TY - JOUR

T1 - Cluster consensus with point group symmetries

AU - Chen, Xudong

AU - Belabbas, Mohamed Ali

AU - Basar, Tamer

N1 - Funding Information:
∗Received by the editors April 18, 2016; accepted for publication (in revised form) September 5, 2017; published electronically December 6, 2017. A preliminary version of this paper was presented at the 22nd International Symposium on Mathematical Theory of Networks and Systems, Springer, Berlin, 2016. http://www.siam.org/journals/sicon/55-6/M107087.html Funding: The second author’s work was partially supported by NSF ECCS 13-07791 and NSF ECCS CAREER 13-51586. The third author’s work was partially supported by the U.S. Air Force Office of Scientific Research (AFOSR) MURI grant FA9550-10-1-0573. †Corresponding author. ECEE Department, University of Colorado at Boulder, Boulder, CO 80309 (xudong.chen@colorado.edu). ‡Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 (belabbas@illinois.edu, basar1@illinois.edu).

PY - 2017

Y1 - 2017

N2 - A cluster consensus system is a multiagent system in which autonomous agents communicate to form groups, and agents within the same group converge to the same point, called the clustering point. We introduce in this paper a class of cluster consensus dynamics, termed G- clustering dynamics for G a point group, in which the autonomous agents can form as many as jGj clusters and, moreover, the associated jGj clustering points exhibit a geometric symmetry induced by the point group. The definition of a G-clustering dynamics relies on the use of the so-called voltage graph: A G-voltage graph is a directed graph (digraph) together with a map assigning elements of a group G to the edges of the digraph. For example, in the case when G = (-1; 1), i.e., the cyclic group of order 2, a voltage graph is nothing but a signed graph. G-clustering dynamics can then be viewed as a generalization of the so-called Altafini's model, which was originally defined over a signed graph, by defining the dynamics over a voltage graph. One of the main contributions of this paper is to identify a necessary and suficient condition for the exponential convergence of a G-clustering dynamics. Various properties of voltage graphs that are necessary for establishing the convergence result are also investigated, some of which might be of independent interest in topological graph theory.

AB - A cluster consensus system is a multiagent system in which autonomous agents communicate to form groups, and agents within the same group converge to the same point, called the clustering point. We introduce in this paper a class of cluster consensus dynamics, termed G- clustering dynamics for G a point group, in which the autonomous agents can form as many as jGj clusters and, moreover, the associated jGj clustering points exhibit a geometric symmetry induced by the point group. The definition of a G-clustering dynamics relies on the use of the so-called voltage graph: A G-voltage graph is a directed graph (digraph) together with a map assigning elements of a group G to the edges of the digraph. For example, in the case when G = (-1; 1), i.e., the cyclic group of order 2, a voltage graph is nothing but a signed graph. G-clustering dynamics can then be viewed as a generalization of the so-called Altafini's model, which was originally defined over a signed graph, by defining the dynamics over a voltage graph. One of the main contributions of this paper is to identify a necessary and suficient condition for the exponential convergence of a G-clustering dynamics. Various properties of voltage graphs that are necessary for establishing the convergence result are also investigated, some of which might be of independent interest in topological graph theory.

KW - Cluster consensus

KW - Decentralized systems

KW - Exponential convergence

KW - Point groups

KW - Voltage graphs

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U2 - 10.1137/16M1070876

DO - 10.1137/16M1070876

M3 - Article

AN - SCOPUS:85039963970

SN - 0363-0129

VL - 55

SP - 3869

EP - 3889

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 6

ER -