We study the emerging large-scale structures in networks subject to selective pressures that simultaneously drive toward higher modularity and robustness against random failures. We construct maximum-entropy null models that isolate the effects of the joint optimization on the network structure from any kind of evolutionary dynamics. Our analysis reveals a rich phase diagram of optimized structures, composed of many combinations of modular, core-periphery, and bipartite patterns. Furthermore, we observe parameter regions where the simultaneous optimization can be either synergistic or antagonistic, with the improvement of one criterion directly aiding or hindering the other, respectively. Our results show how interactions between different selective pressures can be pivotal in determining the emerging network structure, and that these interactions can be captured by simple network models.

中文翻译：

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空模型用于多重优化的大型网络结构

我们研究网络中正在出现的大规模结构，这些结构会受到选择性压力的影响，这些压力会同时推动更高的模块性和针对随机故障的鲁棒性。我们构建了最大熵零模型，该模型将联合优化对网络结构的影响与任何种类的演化动力学隔离开来。我们的分析揭示了优化结构的丰富相图，该结构由模块化，核心外围和二分模式的许多组合组成。此外，我们观察到参数区域，其中同时优化可以是协同的或拮抗的，其中一种准则的改进分别直接辅助或阻碍了另一种准则。我们的结果表明，不同选择压力之间的相互作用如何对确定新兴网络结构至关重要，