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Averaging in Time-Invariant Networks

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Introduction to Averaging Dynamics over Networks

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

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Abstract

This chapter studies the basic averaging dynamics on a fixed network. This linear dynamics is also called “consensus” dynamics, because under suitable assumptions it brings the states associated with the nodes to converge to the same value. Section 2.1 introduces the rendezvous problem, which serves us as the main motivation to seek consensus, and states the main results of the chapter. Section 2.2 solves the consensus problem in the special case of symmetric regular graphs, while the general solution, which is based on the notion of stochastic matrix, is presented in Sect. 2.3. The subsequent sections provide further insights into the averaging dynamics, namely about its speed of convergence (Sects. 2.4 and 2.7) and its consensus value (Sect. 2.5). Meanwhile, Sect. 2.6 presents some classical examples of stochastic matrices associated with a graph, such as simple random walks and Metropolis walks. Finally, Sect. 2.8 concentrates on reversible stochastic matrices and their properties.

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Notes

  1. 1.

    Here and throughout the book, we will make use of the standard asymptotic notation. If \(f_n\) and \(g_n\) are two positive sequences, we say that \(f_n=O(g_n)\) if \(f_n/g_n\) is upper bounded in n; that \(f_n=\varTheta (g_n)\) if \(f_n/g_n\) is both lower and upper bounded in n; that \(f_n=o(g_n)\) if \(f_n/g_n\rightarrow 0\) as \(n\rightarrow \infty \); and that \(f_n\sim g_n\) if \(f_n/g_n\rightarrow 1\) as \(n\rightarrow \infty \).

  2. 2.

    A matrix A is said to be normal when \(A^* A = AA^*\). Normal matrices are precisely those for which a complete basis of eigenvectors exists. Symmetric matrices are normal.

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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

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  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-Invariant 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_2

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  • DOI: https://doi.org/10.1007/978-3-319-68022-4_2

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