Statistical mechanics of random geometric graphs: Geometry-induced first-order phase transition

Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs

The vertex-cover problem is studied for random graphs GN,cN having N vertices and cN edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a c-dependent threshold x = xc(c) appears, where xN is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical effort, which is need...

Core percolation: a new geometric phase transition in random graphs

We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the n...

Softening of First-Order Phase Transition on Quenched Random Gravity Graphs

We perform extensive Monte Carlo simulations of the 10-state Potts model on quenched two-dimensional Φ3 gravity graphs to study the effect of quenched coordination number randomness on the nature of the phase transition, which is strongly first order on regular lattices. The numerical data provides strong evidence that, due to the quenched randomness, the discontinuous first-order phase transit...

Equilibrium statistical mechanics on correlated random graphs

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes curren...

Autoregressive Gaussian Random Vectors of First Order