Topic
Connectivity
About: Connectivity is a research topic. Over the lifetime, 5418 publications have been published within this topic receiving 108611 citations.
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TL;DR: The exponential convergence of the proposed algorithm under strongly connected and weight-balanced digraph topologies when the local costs are strongly convex with globally Lipschitz gradients is established, and an upper bound on the stepsize is provided that guarantees exponential convergence over connected graphs for implementations with periodic communication.
543 citations
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TL;DR: In this article, an analogous difference operator is studied for an arbitrary graph, and it is shown that many properties of the Laplacian in the continuous setting (e.g., the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator.
Abstract: The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference oper- ator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient. The random walks we consider are defined as follows. Let K be a connected graph (i.e. a one dimensional simplicial complex). For a vertex x E K, let m(x) denote the number of edges emanating from x. The probability that a particle moves from x to another vertex y E K is l/m(x) if x and y are connected by an edge and it is zero otherwise. As observed by Courant, Friedrichs and Lewy (CFL) for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the Laplacian
527 citations
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TL;DR: A decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced.
Abstract: In this note, a decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced. The proposed nonholonomic control law is discontinuous and time-invariant and using tools from nonsmooth Lyapunov theory and graph theory the stability of the overall system is examined. Similarly to the linear case, the convergence of the multi-agent system relies on the connectivity of the communication graph that represents the inter-agent communication topology. The control law is first defined in order to guarantee connectivity maintenance for an initially connected communication graph. Moreover, the cases of static and dynamic communication topologies are treated as corollaries of the proposed framework
509 citations
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TL;DR: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity.
Abstract: Deterministic fully dynamic graph algorithms are presented for connectivity, minimum spanning tree, 2-edge connectivity, and biconnectivity. Assuming that we start with no edges in a graph with n vertices, the amortized operation costs are O(log2n) for connectivity, O(log4n) for minimum spanning forest, 2-edge connectivity, and O(log5n) biconnectivity.
501 citations
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TL;DR: The main result is a theorem: the maximum possible flow from left to right through a network is equal to the minimum value among all simple cut-sets.
Abstract: This note discusses the problem of maximizing the rate of flow from one terminal to another, through a network which consists of a number of branches, each of which has a limited capacity. The main result is a theorem: The maximum possible flow from left to right through a network is equal to the minimum value among all simple cut-sets. This theorem is applied to solve a more general problem, in which a number of input nodes and a number of output nodes are used.
493 citations