Probability of graphs with large spectral gap by multicanonical Monte Carlo

TL;DR

This paper introduces a multicanonical Monte Carlo method to estimate the probability of graphs with large spectral gaps, enabling analysis of rare graph configurations in various scientific fields.

Contribution

The paper presents a novel application of multicanonical Monte Carlo to quantify the probability of large spectral gap graphs, especially in the extreme tail regions.

Findings

Successfully applied to random 3-regular graphs

Estimated large deviation probabilities

Demonstrated effectiveness in analyzing rare graph structures

Abstract

Graphs with large spectral gap are important in various fields such as biology, sociology and computer science. In designing such graphs, an important question is how the probability of graphs with large spectral gap behaves. A method based on multicanonical Monte Carlo is introduced to quantify the behavior of this probability, which enables us to calculate extreme tails of the distribution. The proposed method is successfully applied to random 3-regular graphs and large deviation probability is estimated.