Randomized Rumor Spreading in Poorly Connected Small-World Networks

Abstract

The Push-Pull protocol is a well-studied round-robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread the rumor to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random k-trees, a class of power law graphs which are small-world and have large clustering coefficients, built as follows: initially we have a k-clique. In every step a new node is born, a random k-clique of the current graph is chosen, and the new node is joined to all nodes of the k-clique. When k > 2 is fixed, we show that if initially a random node is aware of the rumor, then with probability 1 − o(1) after $\mathcal{O}\left( (\log n)^{{(k+3)}/{(k+1)}} \cdot \log \log n\cdot f(n) \right)$ rounds the rumor propagates to n − o(n) nodes, where n is the number of nodes and f(n) is any slowly growing function. When k = 2, the previous statement holds for $\mathcal{O} \left( \log ^2n\cdot \log \log n \cdot f(n) \right)$ many rounds. Since these graphs have polynomially small conductance, vertex expansion $\mathcal{O}(1/n)$ and constant treewidth, these results demonstrate that Push-Pull can be efficient even on poorly connected networks.

On the negative side, we prove that with probability 1 − o(1) the protocol needs at least $\Omega\left(n^{({k-1})/({k^2+k-1})}/f^2(n)\right)$ rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random k-Apollonian networks, for which we prove an upper bound of $\mathcal{O}\left( (\log n) ^{{{(k^2-3)}/{(k-1)^2}}} \cdot \log \log n \cdot f(n) \right)$ rounds for informing n − o(n) nodes with probability 1 − o(1), when k > 2 is a constant.