Robustness of Randomized Rumour Spreading

In this work we consider three well-studied broadcast protocols: Push, Pull and Push&Pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized, and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures. We explore in particular the following notion of Local Resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary has to be allowed to delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. It turns out that Pull is robust with respect to all parameters that we consider. On the other hand, Push may slow down significantly, even if the adversary is allowed to modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, Push&Pull is robust when no message transmission failures are considered, otherwise it may be slowed down. On the technical side, we develop two novel methods for the analysis of randomized rumour spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemer\`edi's Regularity Lemma.


Introduction
Randomized broadcast protocols are highly relevant for data distribution in large networks of various kinds, including technological, social and biological networks. Among many others there are three basic models in the literature, introduced in [18,9,26], namely push, pull and push&pull (or short pp). Consider a connected graph in which some vertex holds a piece of information; we call this vertex (initially) informed. All three models have the common characteristic that they proceed in rounds. In the push model, in every round every informed vertex chooses a neighbour independently and uniformly at random (iuar) and informs it; this of course has only an effect if the target vertex was previously uninformed. Contrary, in the pull model every round every uninformed vertex chooses a neighbour iuar and asks for the information. If the asked vertex has the information, then the asking vertex becomes informed as well. The third model push&pull combines both worlds: in each round, each vertex chooses a neighbour iuar, and if one of both vertices is informed, then afterwards both become so. We additionally assume that each message transmission succeeds independently with probability q ∈ (0, 1]. For these algorithms, the main parameter that we consider is the random variable that counts how many rounds are needed until all vertices are informed, and we call these quantities the runtimes of the respective algorithms. In the remainder we will denote the runtime of push by T push (G, v, q) where G is the underlying graph, initially the vertex v is informed and we have a transmission success probability of q ∈ (0, 1]. Analogously we denote the runtimes of pull and push&pull by T pull (G, v, q) and T pp (G, v, q) respectively. If the choice of v does not matter we will omit it in our notation. The most basic case is when G is the complete graph K n with n vertices. Then, see for example Doerr and Kostrygin [11], it is known that for P ∈ {push, pull, pp} and q ∈ (0, 1] in expectation and with probability tending to 1 as n → ∞ T P (K n , q) = c P (q) log n + o(log n), where, for q ∈ (0, 1), c push (q) := 1 log(1 + q) , c pp (q) := 1 log(1 + 2q) and where we set c P (1) := lim q→1 c P (q). If q is clear from the context, we write c P instead of c P (q). Actually, the results in [11] and also [12] are much more precise, but the stated forms will be sufficient for what follows.

Contribution & Related Work
In this article our focus is on quantifying the robustness of all three models. Indeed, robustness is a key property that is often attributed to them, since they are simple, randomized, and, crucially, do not exploit explicitly the structure of the underlying graph (apart, of course, from considering the neighborhoods of the vertices). Clearly, the runtime can vary tremendously between different graphs with the same number of vertices. Hence it is essential to understand which structural characteristics of a graph influence in what way the runtime of rumour spreading algorithms. One result in this spirit for the push model was shown in [27]. Roughly speaking, in that paper it is shown that even on graphs with low density, if the edges are distributed rather uniformly, then push is as fast as on the complete graph. This can be interpreted as a robustness result: starting with a complete graph, one can delete a vast amount of edges and as long as this is done rather uniformly, the runtime of push is affected insignificantly. To state the result more precisely, we need the following notion. Definition 1.1 ((n, δ, ∆, λ)-graph). Let G be a connected graph with n vertices that has minimum degree δ and maximum degree ∆. Let µ 1 ≥ µ 2 ≥ · · · ≥ µ n be the eigenvalues of the adjacency matrix of G, and set λ = max 2≤i≤n |µ i | = max{|µ 2 |, |µ n |}. We will call G an (n, δ, ∆, λ)-graph.
In this paper we are interested in the case where G gets large, that is, when n → ∞. Hence all asymptotic notation in this paper is with respect to n; in particular "with high probability", or short whp, means with probability 1 − o(1) when n → ∞.
to contact multiple neighbours per round [27,11], vertices not calling the same neighbour twice [10] and asynchronous versions [4,28,1,2]. Finally, besides [11], robustness of these rumor spreading algorithms with respect to message transmission failures was also studied by Elsässer and Sauerwald in [13]. It was shown for any graph that if a message fails with probability 1 − p, then the runtime of push increases at most by a factor of 6/p.
In this work our focus is on three subjects concerning the robustness of rumour spreading. Our first (and not unexpected) result extends the validity of Theorem 1.3 to the runtimes of pull and push&pull . In particular, we show that none of the three protocols slows down or speeds up on graphs with good expansion properties compared to its runtime on the complete graph. This motivates to investigate how severely a graph with good expansion properties has to be modified to increase the respective runtimes.
In our second contribution, which is also the main result and which differs from what was treated in previous works, we propose and study an unworn approach to quantifying robustness. In particular, we investigate the impact of adversarial edge deletions, where we use the well-known concept of local resilience, see e.g. [30,8]. To be specific, we explore up to which fraction of edges an adversary needs to be allowed to delete at each vertex to slow down the process by a significant amount of time, i.e., by Ω(log n) rounds.
Here we discover a surprising dichotomy in the following sense. On the one hand, we show that both pull and push&pull cannot be slowed down by such adversarial edge deletions -in essentially all but trivial cases, where the fraction is so large that the graph may become (almost) disconnected. On the other hand, we demonstrate that even a small number of edge deletions is sufficient to slow down push by Ω(log n) rounds. In other words, we find that in contrast to pull and push&pull , the push protocol is not resilient to adversarial deletions and lacks (in this specific sense) the robustness of the other two protocols.
As our third subject, we generalise the previous results by additionally considering message transmission failures that occur independently with probability 1 − q ∈ [0, 1). On the positive side, we show that for arbitrary q ∈ (0, 1] all three algorithms inform almost all vertices at least as fast as in an expander sequence in spite of adversarial edge deletions. However, if we want to inform all vertices, only pull is not slowed down by adversarial edge deletions for all values of q; push can be slowed down as before; and push&pull is a mixed bag, for q = 1 it can not be slowed down, for q < 1 it can. Furthermore, in general it is also possible to speed push&pull up by deleting edges, which is however not surprising as the star-graph deterministically finishes in at most 2 rounds. Summarizing, this work enhances previous (robustness) results, particularly the ones concerning precise asymptotic runtimes and random transmission failures. Crucially, we introduce and study the concept of local resilience as a method to investigate robustness. However, apart from that, in this paper we develop two new general methods for the analysis of rumour spreading algorithms.
• The most common approach in the current literature for the study of the runtime is to determine the expected number of newly informed vertices in one or more rounds and to show concentration, for example by bounding the variance. Achieving this, however, is often quite complex and makes laborious and lengthy technical arguments necessary. Here we use the theory of self-bounding functions, see Section 2, that allows us to cleanly upper bound the variance by the expected value. The argument works for all three investigated algorithms and the bound is valid for all graphs. We are certain that this method will also facilitate future work on the analysis of rumour spreading algorithms. • Studying the robustness of the protocols is a challenging task, as the adversary (as described previously) has various opportunities to modify the graph, for example by introducing a high variance in the degrees of the vertices; this turns out to be particularly problematic in the case of push&pull . Here we demonstrate that such types of irregularities can be handled universally by applying a powerful tool from a completely different area, namely extremal graph theory. In particular, we use Szemerédi's regularity lemma (see e.g. [29]), which allows us to partition the vertex set of a graph such that nearly all pairs of sets in the partition behave nearly like perfect regular bipartite graphs. This allows us to apply our methods on these regular pairs; eventually we obtain a linear recursion that can be solved by analysing the maximal eigenvalue of the underlying matrix.

Results
Our first result addresses the question about how fast rumours spread on expander graphs; in order to obtain a concise statement also the occurrence of independent message transmission failures is considered.
The first statement is an extension of Theorem 1.3 and its proof is a straigthforward adaption of the proof in [27]. We omit it. The contribution here is the proof of (b) and (c). Next we consider the case with edge deletions in addition to the message transmission failures.
Theorem 1.5. Let 0 < ε < 1/2, q ∈ (0, 1] and G = (G n ) n∈N be an expander sequence. LetG = (G n ) n∈N be such that eachG n is obtained by deleting edges of G n such that each vertex keeps at least a (1/2 + ε) fraction of its edges. Then whp (a) T pull (G n , q) = c pull (q) log n + o(log n).
This result demonstrates uncoditionally the robustness of pull , and conditionally on q = 1 the robustness of push&pull on dense graphs, in the case of edge deletions, that is, the runtime is asymptotically the same as in the complete graph. It even shows that push&pull may potentially profit from edge deletions in contrast to being slowed down. The proof of this result, especially the statement about push&pull , is rather involved, since the original graph may become quite irregular after the edge deletions. Here we use, among many other ingredients, the aforementioned decomposition of the graph given by Szemeredi's regularity lemma. Note that Theorem 1.5 does not consider push and push&pull (when q = 1) at all. Indeed, our next result states that in these cases the behaviour is rather different and that the algorithms may be slowed down. Theorem 1.6. Let ε > 0 and q ∈ (0, 1]. Then there is an expander sequence G = (G n ) n∈N and a sequence of graphsG = (G n ) n∈N with the following properties. EachG n is obtained by deleting edges of G n such that each vertex keeps at least a (1 − ε) fraction of its edges. Moreover, whp (a) T push (G n , q) ≥ c push (q) log n + ε/(2q) log n + o(log n).
Nevertheless, not all hope is lost. On the positive side, the next result states that push and push&pull are able to inform almost all vertices as fast as on the complete graph in spite of adversarial edge deletions. In this sense, we obtain an almost-robustness result for these cases. Theorem 1.7. Let 0 < ε < 1/2, q ∈ (0, 1] and G = (G n ) n∈N be an expander sequence. LetG = (G n ) n∈N be such that eachG n is obtained by deleting edges of G n such that each vertex keeps at least a (1/2 + ε) fraction of its edges. For P ∈ {push, pp} letT P denote the number of rounds needed to inform at least n − n/ log n vertices. Then whp (a)T push (G n ) = log 1+q (n) + o(log n).
We conjecture that there is also a version of Theorem 1.7 (b) that is true for push&pull on sparse graphs; to be precise we conjecture that in the setting of Theorem 1.7 (b)T pp (G n ) ≤ log 1+2q (n) + o(log n), without further restrictions on G n , i.e. that push&pull can not be slowed down informing almost all vertices.
As a final remark note that Theorems 1.5 and 1.7 are tight in the sense that if an adversary is allowed to delete up to half of the edges at each vertex, then there are expander graphs that become disconnected such that their components have linear size. On those graphs a linear fraction of the vertices will remain uninformed forever.
Outline The rest of this paper is structured as follows. In Section 2 we collect and prove several important facts; this part of the paper also contains our technical contribution concerning the analysis through selfbounding functions. In Subsection 3.1 we show that pull is as fast on expanders with (or without) deleted edges as it is on the complete graph. Subsection 3.2 treats push&pull on expanders without deleted edges. In the remaining subsections we focus on the cases that may be slowed down by edge deletions. In Subsection 3.3 we show that adversarial edge deletions cannot slow down the time until push has informed almost all vertices, by giving a coupling to the case without edge deletions. Contrary in Subsection 3.4 we show that the time until push has informed all vertices can be slowed down by edge deletions, even if only few edges are deleted. Then, in Subsection 3.5 we show that push&pull informs almost all vertices of dense graphs fast in spite of adversarial edge deletions. We utilize a version of Szémeredis Regularity Lemma to get a well-behaved partition of the vertex set that is suitable for performing a round based analysis. However, if q < 1, adversarial edge deletions can slow down or speed up the time until push&pull has informed all vertices for nearly all values of q; we show this in Section 3.6.
Further Notation Let G = (V, E) denote a graph with vertex set V and edge set E ⊆ V 2 . Consider v ∈ V and U, W ⊆ V with U ∩ W = ∅. We will denote the set of neighbours of v in G by N G (v) or by N (v) and we will denote its degree by d G (v) := |N G (v)| or by d(v); δ G or δ and ∆ G or ∆ denote minimum and maximum degree of G. Similarly the neighbourhood of any set of vertices S ⊆ V is defined by N G (S) := ∪ v∈S N G (v). Furthermore let E(U, W ) = E G (U, W ) denote the set of edges with one vertex in U and one vertex in W and let e(U, W ) := e G (U, W ) := |E G (U, W )|. With E G (U ) we denote the set of edges with both vertices in U ; e G (U ) = |E G (U )|. For any round t ∈ N and P ∈ {push, pull, pp}, we denote by I (P) t (G) the set of vertices of G informed by push, pull and push&pull respectively at the beginning of round t and |I (P) 1 | = 1; if the underlying graph is clear from the context we will omit it; if we consider a sequence of graphs G = (G n ) n∈N and a sequence of times t = (t(n)) n∈N , then I

Tools & Techniques
In this section we collect and prove statements about our protocols and properties of expander sequences. We begin with applying the previously mentioned notion of self-bounding functions to derive universal and simple-to-apply concentration results for our random variables, i.e., the number of informed vertices after a particular round. Then we extend the concentration results to more than one round. In the last part we recall the well known Expander Mixing Lemma and utilize it to derive properties (weak expansion, path enumeration) for the case where we delete edges from our graphs.
Self-bounding functions. Our main technical new result in this section is the following bound on the variance for the number of informed vertices in any given round; it is true for any graph and any set of informed vertices. Lemma 2.1. Let G be a graph, t ∈ N and I t = I (P) t (G) for P ∈ {push, pull, pp}. Then Lemma 2.1 follows directly from Lemmas 2.3 and 2.4. Before stating them we introduce the notion of self-bounding functions.
Definition 2.2 (Self-bounding function). Let X be a set and m ∈ N. A non-negative function f : X m → R is self-bounding, if there exist functions f i : X m−1 → R such that for all x 1 , ..., x m ∈ X and all i = 1, ..., m As striking property of self-bounding function is the following bound on the variance.
. For a self-bounding function f and independent random variables X 1 , ..., X m , m ∈ N Lemma 2.4. Let G be a graph, t ∈ N, and let I t = I (P) t (G) for P ∈ {push, pull, pp}. Then, conditional on I t , there exist m ∈ N, independent random variables X 1 , ..., X m and a self-bounding function f = f (P) such that |I t+1 | = f (X 1 , ..., X m ).
Proof. We will prove in detail the result for push, and then we show what needs to be modified in order to obtain the statement in the case of pull and push&pull . Let I t = I (push) t , n ∈ N be number of vertices of G and f : [n] |It| → R with Moreover, let (X i ) 1≤i≤|It| be independent random variables, where X i is a uniformly random neighbour of the ith vertex -according to an arbitrary ordering -in I t . We argue that f (X 1 , . . . , X |It| ) = |I t+1 |. Consider v ∈ I t , then v is counted by the |I t | term in f . For v ∈ I t+1 \I t let v 1 , . . . , v s ∈ I t , s ∈ N be the informed vertices with random neighbour v in round t, i.e. X v1 = · · · = X vs = v and X u = v for all other u ∈ I t .
The function f i arises from f by leaving the ith variable out of consideration, i.e., the push of the ith vertex has no effect. Then by definition f − f i ∈ {0, 1} for all 1 ≤ i ≤ |I t |, and actually we have This quantity is precisely the difference in informed vertices after round t, assuming the ith vertex did not push. Furthermore Thus f has the self-bounding property, which establishes the claim in the case of push. The proof for pull is completely analogous, where we use and, similarly, for push&pull we use f (pp) : [n] n → R with Here it is useful to see that the two sums in f (pp) are complementary, i.e. that only one of the summands for index k can be 1. Thus the functions f Remark 2.5. Let G = (V, E) be a graph. Lemma 2.4 also applies to subsets of I t+1 , i.e for any U ⊂ V and conditioned on I t we have that |I t+1 ∩ U | and |(I t+1 ∩ U ) \ I t | are self-bounding.
The following lemma gives a tool that we will use in order to extend our round-wise analysis to longer phases. Proposition 2.6. Let P ∈ {push, pull, pp}, I t = I (P) t and t 1 ≥ t 0 ≥ 1 such that |I t0 | ≥ √ log n. Let further (A i ) i∈N be a sequence of events, c > 1, and δ > 0 such that Proof. Using the definition of conditional probability we obtain, as c > 1, We give two typical example applications of this lemma below. The first example addresses the case where we have a lower bound for the expected number of informed vertices after one round.
Assume that there is some c > 1 such that E t [|I t+1 |] ≥ c |I t | for all t as long as n/f (n) ≤ |I t | ≤ n/g(n) for some functions 1 ≤ f, g ≤ n, f = o(n). Let t 0 be such that |I t0 | ≥ n/f (n). Then according to Lemma 2.1 we have that Var t [|I t+1 |] ≤ E t [|I t+1 |] and applying Chebychev's inequality gives

Consider the events
The intersection of A t0+1 , . . . , A t implies inductively that either |I t | ≥ n/g(n) or We obtain with (2.1) . . , A t , |I t | ≥ n/g(n)] = 1. Choose τ := t − t 0 = log c (f (n)/g(n)) + o(log n) as small as possible such that this lower bound for |I t+1 | is ≥ n/g(n), that is, this lower bound is < n/g(n) for t = t 0 + τ . Combining the two conditional probabilities we obtain for all Applying Proposition 2.6 then yields whp |I t0+τ +1 | ≥ n/g(n).
In the second example we make the stronger assumption that we can determine asymptotically the expected number of informed vertices after one round. Here we assume that we begin with a "small" set of informed vertices, say of size √ log n, and want to reach a set of size nearly linear in n.
Using this notation, the events A t0+1 , . . . , A t+1 imply together inductively that for all t such that the right-hand side is bounded by n/ log n. Moreover, for all such t Thus, as A t only depends on I t it follows with (2.1) Applying Proposition 2.6 then immediately gives that there is τ 1 = log c (n/|I t0 |) + o(log n) such that whp |I t0+τ1 | ≤ n/ log n. Example 2.7, setting f = n/ √ log n and g = log n, gives an additional Expander Sequences. In this section we collect some important properties of expander sequences that we are going to use later. We start by stating a version of the well-known expander mixing lemma applied to our setting of expander sequences.
The following result is a consequence of the Expander Mixing Lemma that applies to graphs in which some edges were removed. It seems very simple but it turns out to be surprisingly useful.
where eachG n it is obtained from G n by deleting edges such that each vertex keeps at least a (1/2+ε) fraction of its edges. For each n ∈ N let further S n ⊆ V n , then there is n 0 ∈ N such that for all n ≥ n 0 eG n (S n , V n \S n ) ≥ εe Gn (S n , V n \S n ). Proof. Without loss of generality we assume that |S n | ≤ n/2. Since at most (1/2 − ε)∆ n edges are deleted at each vertex, we immediately obtain that Using Lemma 2.9 and choosing n 0 large enough such that o(∆n) ∆n n n−|Sn| < ε for all n ≥ n 0 , we obtain that As n − |S n | ≥ n/2 the last expression is > 0. Hence Next we give a lemma that counts the number of paths between two arbitrary vertices of a dense graph satisfying a weak expander property (as for example guaranteed by Lemma 2.10). This will later be used to give a lower bound on the probability of any vertex to be informed after a given constant number of rounds.
Next comes a technical lemma that given a small set quantifies the number of vertices for which only a small fraction of their neighbourhood intersects that given set.
Lemma 2.12. Let G = (G n ) n∈N = ((V n , E n )) n∈N be an expander sequence. Let ε > 0 and letG = (G n ) n∈N , where eachG n it is obtained from G n by deleting edges such that each vertex keeps at least a (1/2+ε) fraction of its edges. Let further A n ⊆ V n with |A n | = o(n).
Proof. Let δ n , ∆ n denote the minimum and maximum degree of G n . Lemma 2.9 yields that As there are a maximum of ∆ n |A n | edges with at least one point in A n , we get that e Gn (A n ) = o(∆ n )|A n |.
Since we obtainG n from G n by deleting edges With this fact at hand we show a). Let η > 0 and call a vertex To see the b) let again η > 0 and call this time We conclude our preparational section by giving a lemma that bounds crudely the time needed until at least ω(1) vertices are informed.
Lemma 2.13. Let 0 < ε ≤ 1/2, q ∈ (0, 1] and G = (G n ) n∈N be an expander sequence. LetG = (G n ) n∈N be such that eachG n is obtained by deleting edges of G n such that each vertex keeps at least a (1/2 + ε) fraction of its edges. Let further P ∈ {push, pull, pp} and suppose that |I t | < √ log n. Then there is τ = o(log n) such that whp |I Similarly we obtain for push The same bound is obviously also true for push&pull . Thus, for all P ∈ {push, pull, pp} As Lemma 2.9 and Lemma 2.10 imply that e(U t ,

Proof of Theorems 1.4 (b), 1.5 (a) -edge deletions do not slow down pull
Let 0 < ε ≤ 1/2. In this section we study the runtime of pull in the case in which the input graph is an expander, and where at each vertex at most an (1/2 − ε) fraction of the edges is deleted. The runtime on expander sequences without edge deletions, that is, the setting in Theorem 1.4 (b), is included as the special case where we set ε = 1/2. In contrast to previous proofs, in the analysis of pull the 'standard' approach that consists of showing, for example, that E t [|I t+1 \ I t |] ≈ |I t | fails. The main reason is that the graph between I t and U t might be quite irregular, so that, depending on the actual state, E t [|I t+1 \ I t |] ≈ c|I t | for some c < 1. However, we discover a different invariant that is preserved, namely that the number of edges between I t and U t behaves in an exponential way. With Lemmas 2.9 and 2.10 we can then relate this to the number of informed vertices.
Proof. We start with a). Let D t = e(U t+1 , I t+1 ) − e(U t , I t ) and for u ∈ U t let X u be the random variable that indicates whether u gets informed in round t + 1. Then To get a lower bound consider a largest setŨ According to Lemmas 2.9 and 2.10 we have that e(U t , In the next step we bound the variance. For each edge e let X e be the indicator random variable that denotes the events that e ∈ E(U t+1 , I t+1 ). Thus Using that X e and X e ′ are independent for all e, e ′ ∈ E with e ∩ e ′ = ∅, .
Next we show b). We bound the expected number of uninformed vertices after one additional round. Lemma 2.12 (a) asserts that there is a setŨ As Proof. We start with a). Let |I t | ∈ [log n, n/ log n]. First note that any bound on e(U t , I t ) translates to a bound for |I t |, as with Lemmas 2.9, 2.10 we obtain (3.1) In particular, up to constant factors, |I t | is e(U t , I t )/∆ n and vice versa. From Lemma 3.1 (a) we obtain that e(U t+1 , or |I t | ≥ n/ log n" and "e(U t+1 , I t+1 ) = (1 + q ± |I t | −1/3 )e(U t , I t )" we obtain the statement.
Note that for q = 1 this already implies Theorems 1.4 (b), 1.5 (a). This leaves the case for q = 1.
Proof. We consider a modified process in which vertices have a higher chance of getting informed. In particular, note that the probability that u ∈ U t gets informed is at most q|N (u) ∩ I t |/|N (u)| ≤ q and that all these events are independent; now we assume that each such u gets independently informed with probability exactly q. Then the runtime in this modified model constitutes a lower bound for the runtime in the original model.
Let c < 1, u ∈ U t and E u be the event that u does not get informed in cτ rounds in this model. Thus and as the events E u are independent and |U t | = Θ(n)

Proof of Theorem 1.4 (c) -push&pull is fast on expanders
As we are now in the case without edge deletions, we begin with a lemma that determines the expected number of informed vertices in one round. Intuitively we will show that push and pull do not interact badly and therefore push&pull is given as a straightforward combination of push and pull . (a) Let |I t | ≤ n/ log n.
Proof. To begin with a). The probability that v ∈ U t gets informed by pull is q|N (v) ∩ I t |/|N (v)|. Thus, using Lemma 2.9 Since |I t | = o(n) we obtain that |U t | = (1 − o(1))n and this expression simplifies to (q + o(1))|I t |. The probability that v ∈ U t gets informed by push . Using e −1/n+o(1/n) = 1 − 1/n, e −1/n = 1 − 1/n + o(1/n), and |I t | = o(n) we obtain in a similar fashion  , | for all u ∈ I. Since push and pull happen independently Using that |N (u) ∩ I t | = o(|N (u)|) for all u ∈ I we obtain (1))|I t |, as claimed.
Next we show b). Let A u be the event that an uninformed vertex u does not get informed by the push algorithm, let B u be the corresponding event for pull . Then A u and B u are independent and A u ∩ B u is the event that u does not get informed in the current round. We obtain According to Lemma 2.12 (a) there is a set U ⊆ U t , For the lower bound we need to find a lower bound on the probability of a single uninformed vertex not getting informed in one round by push. Indeed, for any u ∈ U t and sufficiently large n Combining this inequality with the trivial bound P [B u ] ≥ 1 − q, we get a lower bound on the expected number of uninformed vertices after one round using push&pull : Next we show upper and lower bounds that together with Lemma 2.13 imply Theorem 1.4 (c). . Let q ∈ (0, 1]. Then the following statements hold whp.

Case q = 1: Then there is
Proof. Since |I t | ≥ |I (pull) t | the statements b) and c) for q = 1 follow immediately from Lemma 3.2. To see a), note that by using Lemma 3.4 we get E t [|I t+1 \I t |] = (2q + o(1))|I t |, and proceeding as in Example 2.8 implies the claim.
Note that for q = 1 this already implies Theorem 1.4 (c). This leaves the case for q = 1.
Proof. We consider a modified process in which vertices have a higher chance of getting informed. In particular, note that the probability that u ∈ U t gets informed by pull is at most q|N (u) ∩ I t |/|N (u)| ≤ q and that all these events are independent; according to (3.5) the probability that u ∈ U t gets informed by push is at most 1 − e −q∆n/δn . Now we assume that each such u gets independently informed with probability exactly 1 − e −q∆n/δn (1 − q). Then the runtime in this modified model constitutes a lower bound for the runtime in the original model. Let u ∈ U t and E u be the event that u does not get informed in this modified model in cτ rounds. Thus for c < 1, and as the events E u are independent and |U t | = Θ(n)

Proof of Theorem 1.7 (a) -push informs almost all vertices fast in spite of edge deletions
To shorten the notation let us call the setting with deleted edges "new model" and the setting without "old model", that is, the term new model corresponds to the graphs inG, while old model refers to the (original) graphs in G. We prove Lemma 3.7 that directly implies Theorem 1.7 (a). We write I t = I (push) t throughout.
Lemma 3.7. Under the assumptions of Theorem 1.7 (a) the following holds for the new model: a) There are τ,τ = log 1+q (n) + o(log n) such that whp |Iτ | < n/ log n < |I τ |.
b) Assume |I t | ≥ n/ log n. Then there is a τ = o(log n) such that whp |I t+τ | ≥ n − n/ log n.
For the proof of Lemma 3.7 we will need the following statements, the first one taken from [27].
Lemma 3.8 (Proof of Lemma 2.5 in [27]). Consider the old model. Assume |I t | < n/ log n and q = 1. Then Lemma 3.9. Consider push on a sequence of graphs (G n ) n∈N , where G n has n vertices. Assume that |I t | = ω(1) and that (3.6) holds for q = 1, that is, assume that Moreover, assume that whenever |I t | < n/ log n, for q = 1, (3.6) holds. Then there are τ,τ = log 1+q (n) + o(log n) such that whp |Iτ | < n/ log n < |I τ |. Proof. For a graph G and for v ∈ I t let X v (G) denote the vertex to which v pushes in round t. Let Note that whenever |I t | < n/ log n whp |N t+1 | = (1 − o(1))|I t | from (3.6). For q ∈ (0, 1] each vertex in N t+1 has a probability of at least q to get informed and all these events are independent; thus (3.7) follows directly by applying the Chernoff bounds whenever |I t | = ω(1).
In order to prove the second statement we call a round t that does not satisfy (3.7) a failed round. Note that we just argued that the probability that a round fails is o(1) whenever |I t | = ω(1) and |I t | < n/ log n, and the events that distinct rounds fail are independent. In particular, the number of failed rounds among the next R rounds, assuming that |I t | stays below n/ log n, is whp o(R). Moreover, if a round does not fail, the number of informed vertices increases by a factor of (1 + q + o(1)) and otherwise it may increase by an arbitrary factor in the interval [1,2]. Finally, Lemma 2.13 yields that there is t * = o(log n) such that whp |I t * | = ω(1), which implies that after R + t * rounds, the number of informed vertices is whp in the interval and choosing R = log 1+q (n) + o(log n) in two ways establishes (3.8).
Proof of Lemma 3.7. We first show a). We assume q = 1 and prove that, for |I t | < n/ log n, denote the set of informed vertices that are being pushed exactly once in round t and the set of informed vertices that are being pushed at least once in round t respectively. Let denote the set of vertices that are being pushed more than once in round t. Let Y t (G) := |Y t (G)| and H t (G) := |H t (G)| and, in slight abuse of notation, let Z t (G) := k≥2 (k − 1) · |{v ∈ V | c v (G) = k}| denote the number of vertices that are being pushed multiple times in round t counted with multiplicity. Note that the quantity Y + Z denotes the number of pushes that have no effect in the respective round, i.e., there are Y + Z pushes that are useless in the sense that even without them, the same number of vertices would become informed in the respective round. In the following paragraphs we condition on I t implicitly, that is, we write P [. . . ] instead of P t [. . . ] etc. to lighten the notation. We want to show that (3.6) does hold in the new model; for contradiction we assume that this is not the case. Hence we can infer that there is a constant Thus, w.l.o.g., we can assume that there is f * > 0 and n 0 ∈ N such that if this is not the case we can restrict ourselves to a suitable subsequence of (n) n∈N on which it is true. Next, we describe an explicit coupling between the new and the old model. For have by construction the correct marginal distribution. Moreover, note that by construction, the family We begin with the case that P [Y t (G n ) ≥ c|I t |] > f * . We will show and therefore, given Y t (G n ), H t (G n ) dominates a binomially distributed random variable Bin(Y t (G n ), 1/2). In particular, this implies with (3. n 1 , . . . , n |Zt(Gn)| ≥ 1/8 (3.11) and then, since by assumption P [Z t (G n ) ≥ c|I t |] > f * , we obtain P [Z t (G n ) ≥ c/8|I t |] ≥ f * /8 which contradicts Lemma 3.8. Due to (3.10) the events are independent. Moreover, for all 1 ≤ i ≤ |Z t (G n )|, 1 ≤ j ≤ n i , Using (3.12) and (3.13), given Z t (G n ), n 1 , . . . , n |Zt(Gn)| , we infer that Z t (G n ) dominates We treat the two sums individually. Note that i∈M1 max{B i − 1, 0} ∼ Bin(|M 1 |, 1/4); in particular, P [ i∈M1 max{B i − 1, 0} ≥ |M 1 |/4] ≥ 1/4 by (3.9). Regarding the second sum, since i∈M2 B i ∼ Bin( i∈M2 n i , 1/2) we obtain P [ i∈M2 B i ≥ 1/2 i∈M2 n i ] ≥ 1/2. Thus, given Z t (G n ), n 1 , . . . , n |Zt(Gn)| and using 2|M 1 | = i∈M1 n i and i∈M2 n i ≥ 3|M 2 |, we infer that with probability at least 1/4 · 1/2 = 1/8 This establishes (3.11). All in all, for q = 1 we have shown that (3.6) does also hold in the new model. Hence claim a) follows directly from Lemma 3.9.

Proof of Theorem 1.6 (a) -edge deletions slow down push
Let I (push) t := I t . In order to show the claim we construct an explicit sequence of graphs that has the desired property. More precisely, for any ε > 0, each q ∈ (0, 1] and n ∈ N we will define a graph G n (ε) that is obtained by deleting edges from the complete graph on n vertices such that each vertex keeps at least an (1 − ε) fraction of its edges and such that push slows down significantly. We . . , ⌊n/2⌋} and V 2 := {⌊n/2⌋ + 1, . . . , n}, as follows. We include in E all pairs of vertices that intersect V 1 and moreover, we add edges (that now have endpoints only in V 2 ) such that all vertices in V 2 have degree ⌈(1 − ε)n⌉ + 1 ± 1. According to Lemma 3.7 a) there is a t = log 1+q (n) + o(log n) such that whp |I t | < n/ log n. It thus suffices to show that it takes whp at least (1 + ε/2)q −1 log n more rounds to inform all remaining vertices. Let As |I t | < n/ log n we have |U ′ t | ≥ n/4 with plenty of room to spare. In the remainder of this proof we will consider a modified process in which vertices have a higher chance of getting informed; in particular we assume that in each round, all vertices choose a neighbour independently and uniformly at random and after this round the chosen vertices are informed. Let E u denote the event that u ∈ U ′ t does not get informed within the next τ := (1 + ε/2)q −1 log n rounds in this modified model. Each vertex u ∈ U ′ t has ⌊n/2⌋ neighbours that have degree n− 1, at most ⌈(1 − ε)n⌉+ 1 ± 1 − ⌊n/2⌋ ≤ (1/2 − ε)n+ 4 neighbours that have at least degree (1 − ε)n and no further neighbours. Therefore, using that for any a ∈ R we have (1 + a/n) n = e a + O(1/n), we obtain for each u ∈ U ′

In this modified model the events {E
and U ⊆ V \ {u} and for some p = ω(n −1 ). This follows immediately from the previous calculation, as conditioning on an event like "{E v : v ∈ U }" only decreases the number of vertices that can push to u. Thus as

Proof of Theorems 1.5 (b), 1.7 (b) -push&pull informs almost all vertices fast in spite of edge deletions
Before we show the actual proof we will first present an informal argument that contains all relevant ideas and important observations. Let √ log n ≤ |I t | ≤ n/ log n and assume q = 1. In Section 3.3 we proved that for push the informed vertices nearly double in every round for an arbitrary expander sequence with edge deletions and an otherwise arbitrary set I t . For pull this is not true; however, we proved in Section 3.1 that the number of edges between the informed and the uninformed vertices nearly doubles in every round. The first attempt towards the proof of Theorems 1.5 (b), 1.7 (b) then seems obvious: one would try to show that either the vertices triple every round, or the the edges do so, or for example that the product of the two quantities increases by a factor of 9. As it turns out, this is in general not the case; indeed, it is possible to choose an expander sequence, to delete edges such that each vertex keeps at least an (1/2 + ε)-fraction of its neighbors, and to choose a (large) set of informed vertices I t such that after one round whp either |I t+1 | < c|I t | or e(I t+1 , U t+1 ) < ce(I t , U t ) or |I t+1 |e(I t+1 , U t+1 ) < c 2 |I t |e(I t , U t ) for some c < 3. On the other hand and although we have no explicit description of these 'malicious' sets, it seems rather unlikely that such sets will occur several times during the execution of push&pull .
In order to show the claimed running time of push&pull we will impose some additional structure. Let ε > 0. In the subsequent exposition we assume that our graph G -obtained from an expander by deleting edges such that each vertex keeps at least an (1/2 + ε) fraction of the edges -has a very special structure. In particular, we assume that there is a partition Π = (V i ) i∈[k] of the vertex set of G into a bounded number k of equal parts such that E G (V i ) = ∅ for all 1 ≤ i ≤ k and such that the induced subgraph (V i , V j ) looks like a random regular bipartite graph for all 1 ≤ i < j ≤ k. Of course, not every relevant G admits such a partition; however, Szemeredi's regularity lemma guarantees that every sufficiently large graph has a partition that is in a well-defined sense almost like the one described previously, and a substantial part of our proof is concerned with showing that being 'almost special' does not hurt significantly.
Assuming that G is very special let us collect some easy facts. Denote the degree of u ∈ V i in the induced subgraph (V i , V j ) with d ij ; this immediately gives that d G (u) = k ℓ=1 d iℓ , and note that d ii = 0 as there are no edges in V i . Moreover, regular bipartite random graphs fulfil an expander property, that is, Obtaining an Appropriate Regular Partition An important ingredient in the previous sketch was the assumption that the given graph has a partition into a bounded number of equal parts, such that the bipartite graph induced by any two different parts looks like a random regular graph. This assumption is quite strong and very much not true in general. However, restricting ourselves to dense graphs we can actually come quite close to that. Let us begin with some definitions; the statements are taken from [29]. Definition 3.10 (Density). Given a graph G = (V, E) and two disjoint non-empty sets of vertices X, Y ⊆ V , we define the density of the pair (X, Y ) as As usual, if the graph is clear from the context the index will be omitted. The next definition gives a partition that is close to the previously described properties; all sets in the partition have nearly the same size and nearly all pairs behave in a well-defined sense like regular bipartite random graphs.
Next we state Szémeredis Regularity Lemma. It guarantees that we will have a Szemerédi partition if the underlying graph is large enough.
The next lemma gives a useful property of regular pairs. In particular, with the exception of a small set only, all other vertices have a degree that is close to dN , where d is the density of the pair and N the number of vertices in each part. Actually, the statement also is true for arbitrary but not too small subsets of the parts.
We call the set E(U, W ) in Lemma 3.13 the exceptional set of U with respect to W . In particular Lemma 3.13 implies that for every ε-regular pair (U, U ′ ) and all W ⊆ U ′ , |W | ≥ (1 − cε)|U ′ |, c > 0 we have Having done these preparations we can now determine a partition that comes close to the initially described properties.
Lemma 3.14. Consider the setting of Theorems 1.5 (b), 1.7 (b). Then for all η > 0 and k 0 > 1/ √ η there exists n 0 , K 0 ∈ N such that for allG n with n ≥ n 0 there is a (η, k 0 , K 0 )-Szemerédi partition Π = {V i } i∈[k] ofG n with the following property. There is F ⊆ Π with |F | ≤ ηk such that for all V i ∈ Π\F • there are at most ηk non-η-regular pairs (V i , V j ), j ∈ [k], and Proof. According to Lemma 3.12, for all ξ > 0 and k 0 > 1/ √ ξ, there are n 0 , K 0 ∈ N such that for allG n with n ≥ n 0 there is a k ∈ N and a (ξ, k 0 , K 0 )-Szemerédi partition Π = {V i } i∈[k] ofG n . Let F ⊆ Π contain the parts V i ∈ Π such that there are at least √ ξk other parts V j ∈ Π such that the pair (V i , V j ) is not ξ-regular. As there are at most ξk 2 non ξ-regular pairs, we infer that be the exceptional set of V i with respect to V j . On top of that let N i ⊆ V i be the set of points in V i that are in at least √ ξk exceptional sets with respect to parts in Π \ A i . As there are at most k exceptional sets and by Lemma 3.13 each exceptional set has at most 2ξ|V i | vertices, we get that By the definition of F and as u ∈ V i \N i we get that | Vj ∈Ai∪B∪{Vi} V j | ≤ ( √ ξk + √ ξk +1)(n/k +1) ≤ 3 √ ξn. With that at hand and by using d(u) ≥ αn/2 and that the sizes of the parts in Π differ by at most one we obtain The Recursion Relation. In this section we exploit the properties of the partition to study the expected number of informed vertices after one additional round; our aim is to establish a precise version of (3.15).
In the remainder let A F = ( 1≤i≤n 1≤j≤n |a i,j | 2 ) 1/2 denote the Frobenius norm of a matrix A ∈ R n×n . For the next lemma consider the setting of Theorems 1.5 (b), 1.7 (b), i.e., we are given an expander sequence (G n ) n∈N with minimal degree δ n ≥ αn for some α > 0 and an ε > 0. We obtain a sequence of graphs (G n ) n∈N by deleting up to a 1/2−ε fraction of the edges at each vertex in G n . Let further η > 0, k 0 ∈ N and Π = {V i } i∈[k] be the (η, k 0 , K 0 )-Szémeredi partition ofG n as given by Lemma 3.14. For that partition define E i,j := E(V i , V j ) as the exceptional set of V i with respect to V j given by Lemma 3.13, i = j ∈ [k], F and N i as the exceptional sets from Lemma 3.14, i ∈ Π\F . Moreover, let Π i = {V j ∈ Π\F : (V i , V j ) is η-regular} and note that Finally, define This definition guarantees that |I t | ≥ X t 1 . The cornerstone of our proof is the following lemma, which bounds the growth of X t = (X t,i ) i∈Π\F after one round.
Lemma 3.15. Consider the situation as described above and assume additionally that |X t,i | ≥ log log n for all i ∈ Π \ F and that |I t | ≤ n/ log n. Then for all ν > 0 and n large enough there exists a symmetric matrix A with biggest eigenvalue λ max ≥ 1 + 2q − ν and an error matrix ∆A with ∆A F ≤ ν such that whp Proof. We set I P,i pull, pp} and let be the vertices in V i newly informed in round t + 1 by operations involving only vertices from V i and V j . Let (i, j) ∈ Π \ F . For all u ∈ U i t we know that d(u) ≥ αn/2. Moreover, |I i t | ≤ |I t | ≤ n/ log n. Thus, the probability of u ∈ U i t being informed by vertices in I j t via pull is q|N (u) ∩ I j t |/|N (u)| = o(1). As the events of u being informed by push and pull are independent P [u ∈ I ]. Thus for any set U ∈ V Let i ∈ Π\F and j ∈ Π i . We start by determining the expected number of vertices informed by pull . Set Since |I i t | ≤ |I t | ≤ n/ log n we get with room to spare that |U i t \ H i,j ′ | ≥ (1 − 5η)n/k for n large enough and all j ′ ∈ Π i . Applying (3.16) As D i = (1 + η)n/k 1≤ℓ≤k d iℓ we get for We continue with push. Let i ∈ Π\F and j, j ′ ∈ Π i , and set (as before) D j = (1 + η) n k 1≤ℓ≤k d ℓj and According to Lemma 3.14 all v ∈ I j t \N j have degree less than D j . Using the inequalities (1 − 1/n) n ≤ e −1 and e −1/n = (1 − 1/n) + o(1/n) we obtain the estimate The remaining steps are similar to the previously considered case of pull . By assumption we have that |I j t \H j,i | = X t,j,i and as |I i t | ≤ |I t | ≤ n/ log n we obtain that |U i t \H i,j ′ | ≥ (1 − 5η)n/k for n large enough and all j ′ ∈ Π i . Using (3.16) we obtain that |d( Using that D j = (1 + η)n/k 1≤ℓ≤k d ℓj , we get for the same constant c 1 as in (3.18) and n large enough With (3.17), we can combine the results for pull , (3.18), and push, (3.20), to get for Next we will show how we can exploit (3.21) to obtain (a lower bound for) Let W ⊆ V . Using (3.17), the previous equation and that Π is a partition we get Choose W = V \ H i,j ′ , then the previous equation implies which in turn, using (3.21) and X t,j,i ≥ X t,j for all j ∈ Π \ F and i ∈ Π j , implies for c : Assume that (3.22) does not only hold in expectation but also for a slightly smaller c, say c − η, with high probability. We are going to show this at the end of the proof. Using this assumption and a union bound over j ′ ∈ Π i gives whp where for i ∈ Π \ F and j ∈ Π i we have (3.24) Let A be the |Π \ F | × |Π \ F | matrix with entries as in the previous equation, i.e., A = (a ij ) (i,j)∈(Π\F ) 2 is given by (3.24) for all (i, j) ∈ (Π \ F ) 2 . Note that A is symmetric. Then we obtain from (3.23) .
As d(u) ≥ αn/2 for all u ∈ V and some α > 0, we also know that and thus ∆A F ≤ 2 √ 2η/α. This leaves us with bounding the biggest eigenvalue λ max of A. Using the well-known inequality for symmetric matrices λ max ≥ (i,j)∈(Π\F ) 2 A ij /|Π \ F | we obtain Thus Choosing η small enough such that 2q(1 − c(1 − 8η/α)), 2 √ 2η/α ≤ ν implies the claim of this lemma. This leaves us with proving that (3.22) also holds with high probability. As |I (pp) t+1 | conditioned on I t is a self-bounding function so is |I (pp),i t+1 \ I i t | for all i ∈ Π \ F and therefore also |I and therefore setting 1≤ℓ≤k d ℓj X t,j for all i ∈ Π \ F and using (3.22), i.e. E t [Y t+1,i,j ′ ] ≥ Z t,i for all i ∈ Π \ F and j ′ ∈ Π i we get with probability at least This and |I i t | ≥ X t,i for all i ∈ Π \ F implies that (3.22) also holds with high probability for a marginally smaller c, as claimed.
Extension. We now solve the linear recurrence relation above and extend it to more than one round to get an upper bound on the runtime of push&pull . We first state a Chernoff Bound that will be very useful in the next lemma.  (a) Let I ⊆ V n satisfying |I| = Θ(n), then there is t = Θ(log log n), such that whp |I t | ≥ |I t ∩ I| ≥ log log n.
(c) Let n/ log n ≤ |I t | ≤ n − n/ log n. Then there is τ = o(log n) such that |I t+τ | > n − n/ log n.
(d) Let |I t | ≥ n − n/ log n and q = 1. Then there is τ = o(log n) such that |I t+τ | = n.
| clearly c) and d) follow from Lemma 3.2. We show a) by determining a lower bound for the probability that an arbitrary vertex gets informed after a constant number of rounds. Set β = min{α, ǫ}, let I 0 = {u} and choose w ∈ V, w = u. By Lemma 2.11 there is d ≤ 8/β 2 + 2 and c = (β 4 /64) 8/β 2 +3 ∈ (0, 1) such that there are at least cn d−1 paths of (edge) length d from u to w. Let γ = (u, v 1 , . . . , v d−1 , w) be such a path from u to w, and denote by A γ the event that w is informed via γ after exactly d rounds performing only push operations, i.e., A γ is the event that in the first round the randomly selected neighbour of u is v 1 , in the second round the randomly selected neighbour of v 1 is v 2 and so forth, until in the dth round the randomly selected neighbour of v d−1 is w. Obviously, the probability of A γ is bounded from below by n −d . Let further γ ′ = γ be another path from u to w with length d. As γ and γ ′ differ by at least one edge we readily obtain that P [A γ ∩ A γ ′ ] = 0. Let Γ denote the set of all paths with length d from u to w. Having done these preparations we use them to conclude for all w ∈ V and t ≥ 0 We define a modified protocol as follows. Wait d := ⌈8/β 2 + 2⌉ rounds, after that with probability c choose one uninformed vertex uniformly at random and set it as informed. Repeat. Call the vertices informed by this algorithm I ⋆ t . Then the probability for any vertex to be informed after d rounds is Thus for any t ≥ 0 P t [v ∈ I t+d |v ∈ U t ] ≥ P t [v ∈ I ⋆ t+d |v / ∈ I ⋆ t ] = c/n. Note that for any s ∈ N the set I ⋆ sd is generated by a very simple procedure: s times independently, with probability c, we choose a random vertex and put it into I ⋆ sd . Thus |I ⋆ sd ∩ I| is binomially distributed with s trials, where each one has success probability c|I|/n = Θ(c); it follows readily that |I ⋆ sd ∩ I| concentrates around a multiple of s for large s, and the claim follows by choosing s = Θ(log log n).
This leaves b) to be shown. Part a) implies that there is some t 0 = o(log n) such that X t0,i = Θ(log log n) for all i ∈ Π \ F by choosing I = V i \ (N i ∪ E i,j ), j ∈ Π i and applying a union bound over i and j. Thus we can apply Lemma 3.15. It gives whp, say with probability 1 − g(n) = 1 − o(1), that X t+1 ≥ (A + ∆A)X t , A has maximal eigenvalue λ max (A) ≥ 1 + 2q − ν and ∆A F ≤ ν. Then B := A + ∆A has maximal eigenvalue λ max (B) ≥ λ max (A) − ∆A F ≥ 1 + 2q − 2ν (Theorem of Wielandt-Hoffmann, compare e.g. [23]) .
Let v be an eigenvector of B to λ max (B). As v = 0 there is an index ℓ such that v ℓ = 0. Without loss of generality we can assume that v ℓ = 1, as v/v ℓ is also an eigenvector to λ max (B).
Our choice of T yields whp |I t+τ | ≥ λ max (B) T ≥ n/log n. Note that, since ν > 0 was chosen arbitrarily, we actually have that τ ≤ log 1+2q (n) + o(log n), and the proof is completed.
3.6 Proof of Theorem 1.6 (b) -edge deletions may slow down push&pull For any 0 < ε < 1/2, q ∈ (0, 1) we consider a sequence of graphs (G n (ε)) n∈N = ((V n , E n )) n∈N that is similar to the one studied in the proof of Theorem 1.6 (a). Let V n = A n ∪ B n with A n := {1, . . . , ⌊n/2⌋}, B n := {⌊n/2⌋ + 1, . . . , n} and deg(v) = n − 1 for all v ∈ A n . Let the induced subgraph of B n be a random graph in which each edge is included independently with probability p = 1 − 2ε. We know and it is easy to show, see for example [14, Section IV], that whp this subgraph is almost regular, i.e., d Bn (v) = (1 + o(1))(1 − 2ε)n/2 for all v ∈ B n , (3.26) and is an expander, which means that for every S n ⊆ B n , 1 ≤ |S n | ≤ n/4 and d Bn := (1 − 2ε)n/2 we have e(S n , B n \S n ) = (1 + o(1)) d Bn |S n ||B n \ S n | |B n | = (1 − 2ε + o(1))|S n ||B n \ S n |. At first we give a statement that describes the expected number of informed vertices after performing one round of push&pull .
Lemma 3.18. Let G n (ε) = (A n ∪ B n , E n ) be as above.
(a) Let √ log n ≤ |I t | ≤ n/ log n and set X t = I Then E t [X t+1 ] = (1 + o(1))M X t , where M = 1 + q q 1 + ε/(2 − 2ε) q 1 + ε/(2 − 2ε) 1 + q 1 − 2ε/(2 − 2ε) . t+1 ∩ J n . We first prove a) by computing the expected number of informed vertices after a single round. Since d(u) = Ω(n) for all u ∈ V n and |I t | ≤ n/ log n, the probability of u ∈ U t being informed by pull is As the events of u being informed by push and pull are independent we have P t [u ∈ (I Proof. We do not give a proof for b) as it follows immediately from Lemma 3.17 (a). For J ∈ {A, B} set U (A) t := U t ∩ J n , I (J) t := I t ∩ J n . We prove a) first. Let t 0 > 0 be the first round such that |I t0 | ≥ log log n and set x t and M as in Lemma 3.18 (a), note that Lemma 3.17 (a) also gives that x t0 ≥ log log n/2. Then for all t ≥ t 0 such that |I t | ≤ n/ log n we obtain from Lemma 3.18 (a) that E t [x t+1 ] = (1 + o(1))M x t and, in particular, E t [(x t+1 ) i ] = Θ(|I t |) for i ∈ {1, 2}. As every component of x t is self-bounding, Lemma 2.1 applies and we get for i ∈ {1, 2} and by union bound, provided that |I t | ≤ n/ log n, (3.28) Using (3.28) we want to find a bound on |I t+1 |. We get as long as |I t | ≤ n/ log n that As seen in Remark 3.19, M has maximal eigenvalue λ max > 1 and as M is a positive matrix there is a positive eigenvector v to λ max , compare [31]. This gives constants c 1 , c 2 > 0 such that c 1 v log log n ≤ x t0 ≤ c 2 v log log n and for t large enough x t0 , and therefore as long as the right hand side is bounded by n/ log n. For all these t we get additionally |I t+1 | ≥ c 2 c 1 ((1 − o(1))λ max ) t−t0 |I t0 |.
Next we show c). The assumption guarantees that less than n/ log n vertices are informed. Thus |U (B) t | ≥ n/2 − |I t | ≥ (1/2 − 1/log n) n. We consider a modified dissemination process, where in each round, each uninformed vertex always chooses an informed neighbour (but does not necessarily get informed as the message transmission may fail), and additionally each vertex chooses a neighbour iuar and after this round the chosen vertex is informed with probability q; in other words, we assume that also uninformed vertices can inform other vertices. In this modified process the probability of an uninformed vertex u ∈ U (B) t staying uninformed after performing one round is given by the product of the probabilities of not being informed by pull or via push by a vertex in A n or B n . Using (3.27) and (1 − 1/n) n = e −1+o(1) we get g(n) = o(1) such that As we have seen in the proof of Lemma 3.18 (b), the probability to be in formed by push&pull is greater for a vertex in A n than for a vertex in B n . Therefore it is sensible to expect that some vertices in B n we will be the last to be informed. Consequently denote by E u the event that a currently uninformed vertex u ∈ U (B) t does not get informed in this modified version within the next τ := 1 log((1 − q) −1 exp(q(1/2 + (1/2 − ε)/(1 − ε) − g(n)))) log(n) − h(n) rounds where h = o(log n) and h = ω(1). Therefore we have (1) .