A Random-Walk Concentration Principle for Occupancy Processes on Finite Graphs

TL;DR

This paper introduces a concentration principle for occupancy processes on finite graphs, linking local state averages and subgraph densities to random walks, applicable to both dense and sparse dynamic graphs.

Contribution

It establishes two theorems connecting concentration phenomena to random walks, extending analysis to edge processes and general subgraph densities in dynamic graphs.

Findings

Concentration of local averages is governed by a random walk on the graph.

Provides bounds for deviations in edge and subgraph densities.

Applicable to both dense and sparse dynamic graphs.

Abstract

This paper concerns discrete-time occupancy processes on a finite graph. Our results can be formulated in two theorems, which are stated for vertex processes, but also applied to edge process (e.g., dynamic random graphs). The first theorem shows that concentration of local state averages is controlled by a random walk on the graph. The second theorem concerns concentration of polynomials of the vertex states. For dynamic random graphs, this allows to estimate deviations of edge density, triangle density, and more general subgraph densities. Our results only require Lipschitz continuity and hold for both dense and sparse graphs.