Resume : Spontaneous synchronization, in which individual dynamical units keep in pace with each other in a decentralized fashion, is a fascinating phenomenon found in systems as diverse as chirping crickets, flashing fireflies, bursting neurons, and power-generating devices. Like other forms of collective behavior, synchronization is known to depend 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 will also facilitate collective synchronization. In this colloquium, 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 and that certain network structures would facilitate while others would inhibit synchronization, are both 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, and hence that "less can be more" in network synchronization. Recent related publications: [1] T. Nishikawa and A.E. Motter, Network Synchronization Landscape Reveals Compensatory Structures, Quantization, and the Positive Effect of Negative Interactions, Proc. Natl. Acad. Sci. USA 2010. [2] A.E. Motter, Spontaneous Synchrony Breaking, Nature Physics 2010.
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