Observability and coarse graining of consensus dynamics through the external equitable partition

Neave, Guy; Yuan, Ye; Stan, Guy-Bart; Barahona, Mauricio

doi:10.1103/physreve.88.042805

Cited by 67 publications

(73 citation statements)

References 36 publications

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“…For a Laplacian diffusion dynamics (A = −L) this idea can be made more precise using so-called externally equitable partitions [25], which explicitly relate our similarity measure to model reduction. Consider an external equitable partition (EEP) characterized by the relation…”

Section: Integrated Dynamical Similarity and Control-theoretic Interpmentioning

confidence: 99%

“…which shows that Ψ will be block-structured. Consequently the dynamics of the full system within the subspace spanned by the partition can be described exactly by a reduced model [25,26], which is governed here by A = L, C = I and has only a single input per group, equal to the average input within the original group. It is instructive to compare the above dynamical blockstructure to the notions like stochastic block-models [6,7], in which each node in a group has statistically the same (static) connection profile.…”

Section: Integrated Dynamical Similarity and Control-theoretic Interpmentioning

confidence: 99%

“…Brown University (16) 15. University of Virginia (25) History Business Ranking Figure 4. Analysing academic influence using low-dimensional embeddings A Low-dimensional embeddings based on the influence dynamics in the hiring network, as described in the text (see Figure 3) for the discipline History.…”

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confidence: 99%

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Multiscale dynamical embeddings of complex networks

2019

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Complex systems and relational data are often abstracted as dynamical processes on networks. To understand, predict and control their behavior, a crucial step is to extract reduced descriptions of such networks. Inspired by notions from Control Theory, we propose a time-dependent dynamical similarity measure between nodes, which quantifies the effect a node-input has on the network. This dynamical similarity induces an embedding that can be employed for several analysis tasks. Here we focus on (i) dimensionality reduction, i.e., projecting nodes onto a low dimensional space that captures dynamic similarity at different time scales, and (ii) how to exploit our embeddings to uncover functional modules. We exemplify our ideas through case studies focusing on directed networks without strong connectivity, and signed networks. We further highlight how certain ideas from community detection can be generalized and linked to Control Theory, by using the here developed dynamical perspective.

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Section: Integrated Dynamical Similarity and Control-theoretic Interpmentioning

confidence: 99%

Section: Integrated Dynamical Similarity and Control-theoretic Interpmentioning

confidence: 99%

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Multiscale dynamical embeddings of complex networks

2019

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“…Here the k × n matrix C + is the (left) Moore-Penrose pseudoinverse of C. Observe that multiplying a vector x ∈ R n by C from the left sums up the components within each cell, and that C C is a diagonal matrix with the number of nodes per cell on the diagonal. Hence, C + = (C C) −1 C can be simply interpreted as a cell averaging operator 59 . After straightforward algebraic manipulations it is easy to show that:…”

Section: B Strictly Invariant Subspaces Of the Network Dynamics And mentioning

confidence: 99%

Structured Networks and Coarse‐Grained Descriptions

Delvenne

Lambiotte

Barahona

2019

Advances in Network Clustering and Blockmodeling

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This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics.In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance.In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.

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“…Email: longwang@pku.edu.cn is equivalent to that of a pair of submatrices of Laplacian matrix. In what follows, some related results on the controllability of continuous-time systems are proposed, such as first-order and high-order multi-agent systems in Wang, Jiang, Xie, and Ji (2009), switching topology and time delay in Ji, Wang, Lin, and Wang (2010), equitable and relaxed equitable partition in Rahmani, Ji, Mesbahi, and Egerstedt (2009) and external equitable partition in O'Clery, Yuan, Stan, and Barahona (2013). For the controllability of discrete-time systems, the controllability conditions for first-order multi-agent systems with switching topology are proposed in Liu, Chu, Wang, and Xie (2008).…”

Section: Introductionmentioning

confidence: 98%

Controllability of discrete-time multi-agent systems with directed topology and input delay

Zhang

et al. 2015

International Journal of Control

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This paper investigates the controllability of first-order and second-order discrete-time multi-agent systems with directed topology and input delay. The problem is studied in the leader-follower framework where a number of agents are selected to be leaders and serve as control inputs to all other agents. Sufficient and necessary conditions are derived for the controllability of first-order discrete-time multi-agent systems. With sampling period and feedback gain satisfying certain conditions, it is proved under three different protocols that the controllability of second-order discrete-time multi-agent systems is equivalent to that of a pair of submatrices of Laplacian matrix. In addition, the controllability of both first-order and second-order discrete-time multi-agent systems with input delay is shown, through some transformations, to be equivalent to that of the transformed non-delayed systems. Finally, some simulation examples are given to illustrate the theoretical results.

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Observability and coarse graining of consensus dynamics through the external equitable partition

Cited by 67 publications

References 36 publications

Multiscale dynamical embeddings of complex networks

Multiscale dynamical embeddings of complex networks

Structured Networks and Coarse‐Grained Descriptions

Controllability of discrete-time multi-agent systems with directed topology and input delay

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