The Distribution of Path Lengths On Directed Weighted Graphs

TL;DR

This paper derives explicit formulas for the asymptotic growth of path counting functions on directed weighted graphs, under certain irrationality conditions, and analyzes associated random walk statistics.

Contribution

It provides new explicit formulas for path counting asymptotics and probabilistic analysis on directed weighted graphs with irrationality assumptions.

Findings

Explicit formulas for asymptotic growth rates of path counting functions

Computed statistics of random walks on directed weighted graphs

Reviewed applications and examples

Abstract

We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed.