Statistics of paths on graphs with two heavy roots

TL;DR

This paper investigates the behavior of path counts on graphs with two heavy roots, revealing localization phenomena and entropy interactions through numerical analysis.

Contribution

It introduces a detailed numerical study of path statistics on graphs with two heavy roots, highlighting localization effects and entropy interactions.

Findings

Localization occurs with the same degree ratio as a single root

Numerical analysis confirms entropic traps influence path counts

Entropy forces of interaction between two roots are characterized

Abstract

The paper considers the behaviour of the number of paths of length $N$ on graphs with two heavy roots. Such vertices can be entropic traps. Numerical analysis is carried out for graphs with different degrees of root vertices. In the symmetric case, a numerical analysis of the number of paths is performed. It is found that localisation is observed with the same ratio of vertex degrees as in the case of a single root. An analysis of the entropy forces of interaction was also carried out for two roots.