# Imaginary replica analysis of loopy regular random graphs

## TL;DR

This paper develops an analytical method using imaginary replica indices to analyze the eigenvalue spectra of regular loopy graphs under spectral constraints, revealing phase transitions and validating results with simulations.

## Contribution

Introduces a novel imaginary replica approach for spectral analysis of loopy graphs, enabling the study of phase transitions in constrained random graph ensembles.

## Key findings

- Derived expressions for eigenvalue spectra in weak loop interactions
- Identified shattering phase transitions in graph structures
- Validated analytical predictions with MCMC simulations

## Abstract

We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two orders of the expected eigenvalue spectrum, through the use of infinitely many replica indices taking imaginary values. We apply the method to models in which the spectral constraint reduces to a soft constraint on the number of triangles, which exhibit `shattering' transitions to phases with extensively many disconnected cliques, to models with controlled numbers of triangles and squares, and to models where the spectral constraint reduces to a count of the number of adjacency matrix eigenvalues in a given interval. Our predictions are supported by MCMC simulations based on edge swaps with nontrivial acceptance probabilities.