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.
The telephone model described above was first introduced by Frieze and Grimmett @cite_0 . A key problem in this model is rumor spreading : a rumor must spread from a single source to the whole network. In studying this problem, algorithmic simplicity is typically prioritized over absolute optimality. The PUSH algorithm (first mentioned @cite_0 ), for example, simply has every node with the message choose a neighbor with uniform randomness and send it the message. The PULL algorithm (first mentioned @cite_2 ), by contrast, has every node without the message choose a neighbor with uniform randomness and ask for the message. The PUSH-PULL algorithm combines those two strategies. In a complete graph, both PUSH and PULL complete in @math rounds, with high probability---leveraging epidemic-style spreading behavior. Karp et al. @cite_12 proved that the average number of connections per node when running PUSH-PULL in the complete graph is bounded at @math .
@cite_0 @cite_12 @cite_2
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We analyze the popular push-pull protocol for spreading a rumor in networks. Initially, a single node knows of a rumor. In each succeeding round, every node chooses a random neighbor, and the two nodes share the rumor if one of them is already aware of it. We present the first theoretical analysis of this protocol on random graphs that have a power law degree distribution with an arbitrary exponent β > 2. Our main findings reveal a striking dichotomy in the performance of the protocol that depends on the exponent of the power law. More specifically, we show that if 2 3, then Ω(log n) rounds are necessary. We also investigate the asynchronous version of the push-pull protocol, where the nodes do not operate in rounds, but exchange information according to a Poisson process with rate 1. Surprisingly, we are able to show that, if 2 < β < 3, the rumor spreads even in constant time, which is much smaller than the typical distance of two nodes. To the best of our knowledge, this is the first result that establishes a gap between the synchronous and the asynchronous protocol. We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c (c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see [11]) imply that the push model is faster on hypercubes than on a random graph G(n, clog n n), where c is sufficiently close to 1. We propose and analyse a quasirandom analogue to the classical push model for disseminating information in networks ("randomized rumor spreading"). In the classical model, in each round each informed node chooses a neighbor at random and informs it. Results of Frieze and Grimmett (Discrete Appl. Math. 1985) show that this simple protocol succeeds in spreading a rumor from one node of a complete graph to all others within O(log n) rounds. For the network being a hypercube or a random graph G(n, p) with p ≥ (1 +e)(log n) n, also O(log n) rounds suffice (Feige, Peleg. Raghavan, and Upfal, Random Struct. Algorithms 1990). In the quasirandom model, we assume that each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above mentioned bounds still hold. In addition, we also show a O(log n) bound for sparsely connected random graphs G(n, p) with p = (log n + f(n)) n, where f(n) → ∞ and f(n) = O(log log n). Here, the classical model needs Θ(log2(n)) rounds. Hence the quasirandom model achieves similar or better broadcasting times with a greatly reduced use of random bits. We show that the asynchronous push-pull protocol spreads rumors in preferential attachment graphs (as defined by Barabasi and Albert) in time @math to all but a lower order fraction of the nodes with high probability. This is significantly faster than what synchronized protocols can achieve; an obvious lower bound for these is the average distance, which is known to be Θ(logn loglogn).