Phase transitions in atypical systems induced by a condensation transition on graphs

Abstract

Random graphs undergo structural phase transitions that are crucial for dynamical processes and cooperative behavior of models defined on graphs. In this work we investigate the impact of a first-order structural transition on the thermodynamics of the Ising model defined on Erdös-Rényi random graphs, as well as on the eigenvalue distribution of the adjacency matrix of the same graphical model. The structural transition in question yields graph samples exhibiting condensation, characterized by a large number of nodes having degrees in a narrow interval. We show that this condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model. The condensation transition also leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigenvalue distribution discontinuous. As a side result, we derive the critical line determining the percolation transition in Erdös-Rényi graph samples that feature condensation of degrees.

11 pages, 4 figures

References in corpus (11)

• Critical phenomena in complex networks
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• Synchronization transition in scale-free networks: Clusters of synchrony
• On the localization transition in symmetric random matrices
• Eigenvalue tunnelling and decay of quenched random networks
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Condensation of Fluctuations in and out of Equilibrium
• Probability distributions with singularities
• Condensation of degrees emerging through a first-order phase transition in classical random graphs
• Finite-size effects in exponential random graphs and cluster evaporation