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Introduction to Averaging Dynamics over Networks pp 69–92Cite as

Averaging in Time-Varying Networks

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 472))

Abstract

This chapter studies averaging dynamics in which the update matrix and, possibly, the underlying graph may be different at each time step. This extension is particularly important for the applications. Indeed, in realistic models of sensor and robotic networks, units, and links may be occasionally off due to environmental reasons or for energy saving purposes. Similarly, social dynamics may involve complex patterns of interactions that change over time. We are going to show that time-dependent consensus algorithms converge under relatively mild assumptions involving suitable notions of connectivity. Actually, the underlying graph needs not to be connected at any time, but the sequence of graphs must be “sufficiently connected” over time. More specifically, in Sects. 3.1 and 3.2 we provide two families of results, corresponding to two sufficient connectivity assumptions. Our presentation also includes, in Sect. 3.3, cases when the matrix evolves randomly in time. These randomized dynamics encompass the so-called gossip algorithms, which have attracted much attention in the last decade.

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Notes

  1. 1.

    In this result, inequalities between matrices like \(A\le B\) have to be intended as \(A-B\) being negative semidefinite.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Turin, Italy

    Fabio Fagnani

  2. GIPSA-lab, CNRS, Grenoble, France

    Paolo Frasca

Authors
  1. Fabio Fagnani
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  2. Paolo Frasca
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Correspondence to Paolo Frasca .

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Fagnani, F., Frasca, P. (2018). Averaging in Time-Varying Networks. In: Introduction to Averaging Dynamics over Networks. Lecture Notes in Control and Information Sciences, vol 472. Springer, Cham. https://doi.org/10.1007/978-3-319-68022-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-68022-4_3

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