Rumor Spreading on Percolation Graphs

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).

💡 Research Summary

The paper investigates how the classic randomized “push” rumor‑spreading protocol behaves when the underlying network undergoes random edge deletions, i.e., percolation. Let G be a d‑regular graph on n vertices and let Gₚ be the subgraph obtained by keeping each edge independently with probability p. The push protocol starts from a single informed vertex; in every synchronous round each informed vertex contacts a uniformly random neighbor and transmits the rumor. The central question is how many rounds are required until every vertex becomes informed.

The authors first recall known results: on the complete graph the broadcast time is (1+o(1))(log₂ n + ln n) w.h.p.; on general graphs the time can be bounded in terms of conductance or vertex expansion. A recent work by Fou‑ntoulakis, Huber and Panagiotou showed that on the Erdős–Rényi random graph G(n,p) with p≫(ln n)/n the broadcast time is essentially unchanged compared with the complete graph. The present work seeks to extend this robustness to arbitrary percolated graphs.

The main technical contribution is Theorem 1.1: if the push protocol on G finishes in T rounds with high probability (w.h.p.) and T = o(p·d), then for any ε>0 the same protocol on Gₚ finishes in at most (1+ε)·T rounds w.h.p. The condition T = o(p·d) is necessary in general (the complete graph shows that the bound cannot be removed).

To prove the theorem the authors introduce a variant called “push without replacement” (PWR). In PWR each informed vertex never contacts the same neighbor twice; this process is stochastically faster than the original push, i.e., J(G) ≼ T(G). By coupling the two processes they obtain that if push on G succeeds within T rounds then PWR also succeeds within T rounds.

The second, more delicate, part of the proof couples the push on Gₚ with PWR on G. For each directed edge (u→v) two independent Bernoulli variables are defined: A_{u→v} indicates whether the edge survives percolation (probability p) and I_{u→v} indicates whether u selects v in a given round (probability C·T/(p·d) for a suitable constant C). The set of “effective neighbors” of u in a given round is N(u) = {v : (u,v)∈E(G), A_{u→v}=1, I_{u→v}=1}. The size of N(u) can be modeled as a sum of geometric random variables, via a domination argument that replaces a negative hypergeometric distribution (sampling without replacement) by a sum of independent geometric variables (Lemma 2.1 and Lemma 2.2). Lemma 2.3 then provides a Chernoff‑type bound for the sum of such near‑deterministic geometric variables, showing that the probability that |N(u)| falls below T is exponentially small when T≪p·d.

Because each vertex needs at most T distinct neighbors to complete the broadcast, the coupling guarantees that, with high probability, the push on Gₚ never lags behind the push on G by more than an ε‑fraction of the rounds. Consequently the broadcast time on the percolated graph is at most (1+ε)·T.

The paper also discusses the tightness of the condition T = o(p·d). In the extreme case G = Kₙ, the broadcast time on G is Θ(log n) while p·d = p·(n‑1). If p is close to the connectivity threshold (≈ln n/n), the condition fails and the broadcast time can increase dramatically, showing that the theorem’s hypothesis cannot be dropped in full generality.

In summary, the authors establish that the push rumor‑spreading protocol is robust against random edge failures: as long as the original graph spreads the rumor quickly relative to the expected degree after percolation, the percolated graph incurs only a negligible (1+ε) multiplicative slowdown. This robustness is valuable for designing distributed systems that must tolerate random link failures, and the techniques introduced (coupling with PWR, stochastic domination of negative hypergeometric by geometric sums, and refined concentration bounds) may find further applications in the analysis of other randomized network algorithms.