In this talk we provide a unified view of several distributed coordination and consensus algorithms which have appeared in various disciplines, such as distributed systems, statistical physics, computer graphics, robotics, and control theory over the past decade. These algorithms have been proposed as a mechanism for demonstrating emergence of a global collective behavior (such as social aggregation in various species such as fish schooling, bird flocking, etc) using purely local interactions. Next, we extend our results from graphs to simplicial complexes (objects of study in algebraic topology) to verify coverage in mobile sensor networks in a purely decentralized fashion and without localization. These simplicial complexes are combinatorial objects that generalize the proximity graphs formed from binary relations between agents to higher order relations, and their study will allow us to infer the coverage properties of mobile sensor networks with time-varying interconnections without any localization. The enabling mathematical technique for our result is use of higher order Laplacian operators from discrete Hodge theory, which will be presented as a generalization of the graph Laplacian used in the first part of the talk for analysis of synchronization, agreement and consensus problems.
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