We consider the random phone call model introduced by Demers et al. [8], which is a well-studied model for information dissemination on networks. One basic protocol in this model is the so-called Push protocol which proceeds in synchronous rounds. Starting with a single node which knows of a rumor, every informed node calls a random neighbor and informs it of the rumor in each round. The Push-Pull protocol works similarly, but additionally every uninformed node calls a random neighbor and may learn the rumor from that neighbor. While it is well-known that both protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes, we are interested by how much we can speed up the spread of the rumor by enabling nodes to make more than one call in each round. We propose a new model where the number of calls of a node u is chosen independently according to a probability distribution R with bounded mean determined at the beginning of the process. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance. If R follows a power law distribution with exponent ∈ (2, 3), we show that the Push-Pull protocol spreads a rumor in Θ(log log n) rounds.
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