Push is Fast on Sparse Random Graphs

TL;DR

This paper shows that the push broadcast process on sparse random graphs, including power law graphs, quickly informs almost all vertices within logarithmic rounds, contrasting with slower times needed to inform all vertices.

Contribution

It proves that push alone is sufficient to inform nearly all vertices rapidly on a broad class of sparse random graphs, challenging previous assumptions about the benefits of push-pull protocols.

Findings

Whp $O( ext{log } n)$ rounds inform all but an $ ext{epsilon}$-fraction of vertices.

Push alone matches push-pull speed for certain power law graphs with $eta>3$.

Push-pull is exponentially faster than push on power law graphs with $2<eta<3$.

Abstract

We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every $\varepsilon>0$, whp $O(\log n)$ rounds are sufficient to inform all but an $\varepsilon$-fraction of the vertices. It is not hard to see that, e.g. for random power law graphs, the push process needs whp $n^{\Omega(1)}$ rounds to inform all vertices. Fountoulakis, Panagiotou and Sauerwald proved that for random graphs that have power law degree sequences with $\beta>3$, the push-pull protocol needs $\Omega(\log n)$ to inform all but $\varepsilon n$ vertices whp. Our result demonstrates that, for such random graphs, the pull mechanism does not (asymptotically) improve the running time. This is surprising as it is known that, on random power law graphs with $2<\beta<3$, push-pull is exponentially faster than pull.