We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule. We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks. We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a differentiation in the activation probabilities of the links. This, in turn, yields the formation of hubs associated to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can be used to solve a wide range of synchronization and control problems.

This chapter reviews several approaches to study convergence of networks of nonlinear dynamical systems based on the use of contraction theory. Rather than studying the properties of the collective asymptotic solution of interest, the strategy focuses on finding sufficient conditions for any pair of trajectories of two agents in the network to converge towards each other. The key tool is the study, in an appropriate metric, of the matrix measure of the agents' or network Jacobian. The effectiveness of the proposed approach is illustrated via a set of representative examples. IntroductionThe problem of steering the collective behaviour of a network of dynamical agents towards a desired common target solution is a fundamental problem in network science and control theory [1][2][3]. A typical problem is that of achieving consensus or synchronization in a network of linear or nonlinear systems. Here, the key challenge, once the coupling function among the nodes has been selected, is to prove convergence of all nodes towards the desired common asymptotic behaviour. This convergence problem is usually solved locally by means of the Master Stability Function Approach (MSF) [4] or globally via Lyapunov stability theory [5].Global convergence is extremely useful particularly when the agents are subject to high level of noise or perturbations as it is often the case in applications; think for example of the recent application of synchronization to biological systems [6].

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