Expansion and Flooding in Dynamic Random Networks with Node Churn

TL;DR

This paper analyzes how information spreads rapidly in dynamic, sparse random networks with node churn, showing that flooding can inform a large fraction of nodes efficiently under various models.

Contribution

It introduces models of dynamic random networks with node churn and proves rapid flooding and expansion properties, including under edge regeneration and realistic continuous-time assumptions.

Findings

Large fraction of nodes informed in O(log n) time

Networks exhibit large-set expansion properties

Flooding remains efficient with edge regeneration

Abstract

We study expansion and information diffusion in dynamic networks, that is in networks in which nodes and edges are continuously created and destroyed. We consider information diffusion by {\em flooding}, the process by which, once a node is informed, it broadcasts its information to all its neighbors. We study models in which the network is {\em sparse}, meaning that it has $\mathcal{O}(n)$ edges, where $n$ is the number of nodes, and in which edges are created randomly, rather than according to a carefully designed distributed algorithm. In our models, when a node is "born", it connects to $d=\mathcal{O}(1)$ random other nodes. An edge remains alive as long as both its endpoints do. If no further edge creation takes place, we show that, although the network will have $\Omega_d(n)$ isolated nodes, it is possible, with large constant probability, to inform a $1-exp(-\Omega(d))$ fraction of nodes in $\mathcal{O}(\log n)$ time. Furthermore, the graph exhibits, at any given time, a "large-set expansion" property. We also consider models with {\em edge regeneration}, in which if an edge $(v,w)$ chosen by $v$ at birth goes down because of the death of $w$, the edge is replaced by a fresh random edge $(v,z)$. In models with edge regeneration, we prove that the network is, with high probability, a vertex expander at any given time, and flooding takes $\mathcal{O}(\log n)$ time. The above results hold both for a simple but artificial streaming model of node churn, in which at each time step one node is born and the oldest node dies, and in a more realistic continuous-time model in which the time between births is Poisson and the lifetime of each node follows an exponential distribution.