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 Erdős-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 Erdős-Rényi graph samples that feature condensation of degrees. arXiv:1909.10564v1 [cond-mat.dis-nn]

Analytic solution of the two-star model with correlated degrees

“…With the purpose of improving the model, it would be interesting to solve it with a hard constraint in the degree sequence or with a prescribed degree distribution [10]. This work also opens the perspective to explore systematically the role of short-range and long-range degree correlations in dynamical processes occurring on tree-like networks, since in this case the equations for the dynamics are typically determined only by the degree distribution [49]. Finally, we point out that the free energy of ERG models can be mapped in the cumulant generating function of certain structural observables of Erdös-Rényi random graphs [50][51][52] .…”

Analytic solution of the two-star model with correlated degrees

“…With the purpose of improving the model, it would be interesting to solve it with a hard constraint in the degree sequence or with a prescribed degree distribution [10]. This work also opens the perspective to explore systematically the role of short-range and long-range degree correlations in dynamical processes occurring on treelike networks, since in this case the equations for the dynamics are typically determined only by the degree distribution [50]. Finally, we point out that the free energy of ERG models can be mapped in the cumulant generating function of certain structural observables of Erdös-Rényi random graphs [51][52][53].…”