Eigenvalue tunneling and decay of quenched random network

Abstract

We consider the canonical ensemble of N -vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ 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, µc, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of two-zonal support for µ > µc. Eigenvalue tunnelling 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 N p. 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.Investigation of critical and collective effects in graphs and networks has becoming a new rapidly developing interdisciplinary area, with diverse applications and variety of questions to be asked, see [1] for review. Ensembles of random Erdős-Rényi topological graphs (networks) provide an efficient laboratory for testing collective phenomena in statistical physics of complex systems, being also tightly linked to conventional random matrix theory. Besides investigating typical statistical properties of networks, like vertex degree distribution, clustering coefficients, "small world" structure etc, last two decades have been marked by rapidly growing interest in more refined graph characteristics, such as distribution of small subgraphs involving triads of vertices.Triadic interactions, being the simplest interactions beyond the free-field theory, play crucial role in the network statistics. Presence of such interactions is responsible for emergence of phase transitions in complex distributed systems. First example of a phase transition in random networks, known as Strauss clustering [2], has been treated by the Random Matrix Theory (RMT) in [3]. It was argued that, when the increasing fugacity, µ, the system develops two phases with essentially different triad concentrations. At large µ the system falls into the Strauss phase with the single clique of nodes. The condensation of triads is a non-perturbative phenomenon identified in [4] with the 1st order phase transition in the framework of mean-field cavity-like approach.Similar critical behavior was found in [5] for the vertexdegree-conserved ER graphs. It was demonstrated in the framework of the mean-field approach that the phase transition takes place in this case as well. The hysteresis for dependence of the triad concentration on the fugacity, µ also has been observed in [5]. For bi-color networks with conserved vertex degree a new phenomena of a wide plateau formation in concentration of black-white bonds as a function of the fugacity of unicolor triples of bonds has been found in...

Exactly solvable random graph ensemble with extensively many short cycles

“…Yet, overall we find a similar phenomenology. In fact, our present analysis predicts that the parameters in [26] would need a scaling with N , in order for the transition not to be a finite size effect but to persist in the N → ∞ limit.…”

Finite-size effects in exponential random graphs

“…Since all hubs are identical in the "fat star", the eigenvalues at fixed network are degenerate. However since the number of links is different in each ensemble they form two zones which are analogue of the non-perturbative zone found in [12]. Increasing N we increase the number of evaporated nodes which results into the interaction between the star nodes and lifting the degeneration of the eigenvalues.…”

“…Secondly the isolated eigenvalues appear at the clusterization phase transition since the number of the isolated eigenvalues corresponds to the number of clusters. The creation of the cluster has been identified as the eigenvalue instanton in [12].…”

Phase transitions in social networks inspired by the Schelling model

“…The system behaves essentially differently when a vertex degree is strictly conserved during the Metropolis rewiring. We found that above some critical fugacity, γ c , a large network is fragmented into a collection of [p −1 ] almost fully connected sub-graphs (cliques) [29], where p is the bond formation probability in the initial random Erdős-Rényi network and [...] denotes the integer part of the argument. In Fig.8a,b we show typical structure of adjacency matrices at few intermediate stages of network rewiring towards the ground states of constrained ( Fig.8a) and unconstrained (Fig.8b) Erdős-Rényi networks (reproduced from [29]).…”