Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects

Area of Science:

• Graph Theory
• Statistical Physics
• Network Science

Background:

• Maximal entropy random walks and path counting on graphs often exhibit localization near high-degree nodes.
• A simple model for this phenomenon involves a regular tree with a unique root node.

Purpose of the Study:

• To conduct an in-depth study of path counting at the localization transition in a regular tree model.
• To analyze the behavior of paths starting from the root in both infinite trees and regular random graphs (RRGs).

Main Methods:

• Analysis of path endpoint probability distribution functions.
• Mathematical modeling of random walks on infinite trees and finite RRGs.
• Investigation of localization transitions and critical phenomena.

Main Results:

• For infinite trees, the path endpoint distribution is a step function moving at a constant velocity v=(p-2)/p.
• In finite RRGs, a critical slowdown occurs, with relaxation to equilibrium taking O(N) trajectory length.
• Exact calculations of equilibrium distribution, relaxation length, and slowly relaxing modes were performed.

Conclusions:

• The study provides a detailed understanding of localization phenomena in random walks on tree-like structures.
• Findings elucidate the distinct behaviors in infinite versus finite graph settings, particularly at the transition point.