Diffusively coupled networks constitute a ubiquitous class of spatially-distributed models fundamental to understanding the behavior of numerous engineered and biological systems and producing surprisingly rich dynamics. In this talk, we develop analysis methods and distributed algorithms that exploit network structure and individual component dynamics in order to guarantee the desirable behavior of the aggregate network system in the absence of centralized coordination.
We first formulate network design problems to guarantee multi-agent coordination by imposing constraints on the diffusive coupling graph linking agents using semidefinite programming. Our approach identifies critical nodes and edges in a network, and aids in enhancing connectivity and robustness. We next derive conditions that guarantee synchronization in compartmental systems of ODEs and reaction-diffusion PDEs. Conversely, we highlight cases of diffusion driven instability, a phenomenon widely hypothesized as a mechanism behind pattern formation in biological systems. We then discuss the case of space-dependent diffusive coupling, and develop adaptive algorithms in which the weights of the coupling graph are modified according to local disagreements between agents with the goal of reducing time to synchronization. We punctuate our discussion with examples from network control, developmental biology, and coupled oscillators.