### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 3869-3889 |

Number of pages | 21 |

Journal | SIAM Journal on Control and Optimization |

Volume | 55 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2017 |

### Keywords

- Cluster consensus
- Decentralized systems
- Exponential convergence
- Point groups
- Voltage graphs

### ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics

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## Cite this

*SIAM Journal on Control and Optimization*,

*55*(6), 3869-3889. https://doi.org/10.1137/16M1070876