit y . F or instance, the traditional v oter mo del requires only tw o states of memory and communication. It is how ev er kno wn that there are graphs for whic h the probability of error is a strictly p ositive constan t, e.g. prop ortional to the n um b er of no des that initially held the minority state in the case of complete graphs (see [10] for the general setting). Another example is the ternary proto col prop osed in [3] for which it was shown that for complete graphs, the probabilit y of error diminishes to zero exp onen tially with the num b er of no des, but provides no impro v ement o ver the voter mo del for some other graphs (e.g. a path). In this paper, w e pro vide an upp er b ound on the expected con vergence time for solving the binary in terv al consensus on arbitrary connected graphs. This provides a unified approac h for estimating the expected con v ergence time for particular graphs. The b ound is tigh t in the sense that there exists a graph, namely the complete graph, for which the b ound is ac hieved asymptotically for large num b er of no des. W e demonstrate ho w the general upp er b ound can be instan tiated for a range of particular graphs, including complete graphs, paths, cycles, star-shap ed netw orks and Erd¨ os-R ´ en yi random graphs. Notice that the complete graph and the Erd¨ os-R ´ en yi random graph are go o d appro ximations of v arious unstructured and structured p eer-to-p eer net works and that star-shap ed netw orks capture the scenarios where some no de is a h ub for other no des. Our results pro vide insights into how the exp ected conv ergence time dep ends on the net- 2 work structur e and the voting mar gin , where the latter is defined as the difference b etw een the fraction of no des initially holding the ma jorit y state and the fraction of no des initially holding the minority state. F or the net w ork structure, we found that the exp ected con ver- gence time is determined by the sp ectral prop erties of some matrices that dictate the contact rates b etw een no des. F or the v oting margin, w e found that there exist graphs for whic h the v oting margin significantly affects the exp ected conv ergence time. Complete graph example F or concreteness, w e describ e ho w the voting margin affects the exp ected con vergence time for the complete graph of n no des. Let us denote with α > 1 / 2 the fraction of no des that initially held the ma jority state, and th us α − (1 − α ) = 2 α − 1 is the voting margin. W e found that that the conv ergence time T satisfies I E( T ) = 1 2 α − 1 log( n )(1 + o (1)) . Therefore, the exp ected conv ergence time is in v ersely prop ortional to the voting margin, and th us, go es to infinit y as the v oting margin go es to 0. Hence, alb eit the interv al consensus guaran tees con vergence to the correct state, the exp ected conv ergence time can assume large v alues for small v oting margins. Outline of the P ap er In Section 2 we discuss the related w ork. Section 3 introduces the notation and the binary interv al consensus algorithm considered in this pap er. Section 4 presen ts our main result that consists of an upp er b ound on the exp ected conv ergence time that applies to arbitrary connected graphs (Theorem 1). Section 5 instan tiates the upp er b ound for particular graphs, namely complete graphs, paths, cycles, star-shap ed net works and Erd¨ os-R ´ enyi random graphs, and compares with alternative analysis. W e conclude in Section 6. Some of the pro ofs are deferred to the app endix. 2 Related W ork In recent y ears there has b een a large b o dy of research on algorithms for decen tralized computations ov er net w orks, under v arious constraints on the memory of individual no des and communication betw een the no des. F or example, in the so called quantize d c onsensus problem [2, 11], the goal is to appro ximately compute the mean of the v alues that reside at individual no des, in a decentralized fashion, where no des communicate quantized information. In [5], the authors provided b ounds on the conv ergence time in the context of av eraging algorithms where no des exchange quan tized information. The work that is most closely related to ours is [4] where the authors sho w ed that the so called interv al consensus algorithm guarantees correctness for arbitrary finite connected graphs. In particular, their work sho ws that for solving the binary consensus problem, it suffices to use only two extr a states to guarantee conv ergence to the correct consensus in a finite time, for every finite connected graph. Our work adv ances this line of work b y establishing the first tight c haracterizations of the exp ected conv ergence time for the binary in terv al consensus. 3 Previous w ork on the binary consensus problem considered algorithms under more strin- gen t assumptions on the n umber of states stored and communicated by individual no des. The standard voter mo del is an algorithm where eac h no de stores and communicates one of tw o states (0 or 1), where eac h instigator no de switches to the state observed from the con tacted no de. The v oter model has b een studied in the con text of v arious graph top olo- gies [12, 13, 14] and the probabilit y of reaching the correct consensus (i.e. corresp onding to the initial ma jorit y state) is kno wn in closed-form for arbitrary connected graphs [10]. Sp ecif- ically , the probability of reac hing the correct consensus is prop ortional to the sum of degrees of the no des that initially held the initial ma jorit y state. In particular, for the complete graphs, this means that the probabilit y of reac hing an incorrect consensus is prop ortional to the n umber of no des that initially held the minorit y state. Moreov er, for some net work top ologies, the conv ergence time of the voter mo del is known to b e quadratic in the num b er of no des, e.g. for a path [15]. In fact, It was shown in [16] that it is imp ossible to solv e the binary consensus problem without adding extra memory to enco de the states of the nodes. Both [4] and [17] show that adding one additional bit of memory is sufficient. In [3], the authors considered a ternary proto col for binary consensus problem where each no de stores and comm unicates an extra state. It was sho wn that for the complete graph in teractions, the probabilit y of reaching the incorrect consensus is exponentially decreasing to 0 as the num b er of nodes n grows large, with a rate that depends on the v oting margin. Moreo ver, if the algorithm conv erges to the righ t consensus, then the time it tak es to complete is logarithmic in the n umber of no des n , and is indep endent of the v oting margin. Similar results hav e b een derived in [18] for the complete graph where, instead of using an extra state, each no des p olls more than one neigh b our at a time and then c ho oses the opinion held b y the ma jorit y of the p olled neighbours. Notice that this is unlik e to the binary interv al consensus, for which we found that the exp ected con vergence time, for the complete graph of n no des, is logarithmic in n , but with a factor that is dep enden t on the v oting margin and going to infinit y as the voting margin approac hes zero. The main adv an tage of the binary in terv al consensus algorithm o v er b oth these proto cols is the guaranteed con vergence to the correct final state with probabilit y 1, alb eit this seems to b e at some exp ense with respect to the con vergence time for some graphs. Finally , w e would lik e to mention that in a bigger picture, our work relates to the cas- cading phenomena that arise in the context of so cial netw orks [19]; for example, in the viral mark eting where an initial idea or b ehaviour held by a p ortion of the p opulation, spreads through the net work, yielding a wide adoption across the whole p opulation [20]. 3 Algorithm and Notation In this section, w e in tro duce the interv al consensus algorithm for the binary consensus prob- lem. Eac h no de is assumed to b e in one of the follo wing four states 0, e 0 , e 1 and 1, at ev ery time instant. It is assumed that the states satisfy the following order relations 0 < e 0 < e 1 < 1. Let us first describ e a simple example to illustrate the up dating proto col. Example Assume that we ha ve four no des lab elled 1 , 2 , 3 , 4, forming a line netw ork 1 − 2 − 3 − 4, starting in state (1 , 0 , 0 , 0). If the first in teraction o ccurs b etw een no de 1 and 4 no de 2 then, as they disagree, b oth of them b ecome undecided. More precisely , the state of no de 1 turns into e 0 indicating that she saw opinion 0 and the state of no de 2 turns in to e 1 indicating that she saw opinion 1. The new vector of states b ecomes ( e 0 , e 1 , 0 , 0). If the following in teraction is betw een nodes 3 and 4 then nothing happ ens. If no des 1 and 2 in teract again then their states are sw app ed, i.e. the v ector of states b ecomes ( e 1 , e 0 , 0 , 0). No w supp ose that no des 2 and 3 interact then they swap their states, i.e. the vector of states b ecomes ( e 1 , 0 , e 0 , 0). This transition indicates that no de 2 was undecided and sa w opinion 0 so she adopts 0 whereas no de 3 saw an undecided no de so she b ecomes undecided. If no des 1 and 2 interact then the v ector of states b ecomes ( e 0 , 0 , e 0 , 0). This indicates that no de 1 sa w an undecided no de so b ecomes undecided whereas no de 2 w as undecided so it adopted the opinion of node 1. After this stage, the dynamics do es not settle as if a no de in e 0 and a node in 0 in teract then they swap their opinions but the num b er of nodes in eac h of the states e 0 and 0 sta ys constant whic h is indicativ e of the fact that the initial ma jority held the opinion 0. W e no w describ e the set of rules for up dating the states of the no des. State update rules The states held b y the no des are up dated at pairwise contacts b et ween no des according to the follo wing state up date rules: 1. If a no de in state 0 and a no de in state 1 get in contact, they switch their states to state e 1 and state e 0 , resp ectively . 2. If a no de in state e 0 and a no de in state 1 get in contact, they switch their states to state 1 and state e 1 , resp ectively . 3. If a no de in state e 1 and a no de in state 0 get in contact, they switch their states to state 0 and state e 0 , resp ectively . 4. If a no de in state e 0 and a no de in state 0 get in contact, they swap their states to state 0 and e 0 , resp ectively . 5. If a no de in state e 1 and a no de in state 1 get in contact, they swap their states to state 1 and state e 1 , resp ectively . 6. If a no de in state e 0 and a no de in state e 1 get in con tact, they swap their states to state e 1 and state e 0 , resp ectively . F or an y other states of a pair of no des that get in contact, their states remain unc hanged. T emp oral pro cess of pairwise in teractions W e admit the standard asynchronous com- m unication mo del [3, 21] where any pair of no des ( i, j ) interacts at instances of a Poisson pro cess with rate q i,j ≥ 0. W e denote with V = { 1 , 2 , . . . , n } the set of no des. The in teraction rates are sp ecified b y the matrix Q = ( q i,j ) i,j ∈ V assumed to b e symmetric 1 . The transition matrix Q induces an undirected graph G = ( V , E ) where there is an edge ( i, j ) ∈ E if and only if q i,j > 0. W e assume that graph G is connected. 1 W e assume that i contacts j at a rate p i,j ≥ 0 so that i and j interact at a rate q ij = p ij + p j i . Therefore, defining P = ( p ij ) i,j ∈ V , we ha v e Q = P + P T is symmetric matrix, i.e. q i,j = q j,i for every i, j ∈ V and q i,i = 0 for ev ery i ∈ V . 5 Tw o con vergence phases The state update rules ensure that in a finite time, a final state is reac hed in whic h all no des are either in state 0 or state e 0 (state 0 is initial ma jority). W e distinguish t w o phases in the conv ergence to the final state, whic h will b e a key step for our analysis of the exp ected conv ergence time that relies on separately analyzing the tw o phases. The tw o conv ergence phases are defined as follows: Phase 1 (depletion of state 1). This phase b egins at the start of the execution of the algorithm and lasts until none of the no des is in state 1. Whenev er a no de in state 0 and a no de in state 1 get in con tact, they switch to states e 1 and e 0 , resp ectiv ely . It is therefore clear that the n um b er of no des holding the minorit y state (state 1) decreases to 0 in a finite amoun t of time and from that time onw ards, the num b er of no des in state 0 remains equal to the difference of the (initial) num b er of no des in state 0 and state 1. Phase 2 (depletion of state e 1 ). This phase follows the end of phase 1 and lasts until none of the no des is in state e 1 . In this phase, the num b er of no des in state e 1 decreases following eac h contact betw een a no de in state e 1 and a no de in state 0. Since, in this phase, no in teraction betw een a pair of nodes results in increasing the n um b er of no des in state e 1 , there are even tually no no des in state e 1 . The duration of eac h of the tw o phases is ensured to b e finite for arbitrary finite connected graphs by the definition of the state up date rules where sw apping of the states enables that state 1 no des get in contact with state 0 no des and similarly , enables that state e 1 no des get in contact with state 0 no des. Additional notation W e denote b y S i ( t ) the set of no des in state i ∈ { 0 , e 0 , e 1 , 1 } at time t . With a sligh t abuse of notation, in some cases w e will use the compact notation | S i | ≡ | S i (0) | , i = 0 , 1, whic h should b e clear from the con text. W e define α ∈ (1 / 2 , 1] as the fraction of nodes that initially hold state 0, assumed to b e the initial ma jority . Therefore, | S 0 | = αn and | S 1 | = (1 − α ) n . 4 General Bound for the Exp ected Con v ergence Time In this section we presen t our main result that consists of an upper b ound on the exp ected con vergence time for arbitrary connected graphs. The b ound is in terms of eigen v alues of a set of matrices Q S that is defined using the transition matrix Q as follows. Let S b e a non-empt y subset of the set of vertices V of size smaller than n and let S c = V \ S . W e consider the matrix Q S = ( q S i,j ) i,j ∈ V that is deriv ed from the con tact rate matrix Q as follo ws q S i,j =    − P l ∈ V q i,l , i = j q i,j , i ∈ S c , j 6 = i 0 , i ∈ S, j 6 = i. (1) W e first establish that eigenv alues of the matrices Q S , for S ⊂ V non-empty , are strictly negativ e. This will b e a key prop ert y that ensures finiteness of our b ound whic h we presen t later in this section. 6 Lemma 1 F or every finite gr aph G , we define δ ( Q, α ) as fol lows δ ( Q, α ) = min S ⊂ V , | S | n ∈ [2 α − 1 ,α ] | λ Q S | = min S ⊂ V , | S | =(2 α − 1) n | λ Q S | . (2) wher e λ Q S is the lar gest eigenvalue of Q S . Then, for al l S non-empty, λ Q S < 0 and thenc e δ ( Q, α ) > 0 . Note that identit y (2) is a direct consequence of the Cauch y in terlacing theorem for principal submatrices of orthogonal matrices [22, Theorem 4.3.8, p. 185]. W e next presen t our main result that establishes an upp er bound on the exp ected con- v ergence time that holds for arbitrary connected graphs. Before stating the result, notice that at the end of phase 1 none of the no des are in state 1, (2 α − 1) n no des are in state 0, and the remaining 2(1 − α ) n no des are in either state e 0 or state e 1 . At the end of phase 2, there are exactly (2 α − 1) n no des in state 0 and 2(1 − α ) n no des in state e 0 . The follo wing theorem establishes a general b ound for the exp ected duration of eac h con vergence phases in terms of the num b er of nodes n and the parameter δ ( Q, α ), whic h we introduced in Lemma 1. Theorem 1 L et T 1 b e the first instant at which al l the no des in state 1 ar e deplete d. Then, I E( T 1 ) ≤ 1 δ ( Q, α ) (log n + 1) . F urthermor e, letting T 2 b e the time for al l the no des in state e 1 to b e deplete d, starting fr om an initial state with no no des in state 1 , we have I E( T 2 ) ≤ 1 δ ( Q, α ) (log n + 1) . In p articular, if T is the first instant at which none of the no des is in either state e 1 or state 1 , then I E( T ) ≤ 2 δ ( Q, α ) (log n + 1) . It is worth noting that the ab ov e theorem holds for ev ery p ositive integer n and not just asymptotically in n . The pro of of the theorem is presented in Section 4.2 and here w e outline the main ideas. The pro of pro ceeds b y first separately considering the tw o conv ergence phases. F or phase 1, w e characterize the evolution ov er time of the probability that a no de is in state 1, for ev ery no de i ∈ V . This amounts to a “piecewise” linear dynamical system. Similarly , for phase 2, w e c haracterize the ev olution ov er time of the probabilit y that a no de is in state e 1 , for every giv en no de i ∈ V , and show that this also amounts to a “piecewise” linear dynamical system. The pro of is then completed b y using a sp ectral b ound on the exp ected n umber of no des in state 1, for phase 1, and in state e 1 , for phase 2, whic h is then used to establish the asserted results. 7 Tigh tness of the b ounds The b ound for the con vergence time of the first phase asserted in Theorem 1 is tight in the sense that there exist graphs for whic h the asymptotically dominan t terms of the exp ected conv ergence time and the corresp onding b ound are either equal (complete graph) or equal up to a constant factor (star netw ork). The bound for the second phase is not tight as w e prov e an upp er b ound for it using the worst case for an initial configuration for the start of phase 2. In what follo ws w e pro vide pro ofs for Lemma 1 and Theorem 1. 4.1 Pro of of Lemma 1 Let S b e a non-empty subset of V . First, for the trivial case S = V the matrix Q S is diagonal with diagonal elements ( − P j ∈ V q ij ) i ∈ V whic h are all negativ e since the graph is connected (each no de has at least one neighbour). No w let S suc h that | S | < n . Note that ev ery eigen v alue λ and the asso ciated eigenv ector ~ x of the matrix Q S satisfy the following equations λx i = − q i x i , for i ∈ S λx i = − q i x i + P l ∈ V q i,l x l , for i ∈ S c (3) where q i := P l ∈ V q i,l , for ev ery i ∈ V . On the one hand, it is clear from the form of the matrices Q S , giv en b y (1), that for ev ery i ∈ S , λ = − q i is an eigenv alue of Q S . Since by assumption, the transition matrix Q induces a connected graph G , w e hav e that for every i ∈ V there exists a j ∈ V such that q i,j > 0. Hence, it follo ws that λ < 0. On the other hand, if λ 6 = − q i for any i ∈ S , it is clear from (3) that x l = 0 for every l ∈ S . In fact, since Q is symmetric, the remaining eigenv alues of Q S are the eigenv alues of the symmetric matrix M S = ( m S i,j ) i,j ∈ S c defined by m S i,j =  − P l ∈ V q i,l , i = j ∈ S c q i,j , i, j ∈ S c , j 6 = i Let λ b e suc h an eigenv alue of Q S and let ~ x b e the corresp onding eigen vector and, without loss of generality , assume that || ~ x || 2 2 = P i ∈ V x 2 i = P i ∈ S c x 2 i = 1. Note that λ = λ~ x T ~ x = ~ x T Q~ x . Since Q is symmetric, we ha v e − λ = X i ∈ S c ,j ∈ V q i,j x 2 i + X i,j ∈ S c q i,j x i x j = X i ∈ S c ,j ∈ S q i,j x 2 i − X i,j ∈ S c q i,j x i ( x i − x j ) = X i ∈ S c ,j ∈ S q i,j x 2 i − 1 2 X i,j ∈ S c q i,j ( x i − x j ) 2 . (4) Therefore, it is clear that λ ≤ 0 with λ = 0 only if X i ∈ S c ,j ∈ S q i,j x 2 i + 1 2 X i,j ∈ S c q i,j ( x i − x j ) 2 = 0 . 8 Let W ⊂ S c b e suc h that x i 6 = 0, for i ∈ W , and x i = 0, for i ∈ S c \ W . Since ~ x is an eigen vector, then W is non empt y . If λ = 0, then X i ∈ W,j ∈ S q i,j x 2 i + X i ∈ W,j ∈ S c \ W q i,j x 2 i + 1 2 X i,j ∈ W q i,j ( x i − x j ) 2 = 0 . The ab o ve implies that there are no edges b et ween S and W , and that there are no edges b et w een W and S c \ W , i.e. W is an isolated component, which is a con tradiction since Q corresp onds to a connected graph. Therefore, λ < 0, which pro ves the lemma. 4.2 Pro of of Theorem 1 W e first separately consider the tw o conv ergence phases and then complete with a step that applies to b oth phases. Phase 1: Depletion of no des in state 1 W e describ e the dynamics of the first phase through the follo wing indicators of no de states. Let Z i ( t ) and A i ( t ) b e the indicators that no de i is in state 0 and 1 at time t , resp ectively . The indicator of b eing in either state e 0 or state e 1 at time t is enco ded by A i ( t ) = Z i ( t ) = 0. The system state ev olves according to a contin uous-time Mark ov pro cess ( Z ( t ) , A ( t )) t ≥ 0 , where A ( t ) = ( A i ( t )) i ∈ V and Z ( t ) = ( Z i ( t )) i ∈ V , with the transition rates giv en as follo ws ( Z, A ) →    ( Z −  i , A −  j ) with rate q i,j Z i A j ( Z −  i +  j , A ) with rate q i,j Z i (1 − A j − Z j ) ( Z, A −  i +  j ) with rate q i,j A i (1 − A j − Z j ) where i, j ∈ V and  i is the n -dimensional v ector whose elemen ts are all equal to 0 but the i -th element that is equal to 1. Since Q is a symmetric matrix, we ha ve for every i ∈ V and t ≥ 0, d dt I E( A i ( t )) = − X j ∈ V q i,j I E( A i ( t ) Z j ( t )) − X j ∈ V q i,j I E ( A i ( t )(1 − A j ( t ) − Z j ( t ))) + X j ∈ V q i,j I E ( A j ( t )(1 − A i ( t ) − Z i ( t ))) or, equiv alen tly , d dt I E( A i ( t )) = − X l ∈ V q i,l ! I E( A i ( t )) + X j ∈ V q i,j I E ( A j ( t )(1 − Z i ( t ))) . Let us no w consider the b eha viour of the set S 0 ( t ) of no des in state 0, i.e. S 0 ( t ) = { i ∈ V : Z i ( t ) = 1 } . F rom the ab ov e dynamics, we see that there are in terv als [ t k , t k +1 ) during whic h the set S 0 ( t ) do es not change (the instan ts t k are stopping times of the Mark ov c hain describing the evolution of the algorithm). Let S k ⊂ V b e the set of no des in state 0, for 9 t ∈ [ t k , t k +1 ), and let S c k = V \ S k , i.e. S 0 ( t ) = S k and V \ S 0 ( t ) = S c k , for t ∈ [ t k , t k +1 ). W e then can write, for t ∈ [ t k , t k +1 ), d dt I E k ( A i ( t )) = − X l ∈ V q i,l ! I E k ( A i ( t )) +  P j ∈ V q i,j I E k ( A j ( t )) , i ∈ S c k 0 , i ∈ S k (5) where I E k is the exp ectation conditional on the even t { S 0 ( t ) = S k } . In a matrix form, this giv es d dt I E k ( A ( t )) = Q S k I E k ( A ( t )) , for t k ≤ t < t k +1 , where Q S k is given b y (1). Solving the ab o ve differen tial equation, we ha ve I E k ( A ( t )) = e Q S k ( t − t k ) I E k ( A ( t k )) , for t k ≤ t < t k +1 . Using the strong Marko v prop erty , it is not difficult to see that I E( A ( t )) = I E  e λ ( t ) A (0)  , for t ≥ 0 , where λ ( t ) = Q S k ( t − t k ) + k − 1 X l =0 Q S l ( t l +1 − t l ) , for t k ≤ t < t k +1 . Note that λ ( t ) is a random matrix that dep ends on the stopping times t k . Phase 2: Depletion of no des in state e 1 T o describ e the dynamics in the second phase, let B i ( t ) be the indicator that a no de i ∈ V is in state e 1 at time t . The notation Z i ( t ) has the same meaning as in phase 1, th us Z i ( t ) is the indicator that no de i ∈ V is in state 0 at time t . The indicator that a no de i ∈ V is in state e 0 at time t is enco ded by B i ( t ) = Z i ( t ) = 0. The dynamics in this phase reduces to a con tin uous-time Mark ov pro cess ( Z ( t ) , B ( t )) t ≥ 0 , where Z ( t ) = ( Z i ( t )) i ∈ V and B ( t ) = ( B i ( t )) i ∈ V , with the transition rates given as follo ws, for i, j ∈ V , ( Z, B ) →    ( Z −  i +  j , B −  j ) with rate q i,j Z i B j ( Z −  i +  j , B ) with rate q i,j Z i (1 − B j − Z j ) ( Z, B −  i +  j ) with rate q i,j B i (1 − B j − Z j ) . F rom this, we hav e for every i ∈ V and t ≥ 0, d dt I E( B i ( t )) = − X i ∈ V q i,j I E( B i ( t ) Z j ( t )) − X j ∈ V q i,j I E ( B i ( t )(1 − Z j ( t ) − B j ( t ))) + X j ∈ V q i,j I E ( B j ( t )(1 − Z i ( t ) − B i ( t ))) . Therefore, for ev ery i ∈ V and t ≥ 0, d dt I E( B i ( t )) = − X l ∈ V q i,l ! I E( B i ( t )) + X j ∈ V q i,j I E ( B j ( t )(1 − Z i ( t ))) . 10 Similar to the first phase, we see that there are interv als [ t 0 k , t 0 k +1 ) during which the set S 0 ( t ) do es not c hange (the instants t 0 k are stopping times). Let S 0 k b e such that S 0 ( t ) = S 0 k , for t ∈ [ t 0 k , t 0 k +1 ). Similarly to the first phase, we ha ve I E( B ( t )) = I E h e λ 0 ( t ) B ( t 0 0 ) i , for t ≥ 0 , where λ 0 ( t ) is a random matrix giv en by λ 0 ( t ) = Q S 0 k ( t − t 0 k ) + k − 1 X l =0 Q S 0 l ( t 0 l +1 − t 0 l ) , for t 0 k ≤ t < t 0 k +1 . Note that t 0 0 = T 1 is the instan t at which phase 2 starts (phase 1 ends). Duration of a phase In b oth phases, the pro cess of in terest is of the form I E( Y ( t )) = I E  e λ ( t ) Y (0)  , for t ≥ 0 , where for a (random) positive integer m > 0 and a sequence 0 = t 0 ≤ t 1 ≤ · · · ≤ t m , w e ha ve λ ( t ) = Q S k ( t − t k ) + k − 1 X l =0 Q S l ( t l +1 − t l ) , for t k ≤ t < t k +1 , k = 0 , 1 , . . . , m − 1 . F or phase 1, Y ( t ) ≡ A ( t ) while for phase 2, Y ( t ) ≡ B ( t ). Using techniques as in [23, Chapter 8], we ha ve for or every t ≥ 0, || I E( Y ( t )) || 2 ≤ I E      e λ ( t ) Y (0)     2  ≤ I E      e λ ( t )     || Y (0) || 2  ≤ I E " || e Q S k ( t − t k ) || k − 1 Y l =0 || e Q S l ( t l +1 − t l ) || || Y (0) || 2 # ≤ e − δ ( Q,α ) t I E ( || Y (0) || 2 ) ≤ √ n e − δ ( Q,α ) t where || · || denotes the matrix norm asso ciated to the Euclidean norm || · || 2 . In the ab ov e, we used Jensen’s inequality in the first inequality , follo wed by the prop erty of matrix norms for the second and third inequalities, then Lemma 1 and finally the fact that Y is a n -dimensional v ector with elements taking v alues in { 0 , 1 } . F urthermore, com bining with Cauch y-Sc hw artz’s inequality , we ha ve X i ∈ V I E( Y i ( t )) ≤ || I E( Y ( t )) || 2 || 1 || 2 ≤ n e − δ ( Q,α ) t , for ev ery t ≥ 0 . where 1 = (1 , . . . , 1) T . Therefore, we ha ve I P( Y ( t ) 6 = 0 ) ≤ X i ∈ V I E( Y i ( t )) ≤ n e − δ ( Q,α ) t , for ev ery t ≥ 0 . 11 Let T 0 b e the time at whic h Y ( t ) hits 0 = (0 , . . . , 0) T , whic h corresponds to T 1 for the process A ( t ) and T 2 for the pro cess B ( t ). Then, we ha ve I E( T 0 ) = Z ∞ 0 I P( T 0 > t ) dt = Z ∞ 0 I P( Y ( t ) 6 = 0 ) dt ≤ log( n ) δ ( Q, α ) + n Z ∞ log( n ) δ ( Q,α ) e − δ ( Q,α ) t dt = log( n ) + 1 δ ( Q, α ) whic h completes the pro of of the theorem. 5 Application to P articular Graphs In this section we instan tiate the b ound of Theorem 1 for particular netw orks including complete graphs, paths, cycles, star-shaped net w orks and Erd¨ os-R ´ en yi random graphs. F or all these cases, we compare with alternativ e computations and find that our b ound is of exactly the same order as the exp ected conv ergence time with resp ect to the num b er of no des. F or the complete graph, we also examine the exp ected conv ergence time as the voting margin go es to zero. 5.1 Complete Graph W e consider the complete graph of n > 1 no des where eac h edge e ∈ E is activ ated at in- stances of a P oisson pro cess with rate 1 / ( n − 1), i.e. we hav e q i,j = 1 / ( n − 1) for all i, j ∈ V suc h that i 6 = j . Lemma 2 F or the c omplete gr aph of n > 1 no des and every fixe d α ∈ (1 / 2 , 1] , we have δ ( Q, α ) ≥ (2 α − 1) . Pro of: F or the complete graph, the matrix Q S is as follo ws q S i,j =    − 1 , i = j 1 n − 1 , i ∈ S c , j 6 = i 0 , i ∈ S, j 6 = i. First of all − 1 is an eigen v alue of order | S | . In addition it is not difficult to see that the v ector ~ x suc h that x i = 0 for i ∈ S and x i = 1 for i ∈ S c is an eigen vector of matrix Q S with the eigen v alue − | S | n − 1 . Since in each of the tw o conv ergence phases, the matrices Q S k are suc h that | S k | ≥ (2 α − 1) n , we ha ve | S k | n − 1 ≥ (2 α − 1) n n − 1 . Finally note that the remaining eigen v alues are the eigenv alues of the matrix M S = ( m S i,j ) i,j ∈ S c defined by m S i,j =  − 1 , i = j ∈ S c 1 n − 1 , i, j ∈ S c , j 6 = i 12 One can rewrite M S as M S = − n n − 1 I + 1 n − 1 J , where I is the iden tity matrix and J is the matrix with all its en tries equal to 1. Therefore the remaining eigen v alue of Q S is − n n − 1 with m utiplicity n − | S | − 1 and δ ( Q, α ) ≥ (2 α − 1). Com bining the last lemma with Theorem 1, we ha ve the following corollary . Corollary 1 F or the c omplete gr aph of n > 1 no des, the exp e cte d dur ation of phase i = 1 and 2 satisfies I E( T i ) ≤ 1 2 α − 1 (log( n ) + 1) . In the follo wing, w e will sho w that the latter b ound is asymptotically tight, for large num b er of no des n , for con vergence phase 1. Comparison with an alternativ e analysis F or complete graphs, the con vergence time can b e studied by an analysis of the underlying sto c hastic system that w e describ e in the follo wing. W e first consider the conv ergence phase 1. Let 0 = τ 0 ≤ τ 1 ≤ · · · ≤ τ | S 1 | denote the time instances at which a no de in state 0 and a no de in state 1 get in contact. Recall that | S 0 | and | S 1 | denote the initial num b er of no des in state 0 and state 1, resp ectively . It is readily observ ed that | S 0 ( t ) | = | S 0 | − i and | S 1 ( t ) | = | S 1 | − i , for τ i ≤ t < τ i +1 and 1 ≤ i < | S 1 | . It is not difficult to observe that τ i +1 − τ i , i = 0 , 1 , . . . , | S 1 | − 1, is a sequence of indep enden t random v ariables such that for eac h 0 ≤ i < | S 1 | , τ i +1 − τ i is a minim um of a sequence of ( | S 0 | − i )( | S 1 | − i ) i.i.d. random v ariables with exp onential distribution with mean n − 1. Therefore, the distribution of τ i +1 − τ i is exp onen tial with mean 1 /µ i , for 0 ≤ i < | S 1 | , where µ i = ( | S 0 | − i )( | S 1 | − i ) / ( n − 1). In particular, we ha v e I E( T 1 ) = P | S 1 |− 1 i =0 µ − 1 i , i.e. I E( T 1 ) = ( n − 1) | S 1 |− 1 X i =0 1 ( | S 0 | − i )( | S 1 | − i ) . (6) The ab ov e considerations result in the following prop osition. Prop osition 1 F or every c omplete gr aph with n > 1 no des and initial state such that | S 0 | > | S 1 | > 0 , the dur ation of the first c onver genc e phase is a r andom variable T 1 with the fol lowing exp e cte d value I E( T 1 ) = n − 1 | S 0 | − | S 1 |  H | S 1 | + H | S 0 |−| S 1 | − H | S 0 |  (7) wher e H k = P k i =1 1 i . F urthermor e, for every fixe d α ∈ (1 / 2 , 1] , we have I E( T 1 ) = 1 2 α − 1 log( n ) + O (1) . 13 10 0 10 1 10 2 10 3 10 4 10 5 0 5 10 15 20 n E(T 1 ) 10 0 10 1 10 2 10 3 10 4 10 5 0 5 10 15 20 n E(T 2 ) Figure 1: Complete graph: the exp ected duration of con vergence phases vs. the n um b er of no des n . The initial ma jorit y state is held by d αn e no des, where α = 3 / 4. The solid line is the asymptote log ( n ) / (2 α − 1) while the bars are simulation results with 95% confidence in terv als. F rom the result of the prop osition, we observ e that the exp ected duration of the first phase is log( n ) /δ ( Q, α ), asymptotically for large n , where δ ( Q, α ) is giv en in Lemma 2, thus matc hing the upp er b ound of Theorem 1. F or the prev ailing case of a complete graph, w e can c haracterize the exp ected con vergence time of the first phase as α approaches 1 / 2, i.e. as the v oting margin 2 α − 1 approac hes 0 from ab ov e. W e first consider the limit case where initially there is an equal num b er of no des in state 0 and state 1, i.e. | S 0 | = | S 1 | . F rom (6), it is straightforw ard to note I E( T 1 ) = π 2 6 n (1 + o (1)) . Therefore, we observe that in case of an initial draw, i.e. equal num b er of state 0 and state 1 no des, the exp ected duration of the first phase scales linearly with the net w ork size n . Note that in this case, the second phase starts with no des in state e 0 and state e 1 and obviously no ma jorit y can follow. W e now discuss the case where | S 0 | − | S 1 | is strictly p ositive but small. T o this end, let µ n denote the v oting margin, i.e. µ n = ( | S 0 | − | S 1 | ) /n . F rom (7), is easy to observe that I E( T 1 ) = 1 µ n log( nµ n ) + O (1) . Therefore, we note that for the voting margin µ n = O (1 /n ), I E( T 1 ) = Θ( n ) while for µ n > 0 a fixed constant, we hav e I E( T 1 ) = Θ(log ( n )). F or the intermediate v alues of the voting margin, say for µ n = 1 /n a , for 0 < a < 1, w e ha v e I E( T 1 ) = 1 − a 2 n a log( n ). Finally , w e compare the b ound log ( n ) / (2 α − 1) with sim ulation results, in Figure 1. W e observ e that the bound is tigh t for phase 1 and not tigh t for phase 2, due to our choice of the initial condition in phase 2. Note also that Figure 1 indicates that the exp ected duration of conv ergence phase 2 scales as Θ (log(log n )). 14 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 4 10 5 n E(T 1 ) 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 4 10 5 n E(T 2 ) Figure 2: P ath: the exp ected duration of conv ergence phases vs. the n um b er of no des n , for α = 3 / 4. The solid line is the asymptote in Corollary 2 while the bars are sim ulation results with 95% confidence interv als. 5.2 P aths W e consider a path of n > 1 no des where eac h edge is activ ated at instances of a Poisson pro- cess of rate 1. Therefore, the contact rate matrix Q is given b y q i,i +1 = 1, for i = 1 , . . . , n − 1, q i,i − 1 = 1, for i = 2 , 3 , . . . , n , and all other elements equal to 0. Lemma 3 F or a p ath of n > 1 no des, we have, for α ∈ (1 / 2 , 1] , δ ( Q, α ) = 2  1 − cos  π 4(1 − α ) n + 1  = π 2 16(1 − α ) 2 n 2 (1 + o (1)) . The proof is pro vided in Appendix A.1.The previous lemma, together with Theorem 1, yields the follo wing result. Corollary 2 F or a p ath of n > 1 no des and α ∈ (1 / 2 , 1) , we have for phase i = 1 and 2 , I E( T i ) ≤ 16(1 − α ) 2 π 2 n 2 log( n ) + O (1) . Finally , w e compare the asymptotic b ound with simulation results in Figure 2. The results indicate that the b ound is rather tight for phase 1 and is not tight for phase 2. 15 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 4 10 5 n E(T 1 ) 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 4 10 5 n E(T 2 ) Figure 3: Cycle: the exp ected durations of conv ergence phases vs. the num b er of the no des n . The initial state is such that state 0 is held b y a set of d α n e consecutive no des along the cycle with α = 3 / 4. 5.3 Cycles W e consider a cycle of n > 1 no des where each edge is activ ated at instances of a P ois- son pro cess with rate 1. Therefore, the con tact rate matrix Q is given by q i,i +1 = 1, for i = 1 , . . . , n − 1, q i,i − 1 = 1, for i = 2 , 3 , . . . , n , q 1 ,n = q n, 1 = 1, and all other elements equal to 0. Lemma 4 F or a cycle network of n > 1 no des, we have, for α ∈ (1 / 2 , 1] , δ ( Q, α ) = 2  1 − cos  π 2(1 − α ) n + 1  = π 2 4(1 − α ) 2 n 2 (1 + o (1)) . The pro of is provided in App endix A.2. Corollary 3 F or the cycle with α ∈ (1 / 2 , 1) , we have for phase i = 1 and 2 , I E( T i ) ≤ 4(1 − α ) 2 π 2 n 2 log( n ) + O (1) . Finally , w e compare the last b ound with simulation results in Figure 3. Similar as in other cases, w e observ e that the b ound has the same scaling with the num b er of no des as the exp ected duration of con vergence phase 1, and is not tigh t for conv ergence phase 2. 16 5.4 Star-Shap ed Net works W e consider a star-shap ed netw ork that consists of a h ub no de and n − 1 leaf no des. Without loss of generalit y , let the hub no de b e node 1 and let i = 2 , 3 , . . . , n b e the leaf nodes. The con tacts b et ween a leaf no de and the h ub are assumed to o ccur at instances of a Poisson pro cess of rate 1 / ( n − 1). This setting is motiv ated in practice b y netw orks where a designated no de assumes the role of an information aggregator to which other no des are connected and this aggregator no de has access capacit y of rate 1. The elemen ts of matrix Q are giv en b y q 1 ,i = q i, 1 = 1 / ( n − 1), for i = 2 , 3 , . . . , n and other elemen ts equal to 0. W e ha ve the following lemma for the star-shap ed netw ork that w e defined ab ov e. Lemma 5 F or the star network of n > 1 no des, we have δ ( Q, α ) = n 2( n − 1) 1 − r 1 − 4(2 α − 1) n ! ≥ 2 α − 1 n wher e the ine quality is tight for lar ge n . The pro of is pro vided in App endix A.3. The previous lemma yields the follo wing corollary . Corollary 4 F or the star network with n > 1 no des and every fixe d α ∈ (1 / 2 , 1] , the exp e cte d dur ation of phase i = 1 and 2 satisfies I E( T i ) ≤ 1 2 α − 1 n (log( n ) + 1) . Comparison with an alternative analysis for phase 1 F or the star-shap ed netw ork of n no des, we can compute the exact asymptotically dominan t term of the exp ected duration of phase 1, for large n , which is presented in the following prop osition. Prop osition 2 F or the star-shap e d network of n no des, the exp e cte d time to deplete no des in state 1 satisfies I E( T 1 ) = 1 (2 α − 1)(3 − 2 α ) n log( n ) + O ( n ) . (8) The pro of is provided in App endix B. Notice that the dominan t term in Prop osition 2 is smaller than the upp er b ound in Corollary 4 for the factor 1 / (3 − 2 α ). 17 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 2 α −1 E(T 1 ) / n 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 2 α −1 E(T 2 ) / n Figure 4: Star-shap ed net work: exp ected duration of conv ergence phases versus the v oting margin 2 α − 1, for n = 1000. The solid curv es indicate log ( n ) / (2 α − 1); the dashed line indicates log( n ) / [(2 α − 1)(3 − 2 α )]; the bars indicate 95%-confidence in terv als of estimates obtained by sim ulations. Remark W e only consider the exp ected con v ergence time for phase 1. Similar analysis could b e pursued for phase 2 but is more complicated, b ecause the lumping of the states as done in the pro of for phase 1 cannot b e made. Finally , w e compare our b ound with simulation results in Figure 4. The results indicate that the b ound of Corollary 4 is not tight. W e also observ e that the asymptote in Prop osition 2 conforms well with simulation results. 5.5 Erd¨ os-R ´ en yi Random Graphs W e consider random graphs for whic h the matrix of contact rates Q is defined as follows. Giv en a parameter p n ∈ (0 , 1) that corresponds to the probabilit y that a pair of no des in teract with a strictly p ositiv e rate, w e define the contact rate of a pair of no des i, j ∈ V , i 6 = j , as follo ws q i,j = 1 ( n − 1) p n X i,j where X i,j is a sequence of i.i.d. random v ariables suc h that I P( X i,j = 1) = 1 − I P( X i,j = 0) = p n , for every i, j ∈ V , j 6 = i . The rates are normalized with the factor 1 / ( n − 1) p n , so that for eac h no de, the in teraction rate with other no des is 1. F urthermore, w e assume that p n is chosen suc h that, for a constan t c > 1, p n = c log( n ) n whic h ensures that the induced random graph is connected with high probability . W e ha ve the following lemma. Lemma 6 Supp ose c > 2 2 α − 1 and α ∈ (1 / 2 , 1] . We then have, with high pr ob ability that δ ( Q, α ) ≥ (2 α − 1) ϕ − 1  2 c (2 α − 1)  + O  1 log n  , (9) 18 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 2 α −1 E(T 1 ) 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 2 α −1 E(T 2 ) Figure 5: Erd¨ os-R ´ enyi random graphs: the exp ected duration of conv ergence phases vs. the v oting margin 2 α − 1, for n = 1000 and c = 100. The solid curv es indicate the b ound of Corollary 10; the dashed lines indicate log( n ) / (2 α − 1); the bars indicate 95%-confidence estimates. wher e ϕ − 1 ( · ) is the inverse function of ϕ ( x ) = x log ( x ) + 1 − x , for x ∈ [0 , 1] . The pro of is provided in App endix A.4. F rom the last lemma and Theorem 1, we ha ve the following corollary . Corollary 5 Under c > 2 2 α − 1 and α ∈ (1 / 2 , 1] , we have for the dur ation of phase i = 1 and 2 , T i ≤ 1 (2 α − 1) ϕ − 1  2 c (2 α − 1)  log( n ) + O (1) (10) with high pr ob ability. Remark W e note the following intuitiv e observ ation: the asserted b ound for the exp ected con vergence time for eac h of the phases b oils do wn to that of the complete graph, for large exp ected degree of a no de, i.e. large c . Indeed, this holds b ecause for every fixed α ∈ (1 / 2 , 1], the term ϕ − 1 ( 2 c (2 α − 1) ) go es to 1 as c grows large. Finally , we compare the b ound of Corollary 5 with estimates obtained by simulations in Fig. 5. The results confirm that the b ound is indeed a b ound and that it is not tight, whic h is b ecause of a b ounding technique that we used in the pro of. 6 Conclusion W e established an upper b ound on the exp ected conv ergence time of the binary interv al con- sensus that applies to arbitrary connected graphs. W e show ed that for a range of particular 19 graphs, the b ound is of exactly the same order as the expected conv ergence time with resp ect to the netw ork size. The b ound provides insights into how the netw ork top ology and the v oting margin affect the exp ected con vergence time. In particular, w e sho wed that there exist net work graphs for which the exp ected con v ergence time b ecomes muc h larger when the v oting margin approac hes zero. The established b ound provides a unifying approach to b ound the exp ected con vergence time of binary in terv al consensus on arbitrary finite and connected graphs. An imp ortan t direction of future work is to consider low er b ounds on the con vergence time. In particular, it w ould b e of in terest to better understand ho w to fine tune the in teraction parameters q ij to ac hieve the b est p ossible conv ergence time for a giv en connected graph under given memory and communication constrain ts. Ac k o wledgemen t MD is supp orted b y QNRF through gran t NPRP 09-1150-2-448. MD holds a Leverh ulme T rust Researc h F ellowship RF/9/RF G/2010/02/08. References [1] M. Draief and M. V o jnovic. 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Applie d Mathematics E-Notes , 5:66–74, 2005. 21 A Characterization of δ for P articular Graphs A.1 Pro of of Lemma 3 (P ath) Recall that the matrix Q is the tridiagonal matrix with q i,i +1 = 1, for i = 1 , 2 , . . . , n − 1, q i,i − 1 = 1, for i = 2 , 3 , . . . , n , and all other elements equal to 0. W e will separately consider four cases depending on whether the resp ective end-no de 1 and n is in S or S c . In the following, we denote with ξ A ( λ ), the characteristic p olynomial of a matrix A . Case 1 : 1 ∈ S and n ∈ S c . In this case, w e rep eatedly expand the matrix λI − Q S along the ro ws i ∈ S , using Laplace’s formula, to obtain ξ Q S ( λ ) = ( λ + 1)( λ + 2) | S |− 1 ξ B ( λ ) (11) where the matrix B is a blo ck-diagonal matrix with blo cks B 1 , B 2 , . . . , B b , for 1 < b ≤ | S | that are symmetric tridiagonal matrices of the form       − 2 + c 1 1 0 0 · · · 0 0 1 − 2 1 0 · · · 0 0 0 1 − 2 1 · · · 0 0 · · · · · · · · · · · · · · · · · · · · · 0 0 0 0 · · · 1 − 2 + c 2       (12) where c 1 = c 2 = 0, for i = 1 , 2 , . . . , b − 1 (t yp e 1), and c 1 = 0 and c 2 = 1, for i = b (type 2). Since B is a blo c k-diagonal matrix, notice that ξ B ( λ ) = Q b i =1 ξ B i ( λ ). Hence, together with (11), we hav e that Q S the largest eigenv alue of a blo c k that is either of t yp e 1 or type 2. The eigenv alues of tridiagonal matrices of the form (12) are well known for some v alues of the parameters c 1 and c 2 , see e.g. [24]. In particular, for a m × m tridiagonal matrix of t yp e 1, i.e. for c 1 = c 2 = 0, we ha ve eigen v alues λ k = − 2  1 − cos  π k m + 1  , k = 1 , 2 , . . . , m. (13) F or a m × m tridiagonal matrix of t yp e 2, i.e. for c 1 = 0 and c 2 = 1, we ha ve eigen v alues κ k = − 2  1 − cos  (2 k − 1) π 2 m + 1  , k = 1 , 2 , . . . , m. It is readily chec ked that the largest eigenv alue is κ 1 with m = | S c | . This corresp onds to the case where the no des in the set S are 1 , 2 , . . . , | S | (i.e. form a cluster). Case 2 : 1 ∈ S c and n ∈ S . In this case, w e ha ve that (11) holds but the blo cks of matrix B redefined so that c 1 = 1 and c 2 = 0, for B 1 , and c 1 = c 2 = 0, for B 2 , B 3 , . . . , and B b . It is readily observed that the eigen v alues of the blo ck matrix B 1 are the same as for c 1 = 0 and c 2 = 1 (type 1 tridiagonal matrix in Case 1). Hence, the largest eigenv alue is same as under 22 Case 1. Notice that in this case, it corresp onds to taking no des n − | S | + 1, n − | S | + 2, . . . , and n to b e in the set S . Case 3 : 1 ∈ S c and n ∈ S c . In this case, by the same argumen ts as in Case 1, we ha ve ξ Q S ( λ ) = ( λ + 2) | S | ξ B ( λ ) where B is a blo ck-diagonal matrix with blocks B 1 , B 2 , . . . , B b , 1 < b ≤ | S c | , whic h are of the form (12) such that c 1 = 1 and c 2 = 0 for B 1 , c 1 = c 2 = 0, for B 2 , B 3 , . . . , and B b , and c 1 = 0 and c 2 = 1, for B b . In this case, the largest eigenv alue is κ 1 with m = | S c | − 1. Case 4 : 1 ∈ S and n ∈ S . In this case, we ha ve ξ Q S ( λ ) = ( λ + 1) 2 ( λ + 2) | S |− 2 ξ B ( λ ) where B is a blo c k-diagonal matrix with blo cks B 1 , B 2 , . . . , B b , 1 < b ≤ | S c | that are all of the form (12) with c 1 = c 2 = 0. In this case, the largest eigenv alue is λ 1 with m = | S c | . Finally , we observ e that for each of the four cases, since | S c | ≤ 2(1 − α ) n , w e can take δ ( Q, α ) as asserted in the lemma, whic h completes the pro of. A.2 Pro of of Lemma 4 (Cycle) The pro of is similar to that for a path in Section A.1. W e will see that for the cycle, w e will deal with blo c ks of tridiagonal matrices of the form (12) with c 1 = c 2 = 0. Recall that for the cycle, we ha ve matrix Q suc h that q i,i +1 = 1, for i = 1 , 2 , . . . , n − 1, q i − 1 ,i = 1, for i = 2 , 3 , . . . , n , q 1 ,n = q n, 1 = 1, and all other elemen ts equal to 0. Again, we separately consider the follo wing four cases. Case 1 : 1 ∈ S and n ∈ S c . By successive expansion along ro ws i ∈ S , we obtain ξ Q S ( λ ) = ( λ + 2) | S | ξ B ( λ ) (14) where B is a blo ck-diagonal matrix with blo cks of the form (12) with c 1 = c 2 = 0 (referred to as type 1). Since B is a block-diagonal matrix, we ha v e ξ B ( λ ) = Q b i =1 ξ B i ( λ ), 1 < b ≤ | S c | , where B i is a matrix of type 1. The largest eigenv alue is λ 1 , given in (13), for m = | S c | . Case 2 : 1 ∈ S c and n ∈ S . In this case, the same argumen ts hold as in Case 1. Case 3 : 1 ∈ S c and n ∈ S c . In this case, (14) holds, with matrix B with diagonal blo c ks and other elemen ts as in Case 1, except that b 1 , | S c | = b | S c | , 1 = 1. This matrix can b e transformed in to a blo ck-diagonal matrix of the same form as in Case 1 b y p erm uting the rows of the matrix, hence, it has the same sp ectral prop erties as the matrix B under Case 1. Sp ecifically , this can b e done b y moving the block of ro ws that corresp ond to B 1 to the b ottom of the matrix B . Case 4 : 1 ∈ S and n ∈ S . In this case, the same arguments apply as in Case 1. Finally , w e observ e that for each of the four cases, since | S c | ≤ 2(1 − α ), w e can take δ ( Q, α ) as asserted in the lemma, which completes the pro of. 23 A.3 Pro of of Lemma 5 (Star) W e separately consider the t wo cases for whic h either the h ub is in the set S or not. Case 1 Supp ose that the hub is in the set S , i.e. 1 ∈ S . It is easy to observe that in this case, the matrix Q S is a triangular matrix with all upp er diagonal elements equal to 0, and the diagonal elements equal to ( − 1 , − 1 n − 1 , . . . , − 1 n − 1 ). Hence, the largest eigenv alue is − 1 n − 1 . Case 2 Supp ose no w that the hub is not in the set S , i.e. 1 ∈ S c . If λ is an eigenv alue of Q S with an eigen vector ~ x , then we ha ve λx 1 = − x 1 + 1 n − 1 X i ∈ S c \{ 1 } x i λx i = − x i n − 1 + x 1 n − 1 , for i ∈ S c \ { 1 } λx i = − 1 n − 1 x i , for i ∈ S. This implies λx 1 = − x 1 + 1 n − 1 X i ∈ S c \{ 1 } x i x 1 = (( n − 1) λ + 1) x i , for i ∈ S c \ { 1 } λx i = − 1 n − 1 x i , for i ∈ S. Supp ose that ~ x is such that x i = 0, for every i ∈ S . F rom the last ab ov e iden tities, it readily follo ws that λ is a solution of the quadratic equation λ 2 + n n − 1 λ + 1 n − 1  1 − | S c | − 1 n − 1  = 0 . It is straigh tforward to show that the tw o solutions are λ 1 = − 1 2 n n − 1 1 − r 1 − 4 | S | n 2 ! and λ 2 = − 1 2 n n − 1 1 + r 1 − 4 | S | n 2 ! . Clearly , the largest eigen v alue is λ 1 and since | S | ≥ (2 α − 1) n , it is maximized for | S | = (2 α − 1) n . Finally , w e note that the largest eigen v alue is attained in Case 2, which establishes the first equalit y in the lemma, from whic h the asserted inequalit y and its tightness readily follow. A.4 Pro of of Lemma 6 (Erd¨ os-R ´ en yi) F rom (4), note that for every S ⊂ V such that 0 < | S | < n , if λ is an eigen v alue of matrix Q S , then λ = − X i ∈ S c ,j ∈ S q i,j x 2 i − 1 2 X i,j ∈ S c q i,j ( x i − x j ) 2 ≤ − min i ∈ S c ( X j ∈ S q i,j ) . 24 since P i ∈ S c x 2 i = 1. In the following, we would like to find a v alue x n > 0 suc h that min i ∈ S c n P j ∈ S q i,j o > x n holds with high probability . T o this end, w e consider the proba- bilit y that the latter even t do es not hold, i.e., for x n > 0, w e consider p e := I P min i ∈ S c ( X j ∈ S q i,j ) ≤ x n ! . W e first show that the following b ound holds p e ≤ ¯ p e with ¯ p e = 2(1 − α ) n exp  − (2 α − 1)( n − 1) p n ϕ  x n 2 α − 1  (15) where we define ϕ ( x ) = x log ( x ) + 1 − x , for x ≥ 0. T o see this, note that for ev ery fixed θ > 0, p e = I P [ i ∈ S c ( X j ∈ S q i,j < x n )! ≤ | S c | I P X j ∈ S q i,j < x n ! ≤ | S c | e θx n I E  e − θ ( n − 1) p n X i,j  | S | = | S c | e θx n  1 + p n  e − θ ( n − 1) p n − 1  | S | where the first inequalit y follows b y the union b ound, the second inequalit y b y the Chernoff ’s inequalit y , and the third equalit y b y the fact that X i,j is a Bernoulli random v ariable with mean p n . Since | S c | ≤ 2(1 − α ) n and | S | ≥ (2 α − 1) n , we ha ve p e ≤ 2(1 − α ) ne θx n  1 + p n  e − θ ( n − 1) p n − 1  (2 α − 1) n . F urthermore, using the fact 1 + p n ( e − θ ( n − 1) p n − 1) ≤ exp( p n ( e − θ ( n − 1) p n − 1)), it follo ws p e ≤ 2(1 − α ) ne θx n +(2 α − 1) np n  e − θ ( n − 1) p n − 1  . It is straigh tforw ard to c heck that the righ t-hand side in the last inequalit y is minimized for θ = − ( n − 1) p n log  x n 2 α − 1  and for this v alue is equal to ¯ p e giv en in (15). Requiring ¯ p e ≤ 1 n is equiv alen t to ϕ  x n 2 α − 1  ≥ 2 log( n ) + log(2(1 − α )) (2 α − 1) np n . F rom this it follo ws that λ ≤ − x n holds with high probability provided that x n ≥ 0 can b e c hosen suc h that ϕ  x n 2 α − 1  ≥ 2 c (2 α − 1)  1 + log(2(1 − α )) 2 log( n )  . (16) 25 Suc h a v alue x n exists as ϕ ( x ) is a decreasing function on [0 , 1], with b oundary v alues ϕ (0) = 0 and ϕ (1) = 0, and under our assumption, c (2 α − 1) > 2, the right-hand side in (16) is smaller than 1 for large enough n . Finally , from (16), we note x n ≥ (2 α − 1) ϕ − 1  2 c (2 α − 1)  + O  1 log( n )  from which the asserted result follo ws. B Pro of of Prop osition 2 Let H ( t ) denote the state of the h ub at time t . Due to the symmetry of the considered graph, it is not difficult to observ e that the dynamics is fully describ ed b y a contin uous-time Mark ov pro cess ( H ( t ) , | S 0 ( t ) | , | S 1 ( t ) | , | S e 0 ( t ) | , | S e 1 ( t ) | ) t ≥ 0 . W e need to compute the exp ected v alue of the smallest time t such that | S 1 ( t ) | = 0, i.e. the time when the state 1 becomes depleted. T o this end, it suffices to consider ( H ( t ) , | S 0 ( t ) | , | S 1 ( t ) | , | S e ( t ) | ) where S e ( t ) = S e 0 ( t ) ∪ S e 1 ( t ), i.e. the system states e 0 and e 1 are lump ed in to one state, which w e denote with e . The system will be said to b e in mo de i at time t , whenever the num b er of depleted state 1 no des before time t is equal to i , for i = 0 , 1 , . . . , | S 1 (0) | − 1. Notice that if the system is in mo de i at time t , then | S 0 ( t ) | = | S 0 (0) | − i , | S 1 ( t ) | = | S 1 (0) | − i , and | S e ( t ) | = | S e (0) | + 2 i . F or simplicit y , w e will use the following notation x i 0 = | S 0 (0) | − i , x i 1 = | S 1 (0) | − i , and x i e = | S e (0) | + 2 i , for ev ery giv en mo de i . W e will compute the exp ected so journ time in each of the mo des by analyzing a discrete- time Marko v chain φ i = ( φ i k ) k ≥ 0 , for given mo de i , defined as follows. This Marko v chain is embedded at time instances at whic h the hub no de interacts with a leaf no de. The state space of φ i consists of the states 0, 1, e , e ∗ with the transition probabilities given in Figure 6. The state of the Mark ov c hain φ i indicates the state of the hub at contact instances of the h ub with the leaf no des, where we introduced an extra state e ∗ to enco de the ev ent where the hub is in state e and that this state was reac hed by the hub from either state 0 or 1, th us indicating a depletion of state 1. Note that the exp ected duration of mo de 0 is equal to the mean hitting time of state e ∗ for the Marko v chain φ 0 started at either state 0 or 1, while the exp ected duration of mo de i , for 0 < i < | S 1 (0) | , is equal to the mean hitting time of state e ∗ for the Marko v chain φ i started at state e . W e compute these mean hitting times in the follo wing. Fix an arbitrary mode i , and then let ϕ s ( i ) b e the mean hitting time of state e ∗ for the Marko v c hain φ i started at state s = 0, 1, and e . By the first-step analysis, w e ha ve that the latter mean hitting times are the solution of the following system of linear equations ϕ 0 ( i ) = x i 0 − 1 n − 1 ϕ 0 ( i ) + x i e n − 1 ϕ e ( i ) + 1 ϕ e ( i ) = x i 0 n − 1 ϕ 0 ( i ) + x i e − 1 n − 1 ϕ e ( i ) + x i 1 n − 1 ϕ 1 ( i ) + 1 ϕ 1 ( i ) = x i e n − 1 ϕ e ( i ) + x i 1 − 1 n − 1 ϕ 1 ( i ) + 1 . (17) 26 Figure 6: Star-shap ed net w ork: the transition probabilities of φ i . F rom this, it is straigh tforw ard to derive ϕ 0 ( i ) = ( n − 1) n 2 x i e + x i 0 x i 1 x i 0 x i 1 ( n − x i 0 )( n + x i e ) , (18) ϕ 1 ( i ) = ( n − 1) n 2 x i e + x i 0 x i 1 x i 0 x i 1 ( n − x i 1 )( n + x i e ) , (19) ϕ e ( i ) = ( n − 1) n 2 − x i 0 x i 1 x i 0 x i 1 ( n + x i e ) . (20) The exp ected so journ time in eac h giv en mode is as follo ws. F or mode i = 0, it holds x 0 0 = αn , x 0 1 = (1 − α ) n , x 0 e = 0, and th us, from (18) and (19), the exp ected so journ in mo de 0 is ϕ 0 (0) and ϕ 1 (0), for the initial state of the h ub equal to 0 and 1, resp ectively , where ϕ 0 (0) = n − 1 (1 − α ) n and ϕ 1 (0) = n − 1 αn . (21) On the other hand, for 0 < i < | S 1 (0) | , the exp ected so journ time in mo de i is ϕ e ( i ), and from (20) and x i 0 = | S 0 (0) | − i , x i 1 = | S 1 (0) | − i , x i e = | S e (0) | + 2 i , we ha ve ϕ e ( i ) = ( n − 1)  n 2 ( αn − i )((1 − α ) n − i )( n + 2 i ) − 1 n + 2 i  . (22) Finally , the exp ected duration of phase 1 is equal to ϕ s (0) + P (1 − α ) n − 1 i =1 ϕ e ( i ), where s denotes the initial state of the hub, either 0 or 1. On the one hand, from (21), w e hav e that for ev ery fixed α ∈ (1 / 2 , 1], b oth ϕ 0 (0) and ϕ 1 (0) are asymptotically constants, as the n umber of no des n grows large, th us ϕ 0 (0) = Θ(1) and ϕ 1 (0) = Θ(1). On the other hand, using (22) 27 and some elemen tary calculus, we obtain (1 − α ) n − 1 X i =1 ϕ e ( i ) = n − 1 (2 α − 1)(3 − 2 α ) H (1 − α ) n − 1 − n − 1 (2 α − 1)(1 + 2 α ) [ H αn − 1 − H (2 α − 1) n ] +  2 (2 α − 1)(3 − 2 α ) − 2 (2 α − 1)(1 + 2 α ) − 1  (1 − α ) n − 1 X i =1 n − 1 n + 2 i . where, recall, H k = P k i =1 1 i . F rom this, it can b e observ ed that (1 − α ) n − 1 X i =1 ϕ e ( i ) = 1 (2 α − 1)(3 − 2 α ) n log( n ) + O ( n ) . (23) This completes the pro of of the prop osition. 28