The principle of least action for random graphs

TL;DR

This paper explores the statistical properties of the physical action in random graphs using a lattice quantum field theory approach, revealing that denser graphs tend toward a Gaussian distribution of action and that the most probable configurations follow a least-action principle.

Contribution

It introduces a novel method applying LQFT to analyze the action distribution in evolving random graphs, linking graph density to classical least-action paths.

Findings

Action distribution approaches Gaussian as graph density increases.

Maximum probability of action corresponds to least-action configurations.

Balanced regular-irregular structure in most probable graph configurations.

Abstract

We study the statistical properties of the physical action $S$ for random graphs, by treating the number of neighbors at each vertex of the graph (degree), as a scalar field. For each configuration (run) of the graph we calculate the Lagrangian of the degree field by using a lattice quantum field theory(LQFT) approach. Then the corresponding action is calculated by integrating the Lagrangian over all the vertices of the graph. We implement an evolution mechanism for the graph by removing one edge per a fundamental quantum of time, resulting in different evolution paths based on the run that is chosen at each evolution step. We calculate the action along each of these evolution paths, which allows us to calculate the probability distribution of $S$. We find that the distribution approaches the normal(Gaussian) form as the graph becomes denser, by adding more edges between its vertices. The maximum of the probability distribution of the action corresponds to graph configurations whose spacing between the values of $S$ becomes zero $\Delta S=0$, corresponding to the least-action (Hamilton) principle, which gives the path that the physical system follows classically. In addition, we calculate the fluctuations(variance) of the degree field showing that the graph configurations corresponding to the maximum probability of $S$, which follow the Hamilton's principle, have a balanced structure between regular and irregular graphs.