The averaging process on infinite graphs

TL;DR

This paper proves that on infinite bounded-degree graphs with i.i.d. initial opinions, the averaging process causes all opinions to converge in mean square to the initial average, using a novel probabilistic approach.

Contribution

It establishes convergence of the averaging process on infinite graphs with i.i.d. initial opinions, extending previous finite or special cases.

Findings

Opinions converge in L^2 to the initial mean.

The convergence holds under finite second moment assumption.

The proof employs the Sharing a drink procedure.

Abstract

We consider the averaging process on an infinite connected graph with bounded degree and independent, identically distributed starting values or initial opinions. Assuming that the law of the initial opinion of a vertex has a finite second moment, we show that the opinions of all vertices converge in $L^2$ to the first moment of the law of the initial opinions. A key tool in the proof is the Sharing a drink procedure introduced by Olle H\"aggstr\"om.