Vertex reinforced non-backtracking random walks: an example of path formation

TL;DR

This paper analyzes vertex reinforced non-backtracking random walks, demonstrating that under certain conditions they tend to localize on a finite set of vertices, modeling path formation phenomena like those observed in ant colonies.

Contribution

It introduces a broad class of reinforced random walks and applies the results specifically to VRNBWs on complete graphs with power-law weights, revealing localization behavior.

Findings

For α>1 and certain m, walks localize on m vertices with positive probability.

Localization on more than a specific number of vertices is almost surely impossible.

Walks tend to visit a finite set of vertices asymptotically equally.

Abstract

This article studies vertex reinforced random walks that are non-backtracking (denoted VRNBW), i.e. U-turns forbidden. With this last property and for a strong reinforcement, the emergence of a path may occur with positive probability. These walks are thus useful to model the path formation phenomenon, observed for example in ant colonies. This study is carried out in two steps. First, a large class of reinforced random walks is introduced and results on the asymptotic behavior of these processes are proved. Second, these results are applied to VRNBWs on complete graphs and for reinforced weights $W(k)=k^\alpha$, with $\alpha\ge 1$. It is proved that for $\alpha>1$ and $3\le m< \frac{3\alpha -1}{\alpha-1}$, the walk localizes on $m$ vertices with positive probability, each of these $m$ vertices being asymptotically equally visited. Moreover the localization on $m>\frac{3\alpha -1}{\alpha-1}$ vertices is a.s. impossible.