Diverse phase transitions in optimized directed network models with distinct inward and outward node weights

We consider growing directed network models that aim at minimizing the weighted connection expenses while at the same time favoring other important network properties such as weighted local node degrees. We employed statistical mechanics methods to study the growth of directed networks under the principle of optimizing some objective function. By mapping the system to an Ising spin model, analytic results are derived for two such models, exhibiting diverse and interesting phase transition behaviors for general edge weight, inward and outward node weight distributions. In addition, the unexplored cases of negative node weights are also investigated. Analytic results for the phase diagrams are derived showing even richer phase transition behavior, such as first-order transition due to symmetry, second-order transitions with possible reentrance, and hybrid phase transitions. We further extend previously developed zero-temperature simulation algorithm for undirected networks to the present directed case and for negative node weights, and we can obtain the minimal cost connection configuration efficiently. All the theoretical results are explicitly verified by simulations. Possible applications and implications are also discussed.

Automated summary

This study utilized statistical mechanics methods to investigate the growth of directed network models, with the aim of minimizing weighted connection expenses while favoring other important network properties such as weighted local node degrees. Analytic results were derived for two models, showing diverse and interesting phase transition behaviors under different weight distributions. The unexplored cases of negative node weights were also explored. The derived phase diagrams exhibited even richer phase transition behavior, including first-order transitions due to symmetry, second-order transitions with possible reentrance, and hybrid phase transitions. The developed zero-temperature simulation algorithm for undirected networks was extended to the directed case and negative node weights, allowing for efficient identification of minimal cost connection configurations. The theoretical results were verified through simulations, and possible applications and implications were discussed.