We study the relation between the performance of the randomized rumor spreading (push model) in a d-regular graph G and the performance of the same algorithm in the percolated graph G_p. We show that if the push model successfully broadcast the rumor within T rounds in the graph G then only (1 + \epsilon)T rounds are needed to spread the rumor in the graph G_p when T = o(pd).
Randomized rumor spreading or randomized broadcasting are simple randomized algorithms to spread information in networks. In this work we consider the classical push model for rumor spreading, which is described as follows. Initially, one arbitrary vertex knows an information. In the succeding time steps each informed vertex chooses a neighbor independently and uniformly at random and forward the information to it. The fundamental question is: how many time steps are needed until every vertex of the network has been informed?
The push model has been extensively studied. Most of the papers analyze the runtime of this algorithm on different graph classes. On the complete graph, Frieze and Grimmett [5] proved that with high probability 1 (1 + o (1))(log 2 n + log n) rounds are necessary and sufficient to inform all vertices. In [3] Feige et al. gave general upper bounds holding for any graph and determine the runtime on random graphs. Also, Chierichetti, Lattanzi and Panconesi [2] proved runtime bounds in terms of the conductance and Sauerwald and Stauffer [7] obtain bounds in terms of the vertex expasion for regular graphs.
The starting point of this paper is a recent article by Fountoulakis, Huber and Panagiotou [4] which analyze the push protocol on the Erdös-Rényi random graph G n,p where p ≫ ln n n . Among other things, they show that the protocol will inform every vertex in (1 + o (1))(log 2 n + ln n) steps w.h.p.. One may restate this result by the runtime of randomized broadcasting on a complete graph is essentially not affected by random edge deletions, at least up to the connectivity threshold p = ln n/n.
In this paper we prove a partial extension of this result to the case of arbitrary percolation graphs. Here one starts with some arbitrary graph G and performs edge percolation on it. Under certain conditions, we show that this will not increase the runtime of the protocol by more than a 1 + o (1) factor. This suggests that the push protocol is robust against random edge failures, which is a desirable quality for applications.
We need some definitions in order to state our main Theorem. Given a graph G = (V, E), we let G p denote the random subgraph of G where each edge is removed independently with probability 1p (the vertex set stays the same). We let T v (G), T v (G p ) denote the runtimes of the push protocol starting at v over G and G p (respectively).
T n with high probability (here the probability is over the choice of G n,pn and over the additional randomness of the push protocol).
One can check in our proofs that the same result holds if G n has minimum degree d n .
The case where G n = K n is complete shows that the condition T n = o (p n d n ) cannot be removed in general. We also remark that proving lower bounds for T vn (G n,pn ) in terms of T vn (G n ) is an interesting open problem, but the upper bound we give seems more interesting for applications
The proof strategy of [4] relies on the geometry of the Erdös-Rényi graph above the connectivity threshold. We have no such information available in our general setting, and instead rely on a very different proof strategy:
Proof strategy: Construct a coupling of:
Then show that the runtimes of push over the two graphs are close under the coupling.
It is not hard to sketch a coupling that solves the analogous problem over oriented graphs. Assume D = (V, F ) is a n-vertex digraph. Define D p as the random digraph obtained from D by deleting each oriented edge with probability 1p. We consider a variant of push over D and D p , where each informed vertex v pushes the rumour along outgoing edges chosen uniformly but without replacement. We will assume that all vertices have the same out-degree d and that this modified push protocol over D typically takes T ≪ pd steps.
We now couple For this we need two Bernoulli (indicator) random variables A v→w and I v→w for each oriented edge v → w. All of those variables will be assumed independent, and we take:
Notice that T ≥ log 2 n as the number of informed vertices can only double at each time step. Chernoff bounds (see [1]) imply that, if C is sufficiently large, then w.h.p., for all v ∈ V , the set
will have at least T elements, as each of the d out-neighbors of v belongs to this set with probability CT /d. This means we can run modified push over D by having each v select the edges v → w with w ∈ N (v) in a random order, up to time T . Notice that this gives the right distribution because, conditionally on |N (v)| = k ≥ T , N (v) is uniform over all k-subsets of out-neighbors of v in D.
We now let D p be the digraph whose oriented edges are the pairs v → w with A v→w = 1. To run modified push on D p , have each v select the edges v → w, w ∈ N (v), in the same order as in the protocol over D. The key points are that:
• This gives the right distribution because conditionally on D p and on |N (v)| = k, N (v) is uniform over k-subsets of the out-neighbors of v in D p .
• The set of info
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