Scaling Properties of Paths on Graphs

TL;DR

This paper investigates how the sum of weights along paths in directed graphs scales asymptotically, revealing that the structure of strongly connected components determines whether the path weights follow a power law or weaker scaling.

Contribution

It generalizes previous results by characterizing the asymptotic behavior of path weights on graphs, including Markov chains, based on their strongly connected components.

Findings

Asymptotic behavior depends on the structure of strongly connected components.

Path weight sequences exhibit power law or weaker scaling.

Generalizes earlier work on Markov chain path distributions.

Abstract

Let $G$ be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on $G$. Fix vertices $u$ and $v$ and consider the set of paths that start at $u$ and end at $v$, self-intersecting in any number of places along the way. For each path, sum the weights of its edges, and then list the path weights in increasing order. The asymptotic behaviour of this sequence is described, in terms of the structure and type of strongly connected components on the graph. As a special case, for a Markov chain the asymptotic probability of paths obeys either a power law scaling or a weaker type of scaling, depending on the structure of the transition matrix. This generalizes previous work by Mandelbrot and others, who established asymptotic power law scaling for special classes of Markov chains.