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**11 - 15**of**15**### Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading

"... We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem: ..."

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We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem: First, we establish a new connection between the rumor spreading process in an arbitrary graph and certain Markov chains. While most previous work analyzed the rumor spreading time in general graphs by studying the rate of the number of (un-)informed nodes after every round, we show that the mixing time of a certain Markov chain suffices to bound the rumor spreading time in an arbitrary graph. Second, we construct a reduction from rumor spreading processes to branching programs. This reduction gives us a general framework to derandomize the rumor spreading and other gossip processes. In particular, we show that, for any n-vertex expander graph, there is a protocol which informs every node in O(logn) rounds with high probability, and uses O(logn · log logn) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O(logn · log log n) random bits almost matches the lower bound of Ω(logn) random bits. We further show that, for many graph families (defined with respect to the expansion and the degree), O(poly logn) random bits in total suffice for fast rumor spreading. These results give us an almost complete understanding of the role of randomness in the rumor spreading process, which was extensively studied over the past years. 1

### Faster rumor spreading with multiple calls

"... We consider the random phone call model introduced by Demers et al.,which is a well-studied model for information dissemination in networks. One basic protocol in this model is the so-called Push protocol that proceeds in synchronous rounds. Starting with a single node which knows of a rumor, every ..."

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We consider the random phone call model introduced by Demers et al.,which is a well-studied model for information dissemination in networks. One basic protocol in this model is the so-called Push protocol that 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 Θ(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 up 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