How to Discreetly Spread a Rumor in a Crowd

Abstract

In this paper, we study PUSH-PULL style rumor spreading algorithms in the mobile telephone model, a variant of the classical telephone model in which each node can participate in at most one connection per round; i.e., you can no longer have multiple nodes pull information from the same source in a single round. Our model also includes two new parameterized generalizations: (1) the network topology can undergo a bounded rate of change (for a parameterized rate that spans from no changes to changes in every round); and (2) in each round, each node can advertise a bounded amount of information to all of its neighbors before connection decisions are made (for a parameterized number of bits that spans from no advertisement to large advertisements). We prove that in the mobile telephone model with no advertisements and no topology changes, PUSH-PULL style algorithms perform poorly with respect to a graph's vertex expansion and graph conductance as compared to the known tight results in the classical telephone model. We then prove, however, that if nodes are allowed to advertise a single bit in each round, a natural variation of PUSH-PULL terminates in time that matches (within logarithmic factors) this strategy's performance in the classical telephone model-even in the presence of frequent topology changes. We also analyze how the performance of this algorithm degrades as the rate of change increases toward the maximum possible amount. We argue that our model matches well the properties of emerging peer-to-peer communication standards for mobile devices, and that our efficient PUSH-PULL variation that leverages small advertisements and adapts well to topology changes is a good choice for rumor spreading in this increasingly important setting.

Random Gossip Processes in Smartphone Peer-to-Peer Networks

“…In this paper, we both study and extend the mobile telephone model introduced in 2016 by Ghaffari and Newport [1]. This model modifies the classical telephone model of wired peer-topeer networks (e.g., [2]- [10]) to better match the constraints and capabilities of the smartphone setting.…”

“…In particular, the mobile telephone model differs from the classical telephone model in that it allows small advertisements but restricts the number of concurrent connections at a given node. As agued in [1], these differences (especially the latter) significantly change achievable results, algorithm strategy, and analysis techniques. The details of this model are inspired, in particular, but the multipeer connectivity framework offered in iOS.…”

“…The previous best known algorithm for this model is crowded bin gossip [13], which is significantly more complicated and requires O((k/α) log 5 n) rounds. 3 To put these time bounds into context, we note that previous work in the mobile telephone model solved rumor spreading [1] and leader election [12] in O(polylog(n)/α) rounds. In the classical telephone model, a series of papers [7]- [10] (each optimizing the previous) established that simple random rumor spreading requires O(log 2 n/α) rounds [10], which is optimal in the sense that for many α values, there exists networks with a diameter in Ω(log 2 n/α).…”

“…5 We now turn our attention to our gossip algorithm's ability to take advantage of the potential productive connections captured by the productive subgraph defined at the beginning of the phase. To do so, we first adapt a useful result on rumor spreading from [1] to the behavior of our gossip algorithm. Notice that it is the proof of the below adapted lemma that requires the use of the done flag in our algorithm.…”

“…Proof Notes. The original version of this theorem from [1] requires that G is a bipartite graph. This follows in our case because it is a subset of a productive subgraph.…”