### Faster Rumor Spreading With Multiple Calls

#### Abstract

We consider the random phone call model introduced by Demers et al., 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 in each round a random neighbor and informs it of the rumor. The Push-Pull protocol works similarly, but additionally every uninformed node calls a random neighbor and may learn the rumor from it.

It is well-known that both protocols need $\Theta(\log n)$ rounds to spread a rumor on a complete network with $n$ nodes. Here we are interested in how much the spread can be speeded 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 is chosen independently according to a probability distribution $R$. 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 they exist). In particular, if $R$ follows a power law distribution with exponent $\beta \in (2,3)$, we show that the Push-Pull protocol spreads a rumor in $\Theta(\log \log n)$ rounds. Moreover when $\beta=3$, the Push-Pull protocol spreads a rumor in $\Theta(\frac{ \log n}{\log\log n})$ rounds.