# Faster Rumor Spreading: Breaking the log n Barrier

## Chen Avin and Robert Elsässer

Abstract. $O(\log{n})$ rounds has been a well known upper bound for rumor spreading using push&pull in the random phone call model (i.e., uniform gossip in the complete graph). A matching lower bound of $\Omega(\log{n})$ is also known for this special case. Under the assumptions of this model and with a natural addition that nodes can call a partner once they learn its address (e.g., its IP address) we present a new distributed, address-oblivious and robust algorithm that uses push&pull with pointer jumping to spread a rumor to all nodes in only $O(\sqrt{\log{n}})$ rounds, w.h.p. This algorithm can also cope with $F = o( n/2^{\sqrt{\log{n}}} )$ node failures, in which case all but $O(F)$ nodes become informed within $O(\sqrt{\log{n}})$ rounds, w.h.p.

Guest: Chen Avin

Host: Shantanu Das

# Ultra-Fast Rumor Spreading in Social Networks

## Nikolaos Fountoulakis, Konstantinos Panagiotou and Thomas Sauerwald

Abstract:
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 graohs 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 < β < 3, then the rumor spreads to almost all nodes in Θ(log log n) rounds with high probability. On the other hand, if β > 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.

Guest: Thomas Sauerwald
Host: Chen Avin

# Rumor Spreading and Vertex Expansion

## George Giakkoupis and Thomas Sauerwald

Abstract:
We study the relation between the rate at which rumors spread throughout a graph and the vertex expansion of the graph. We consider the standard rumor spreading protocol where every node chooses a random neighbor in each round and the two nodes exchange the rumors they know. For any n-node graph with vertex expansion $\alpha$, we show that this protocol spreads a rumor from a single node to all other nodes in $O(\alpha^{-1} \log
^2 n \sqrt{\log n})$
rounds with high probability. Further, we construct graphs for which $\Omega(\alpha^{-1} \log^2 n)$ rounds are needed. Our results complement a long series of works that relate rumor spreading to edge-based notions of expansion, resolving one of the most natural questions on the connection between rumor spreading and expansion.

Guest: George Giakkoupis
Host: Chen Avin