Seminário da Pós-graduacao em Matemática Aplicada

"The Network Analogue of Chaos"

Adilson Motter

Spontaneous synchronization is a process in which individual dynamical units keep in pace with each other in a decentralized fashion. This process depends both on the dynamical units and on the properties of the interaction network. Yet, the role played by the network has resisted comprehensive characterization within the prevailing paradigm that interactions facilitating pair-wise synchronization also facilitate collective synchronization.

In this talk I will challenge this paradigm using networks of diffusively coupled systems. I will show that the synchronization landscape is characterized by a non-monotonic, periodic structure of cusps, which leads to sensitive dependence of the synchronization properties on the specific combination of "nodes and links". Accordingly, the expectations that synchronization would be easier to achieve with more interactions, that larger networks would be more difficult to synchronize, and that certain network structures would facilitate while others would inhibit synchronization, are all false. In particular, I will show that networks with best complete synchronization, least coupling cost, and maximum dynamical robustness, have arbitrary complexity but quantized total interaction strength that constrains the allowed number of connections. It stems from this characterization that negative interactions as well as link removals can be used to systematically improve and optimize synchronization properties in both directed and undirected networks.