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

Abstract

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 hypergraph. 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 intracluster 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 topological measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity and total integration, and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.

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“…However, the optimised networks lacked of hubs and rich-clubs. Dynamical models on modular networks have shown that there is a balanced rate in the number of inter- to intra-modular links that optimises the complexity of the network dynamics1011. This phenomenon has also been observed in contagion spreading, where the contagion threshold depends on the node’s degree12.…”

Assortative and modular networks are shaped by adaptive synchronization processes

“…However, relevant situations may occur in which the network is globally unsynchronized and yet groups of nodes in it display a high level of local synchrony. To measure the degree of local synchronization around node n, here we consider (4) and, consequently, the average degree of local synchronization over the whole network is…”

“…Natural networking systems [1] are vastly characterized by a modular organization of their connectivity structure [2], and by nontrivial correlation features in the way units with a given number of connections (degree) tend to link with members of the same degree (assortativity), or with units with different degrees (disassortativity). Modularity is clearly the result of the need for social, biological, and technological systems to optimize their parallel, yet integrated, functioning [3,4] by means of an organization into mesoscale structures, such as communities, i.e., groups of highly interconnected nodes that are sparsely connected to the rest of the graph [5]. Degreedegree correlation reflects the observed tendency of natural networks to organize the main topology on top of a backbone of nodes that may be starlike (disassortativity) or of highly connected hubs (assortativity).…”

“…Modularity is clearly the result of the need for social, biological, and technological systems to optimize their parallel, yet integrated, functioning [3,4] by means of an organization into mesoscale structures, such as communities, i.e., groups of highly interconnected nodes that are sparsely connected to the rest of the graph [5]. Degreedegree correlation reflects the observed tendency of natural networks to organize the main topology on top of a backbone of nodes that may be starlike (disassortativity) or of highly connected hubs (assortativity).…”

Network analysis: a new approach to study endocrine disorders

“…definite function ( Fig. 1C; Gavin et al 2006, Adjari et al 2012, Szalay-Beko et al 2012. Important concepts in analysis include node centrality and network robustness (Barzel & Biham 2009; Box 1 and Fig.…”