模块化网络中分离与集成有效交叉点的拓扑测度
Topological Measure Locating the Effective Crossover between Segregation and Integration in a Modular Network
摘要：
我们引入了一个易于计算的拓扑测度，它可以定位模块化网络中隔离和集成之间的有效交叉。隔离度对应于网络的模块化程度，而集成度则用关联超图的代数连通度来表示。严格的治疗简化的情况下，集团的大小相等，逐渐重新布线，直到他们成为完全合并，使我们能够表明，这种拓扑交叉可以符合动态交叉从集群到全球同步的系统耦合相位振荡器。动态交叉信号的峰值内集群和全球同步的措施，我们建议作为一个动态的复杂性措施的产品。这个量比熵（振荡器的平均频率）更容易计算，并且显示出一种行为，它非常模仿基于后者的动力学复杂性指数。所提出的拓扑测量同时提供了动态行为的信息，揭示了模块化与总集成之间的相互作用，并显示了这如何影响网络执行本地和分布式动态任务的能力。
英文摘要：
We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hyper-graph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intra-cluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed toplogical measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity vs total integration and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.
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