Eigenvalue tunnelling and decay of quenched random networks

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, $μ_c$, 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 $μ>μ_c$. 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.

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