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Reaching a consensus in a dynamically changing environment: a graphical approach. (English) Zbl 1157.93514
Summary: This paper presents new graph-theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments. The concepts of rooted, strongly rooted, and neighbor-shared are defined, and conditions are derived for compositions of sequences of directed graphs to be of these types. The graph of a stochastic matrix is defined, and it is shown that under certain conditions the graph of a Sarymsakov matrix and a rooted graph are one and the same. As an illustration of the use of the concepts developed in this paper, graph-theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied Vicsek consensus problem.
MSC:
93E03General theory of stochastic systems
05C50Graphs and linear algebra
05C75Structural characterization of families of graphs
15A51Stochastic matrices (MSC2000)
40A20Convergence and divergence of infinite products
68W15Distributed algorithms
93C85Automated control systems (robots, etc.)