Maximum Entropy Random Walks: the Infinite Setting and the Example of Spider Networks with their Scaling Limits

TL;DR

This paper develops a rigorous framework for Maximal Entropy Random Walks on infinite graphs, explores their properties through examples like spider networks, and reveals phenomena such as phase transitions and scaling limits.

Contribution

It introduces a generalized definition of MERWs on infinite graphs, addresses existence and uniqueness, and analyzes their behavior on spider networks with scaling limits and phase transitions.

Findings

MERWs maximize entropy rate even in infinite settings

Scaling limits of spider networks exhibit phase transitions

Unified proof based on submartingale problems

Abstract

In this article, we establish solid foundations for the study of Maximal Entropy Random Walks (MERWs) on infinite graphs. We introduce a generalized definition that extends the original concept, along with rigorous tools for handling this generalization. Unlike conventional simple random walks, which maximize entropy locally, MERWs maximize entropy globally along their paths, marking a significant paradigm shift and presenting substantial computational challenges. Originally introduced by physicists and computer scientists in [1], MERWs have connections to concepts such as Parry measures and Doob h-transforms. Our approach addresses the challenges of existence, uniqueness, and approximation, illustrated through examples and counterexamples. Even in the infinite setting, MERWs continue to maximize the entropy rate, albeit in a less direct manner. Additionally, we conduct an in-depth analysis of weighted spider networks, including scaling limits, revealing various phenomena characteristic of the infinite framework, notably a phase transition. A unified proof of scaling limits based on submartingale problems is presented. Furthermore, we consider some extended models, where the spider lattice provides valuable insights, highlighting the complexity of studying these walks for general infinite weighted graphs.