Symmetry induced group consensus

Abstract

There has been substantial work studying consensus problems for which there is a single common final state, although there are many real-world complex networks for which the complete consensus may be undesirable. More recently, the concept of group consensus whereby subsets of nodes are chosen to reach a common final state distinct from others has been developed, but the methods tend to be independent of the underlying network topology. Here, an alternative type of group consensus is achieved for which nodes that are symmetric achieve a common final state. The dynamic behavior may be distinct between nodes that are not symmetric. We show how group consensus for heterogeneous linear agents can be achieved via a simple coupling protocol that exploits the topology of the network. We see that group consensus is possible on both stable and unstable trajectories.We observe and characterize the phenomenon of isolated group consensus, where one or more clusters may achieve group consensus while the other clusters do not.Consensus problems are an important topic when designing control protocols for distributed systems where it is desirable for uniform behavior between nodes like power generators in the grid, mobile robots, or autonomous vehicles. Sometimes though, complete consensus is not the desired behavior, but rather group consensus whereby some nodes' behaviors will coincide while others do not. This problem has been tackled previously using linear matrix inequalities (LMIs) or Lyapunov functions but the result can only guarantee complete consensus. We instead present a method derived using the automorphism group of the underlying graph which provides more granular information that splits the dynamics of consensus motion from different types of orthogonal, cluster breaking motion.

Symmetries and cluster synchronization in multilayer networks

“…As each row of the matrix T is associated to a specific cluster 40 , each one of the blocksB r corresponds to a set of clusters which are identified by the rows of the matrix T. The trivial representation (r = 1) is associated with all the clusters…”

Distributed Control of Multiconsensus

“…Previous works on the analysis of multi-consensus have investigated the properties of the network of interaction among agents leading to multi-consensus states [14]- [17]. More in details, the criteria found in [14] are based on the use of Markov chains and nonnegative matrix analysis for fixed and switching topologies, while the existence of multi-consensus is related to the presence of symmetries [15] or of external equitable partitions [16] in the topology. Finally, multi-consensus can be observed also in the presence of delays or differentiation of the dynamics of the units, as shown in [17].…”

Dimension-reduction of dynamics on real-world networks with symmetry

“…Symmetries are intimately connected to the spectral peaks of the adjacency and Laplacian matrices [39][40][41], and hence they impact on a wide range of network 'structural measures' [42] and dynamical phenomena. In particular, network symmetries facilitate cluster synchronization [43,44] and can be used to control group consensus [45]. Symmetry in complex networks has received increasing interest recently, in part because it has been shown that many real-world networks have a significant amount of symmetry [46][47][48].…”