Critical phenomena in complex networks: from scale-free to random networks

Within the conventional statistical physics framework, we study critical phenomena in configuration network models with hidden variables controlling links between pairs of nodes. We obtain analytical expressions for the average node degree, the expected number of edges in the graph, and the Landau and Helmholtz free energies. We demonstrate that the network’s temperature controls the average node degree in the whole network. We also show that phase transition in an asymptotically sparse network leads to fundamental structural changes in the network topology. Below the critical temperature, the graph is completely disconnected; above the critical temperature, the graph becomes connected, and a giant component appears. Increasing temperature changes the degree distribution from power-degree for lower temperatures to a Poisson-like distribution for high temperatures. Our findings suggest that temperature might be an inalienable property of real networks. Graphical abstract