Eigenvalue tunnelling and decay of quenched random networks

TL;DR

This paper studies how large Erdős-Rényi networks fragment into cliques as a parameter increases, analyzing the spectral density evolution and eigenvalue tunneling phenomena with implications for physical systems.

Contribution

It introduces a detailed spectral analysis of network fragmentation into cliques at a critical fugacity, revealing eigenvalue tunneling and block-diagonal adjacency matrix structures.

Findings

Network fragments into cliques above critical fugacity.

Spectral density develops two zones with eigenvalue tunneling.

Ground state adjacency matrix exhibits block-diagonal form.

Abstract

We consider the canonical ensemble of $N$-vertex Erd\H{o}s-R\'enyi (ER) random topological graphs with quenched vertex degree, and with fugacity $\mu$ for each closed triple of bonds. We claim complete defragmentation of large-$N$ graphs into the collection of $[p^{-1}]$ almost full subgraphs (cliques) above critical fugacity, $\mu_c$, where $p$ is the ER bond formation probability. Evolution of the spectral density, $\rho(\lambda)$, of the adjacency matrix with increasing $\mu$ leads to the formation of two-zonal support for $\mu>\mu_c$. Eigenvalue tunneling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value $Np$. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.