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Nonlinear Dynamics of Networks

CSIC Building (#406), Seminar Room 4122.

Sensitive Dependence on Network Structure: To Sync or Not to Sync

Adilson Motter

Northwestern University

Abstract:   Synchronization, in which individual dynamical units keep in pace with each other in a decentralized fashion, 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 synchronization properties would change monotonically as the number of available interactions is varied, 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, which 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. These results extend the recently discovered compensatory perturbations in metabolic networks to the realm of oscillator networks and demonstrate why "less can be more" in network synchronization.

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