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

Theorems & Definitions (8)

• proof
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• proof
• proof : Proof of Theorem \ref{['thm:poly']}
• proof : Proof of Corollary \ref{['cor:vertices']}
• proof : Proof of Proposition \ref{['prop:diag_con']}
• proof