Networks of dynamical systems occur in diverse areas of engineering, such as distributed computation, automated vehicular formations, and power systems networks. We investigate the interplay between network coherence and the network's interconnection topology. In such networks, synchronization or more general coherence under stochastic uncertainty is an important measure of performance. For example, in power systems this notion corresponds to phase and frequency coherence of multiple generators (the lack of which may lead to so-called inter-area oscillations), while in vehicular formations this notion corresponds to how well vehicles are collectively tracking their commanded trajectories. We study abstractions of all of the above problems as distributed/cooperative control problems. We investigate the asymptotic limits of performance when network sizes become large as a function of the networks' topologies. This has important implications for the design of future highly-distributed-generation power networks, as well as large vehicular formation schemes such as those in proposed automated highway systems. This talk will specifically address the notion of network coherence under stochastic disturbances, and its dependence on network topology and various notions of network dimension. Regular lattices and fractal networks provide case studies with both integer and fractional dimension. We give asymptotic lower bounds on network disorder and show its dependence on both the complexity of individual node dynamics, as well as network dimension. It turns out that higher connectivity improves coherence, while more complex node dynamics can hinder it. However, in all cases there is a critical network dimension above which purely local interactions can lead to the emergence of global order. We outline the connections between these results and those on the statistical mechanics of harmonic solids. For future highly-distributed-generation power networks, we quantify the cost of synchronization in terms of resistive line losses due to the power flow needed to achieve this synchrony. We show that this cost scales unboundedly with the number of generators in the network and is independent of the underlying connection topology. This result is a further argument for the use of information communication rather than only power flows to enhance synchrony in such large networks.


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