A Notion of Entropy for Stochastic Processes on Marked Rooted Graphs

TL;DR

This paper generalizes the concept of entropy for stochastic processes on marked rooted graphs using local weak limit theory, extending prior work to include vertex and edge marks, with potential broad applications.

Contribution

It introduces a new entropy notion for marked graph processes, expanding the framework of Bordenave and Caputo to more complex graph structures with marks.

Findings

Generalized entropy for marked graphs

Extended local weak limit framework

Potential applications in various fields

Abstract

In this document, we introduce a notion of entropy for stochastic processes on marked rooted graphs. For this, we employ the framework of local weak limit theory for sparse marked graphs, also known as the objective method, due to Benjamini, Schramm, Aldous, Steele and Lyons. Our contribution is a generalization of the notion of entropy introduced by Bordenave and Caputo to graphs which carry marks on their vertices and edges. The theory of time series is the engine driving an enormous range of applications in areas such as control theory, communications, information theory and signal processing. It is to be expected that a theory of stationary stochastic processes indexed by combinatorial structures, in particular graphs, would eventually have a similarly wide-ranging impact.