Ergodic Randomized Algorithms and Dynamics over Networks

TL;DR

This paper studies randomized algorithms over networks, showing that under certain conditions, their oscillations are ergodic and their limits can be recovered through time-averaging, with applications to network localization, PageRank, and opinion dynamics.

Contribution

It establishes a general framework linking stability of expected dynamics to ergodic oscillations and limit recovery in randomized network algorithms.

Findings

Oscillations in randomized network dynamics are ergodic under stability.

Time-averaging can recover the desired limit in randomized algorithms.

Applications include network localization, PageRank, and opinion dynamics.

Abstract

Algorithms and dynamics over networks often involve randomization, and randomization may result in oscillating dynamics which fail to converge in a deterministic sense. In this paper, we observe this undesired feature in three applications, in which the dynamics is the randomized asynchronous counterpart of a well-behaved synchronous one. These three applications are network localization, PageRank computation, and opinion dynamics. Motivated by their formal similarity, we show the following general fact, under the assumptions of independence across time and linearities of the updates: if the expected dynamics is stable and converges to the same limit of the original synchronous dynamics, then the oscillations are ergodic and the desired limit can be locally recovered via time-averaging.