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We introduce an easily computable topological measure which locates theeffective crossover between segregation and integration in a modular network.Segregation corresponds to the degree of network modularity, while integrationis expressed in terms of the algebraic connectivity of an associatedhyper-graph. The rigorous treatment of the simplified case of cliques of equalsize that are gradually rewired until they become completely merged, allows usto show that this topological crossover can be made to coincide with adynamical crossover from cluster to global synchronization of a system ofcoupled phase oscillators. The dynamical crossover is signaled by a peak in theproduct of the measures of intra-cluster and global synchronization, which wepropose as a dynamical measure of complexity. This quantity is much easier tocompute than the entropy (of the average frequencies of the oscillators), anddisplays a behavior which closely mimics that of the dynamical complexity indexbased on the latter. The proposed toplogical measure simultaneously providesinformation on the dynamical behavior, sheds light on the interplay betweenmodularity vs total integration and shows how this affects the capability ofthe network to perform both local and distributed dynamical tasks.

Topological Measure Locating the Effective Crossover between Segregation and Integration in a Modular Network