Modular organization and degree-degree correlations are ubiquitous in the connectivity structure of biological, technological, and social interacting systems. So far most studies have concentrated on unveiling both features in real world networks, but a model that succeeds in generating them simultaneously is needed. We consider a network of interacting phase oscillators, and an adaptation mechanism for the coupling that promotes the connection strengths between those elements that are dynamically correlated. We show that, under these circumstances, the dynamical organization of the oscillators shapes the topology of the graph in such a way that modularity and assortativity features emerge spontaneously and simultaneously. In turn, we prove that such an emergent structure is associated with an asymptotic arrangement of the collective dynamical state of the network into cluster synchronization. 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).Whereas the emphasis typically has been on devising suitable tools and algorithms to unveil, identify, and quantify both modular and correlation features in real world networks [1,2], a model that has proven to be capable of generating these features simultaneously is needed. Modularity and assortativity are usually imposed separately, in a second step, on top of already generated networks by means of ad hoc algorithms and methods [6,7].In this Rapid Communication, we show that all these features may spontaneously emerge in an adaptive network of interacting oscillators as the result of a delicate interplay between synchronization processes and coevolution of the connectivity structure. When the connectivity dynamics is such that links coupling the nodes with synchronous (nonsynchronous) dynamics are promoted (weakened), we prove that an initially unstructured clique configuration evolves in time toward an emerging structured network displaying both modularity and assortativity.Let us then start by considering an initial ensemble of N all-to-all coupled Kuramoto oscillators [8,9]. Each unit of the ensemble n = 1,..., N is characterized by its phase 9", whose dynamics is ruled byw nm sin(0 m -6"), (1) where £2 n is the natural frequency of the wth ...
