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.

中文翻译：

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优化网络模型中的相变

我们考虑网络连接增长模型，该模型旨在最小化布线成本，同时最大化网络连接。通过将系统映射到Ising自旋模型，我们获得了两个这样的模型的分析结果，对于一般的布线成本分布，这两个模型都显示出有趣的但不同的相变行为。还获得了这些过渡的相位图。这些结果也扩展到使用加权节点度进行连接优化的网络。根据边缘和节点权重分布的属性，系统可以显示各种相变，包括一阶，二阶和混合相变，从无连接到有限连接部分的网络。此外，平均场理论为找到在这些模型中完全优化的网络配置提供了一种有效的算法。所有这些结果也通过数值模拟得到了验证。