Optimized network properties in network growing models

Two recently proposed network connection growing models that aim at minimizing the wiring cost while at the same time maximizing the network connections have shown to exhibit a variety of interesting phase transition phenomena [Cheng and Lai (2020)]. Here we derive analytical results for these models on the key properties of the fully optimized network, such as the degree distribution, optimized weight distribution, cluster coefficients and minimal path length for general wiring cost and node weight distributions. For the case of unweighted nodes, the degree distribution of the optimized network is shown to be generically binomial and the cluster coefficient is simply the mean connection probability. Further optimized network characteristics are simulated and measured using a recently proposed effective algorithm and agree well with the analytic results. The optimized networks are discussed in the light of scaling behavior and network design.