Rapid Mixing of Dynamic Graphs With Local Evolution Rules

Abstract

Dynamic graphs arise naturally in many contexts. In peer-to-peer networks, for instance, a participating peer may replace an existing connection with one neighbor by a new connection with a neighbor of that neighbor. Several such local rewiring rules have been proposed to ensure that peer-to-peer networks achieve good connectivity properties (e.g., high expansion) at equilibrium. However, the question of whether there exists such a rule that converges rapidly to equilibrium has remained open. In this paper, we provide an affirmative answer: we exhibit a local rewiring rule that converges to equilibrium after each participating node has undergone only a number of changes that is at most poly-logarithmic in the system size. As a byproduct, we derive new results for random walks on graphs, bounding the spread of their law throughout the transient phase, i.e., prior to mixing. These rely on an extension of Cheeger's inequality, based on generalized isoperimetric constants, and may be of independent interest.