Dynamics on complex networks: modeling the form to shape the substance

Dynamical systems coupled via complex networks are widespread in nature and thus frequently invoked for a large plethora of interesting applications, from physics to ecology, passing through transportation and social sciences. The dynamical behavior of such systems is rooted in the interplay between the network structure and the local dynamics to which each node is subjected. The asymptotic equilibria eventually attained by the systems, and their associated stability, can be assessed by employing standard nonlinear dynamics tools. For many practical applications it is however important to actively drive the system under exam towards a different chosen equilibrium, also requiring its stability to external perturbations. With this goal in mind, I here analyze the reflex of the network structure on the global dynamical behavior of the scrutinized system. Tracing such reflex back to the graph I devise apt control methods to steer the stable equilibria towards the desired states by modifying the network topology and without touching the local dynamics. Such investigation also allows, when dealing with reaction-diffusion systems, to divide the networks in classes which bear the same kind of instability and consequently the same pattern of node activation. Moreover, by making use of a special random walk process I approach the inverse problem of network reconstruction: starting from the naturally obtained equilibrium I try to get some hints about the structure of the underlying network of connections. The implemented methods are mostly based on spectral properties of the connection matrix and our microscopic analysis will enable us to address macroscopic phenomena like oscillators synchronization and ecosystems stability.