2016
DOI: 10.1103/physreve.94.062313
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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 tu… Show more

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Cited by 24 publications
(55 citation statements)
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“…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.…”
Section: Discussionmentioning
confidence: 75%
“…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.…”
Section: Discussionmentioning
confidence: 75%
“…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.…”
Section: Comments On the Spectral Behaviormentioning
confidence: 83%
“…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].…”
Section: Comments On the Spectral Behaviormentioning
confidence: 99%
“…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]).…”
Section: A Favoring Of Triangles (Regime A)mentioning
confidence: 99%