Phase transitions in optimized network models

Abstract

We consider network connection growing models that aim at minimizing the wiring cost while at the same time maximizing the network connections. By mapping the system to an Ising spin model, we obtain analytic results for two such models, both of them show interesting, but different phase transition behaviors for general wiring cost distributions. The phase diagrams for these transitions are also obtained. These results are also extended for networks optimized with weighted node degree for connections. Depending on the properties of the edge and node weight distributions, the system can exhibit a variety of phase transitions, including first-order, second-order and hybrid ones, from no connection to a network of finite fraction of connections. Furthermore, mean-field theory leads to an effective algorithm for finding the fully optimized network configurations in these models. All these results are also verified by numerical simulations.