Statistical mechanics of topological phase transitions in networks

We provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. The associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge-dual graph of a fully connected network. In our studies—based on an exact enumeration method, Monte Carlo simulations, and theoretical considerations—we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure of the network. Depending on the energy function chosen, the observed transitions can be best monitored using the order parameters ${\Phi }_s{=s}_{\max }/M,$ i.e., the size of the largest connected component divided by the number of edges, or ${\Phi }_k{=k}_{\max }/M,$ the largest degree in the network divided by the number of edges. If, for example, the energy is chosen to be $E=-s_{\max },$ the observed transition is analogous to the percolation phase transition of random graphs. For this choice of the energy, the phase diagram in the $(〈k〉,T)$ plane is constructed. Single-vertex energies of the form $E={\sum }_i{f(k}_i),$ where $k_i$ is the degree of vertex i, are also studied. Depending on the form of ${f(k}_i),$ first-order and continuous phase transitions can be observed. In case of ${f(k}_i)=-{(k}_i+\alpha )\ln {(k}_i),$ the transition is continuous, and at the critical temperature scale-free graphs can be recovered. Finally, by abruptly decreasing the temperature, nonequilibrium processes (e.g., nucleation and growth of particular topological phases) can also be interpreted by the present approach.