Traversals of Infinite Graphs with Random Local Orientations

TL;DR

This paper introduces the concept of random basic walks on infinite graphs, compares them to traditional random walks, and explores their recurrence, transience, and visitation properties with various graph classes.

Contribution

It defines and analyzes the properties of random basic walks, including recurrence, transience, and visitation bounds, providing new insights and conjectures in the study of infinite graph traversals.

Findings

Cycles of arbitrary length are possible but unlikely in regular graphs.

Upper bounds on expected visited vertices are established.

Complete graphs have walks that visit a constant fraction of nodes asymptotically.

Abstract

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks. We define analogues in the setting of random basic walks of the notions of recurrence and transience in the theory of simple random walks, and we study the question of which graphs have a cycling random basic walk and which a transient random basic walk. We prove that cycles of arbitrary length are possible in any regular graph, but that they are unlikely. We give upper bounds on the expected number of vertices a random basic walk will visit on the infinite graphs studied and on their finite analogues of sufficiently large size. We then study random basic walks on complete graphs, and prove that the class of complete graphs has random basic walks asymptotically visit a constant fraction of the nodes. We end with numerous conjectures and problems for future study, as well as ideas for how to approach these problems.