Consensus for clusters of agents with cooperative and antagonistic relationships

Tripartite and Sign Consensus for Clustering Balanced Social Networks

“…In this paper we want to extend some recent results obtained for tripartite consensus of a multi-agent system with undirected, signed, weighted, connected and clustering balanced communication graph, by relaxing the homogeneity constraint regarding mutual relationships between agents introduced in [17], [18]. Specifically, in [17], [18] we proved that if each agent in a cluster distributes the same amount of "trust" to the agents in its own group and "distrust" to the agents belonging to adverse clusters, then it is possible to adopt a slightly modified version of DeGroot's algorithm in such a way that agents belonging to the same cluster V i , i ∈ [1,3], asymptotically converge to the same decision, i.e., lim…”

“…, not all of them equal to zero. It is worth noticing that while in the homogeneous case investigated in [17], [18], the desired goal was achieved by suitably choosing a stubbornness degree common to all the agents belonging to the same class, now the degree of stubbornness is individually tuned.…”

“…To the best of our knowledge, the literature on consensus problems over clustering balanced networks, by this meaning the problem of making all agents belonging to the same cluster in a clustering balanced network achieve a common decision, is quite limited [17], [18]. On the other hand, there have been some interesting research efforts aiming to explore the possibility of achieving group consensus for networks whose agents have been partitioned in disjoint groups.…”

“…Proskurnikov and R. Tempo [19] on social networks, and by the works of J. Davis [6] and P. Cisneros-Velarde and F. Bullo [5], where the concept of clustering balance plays a major role. In our recent papers [17], [18] we have investigated consensus problems for clustering balanced networks 1 with 3 or, in general, k ≥ 3 clusters, under some "homogeneity" constraint on the (positive and negative) weights of the communication network, namely by assuming that the amount of trust/mistrust that each agent attributes to its friends/enemies is prefixed for all the agents in the same cluster. In this paper we will focus on networks that are partitioned into three clusters, and we will first show that, even without the homogeneity assumption, (tripartite) consensus can be obtained by means of a slightly revised version of De Groot's distributed feedback control law.…”

“…Lemma 1: [17] Let D ∈ R n×n be a diagonal matrix and let A ∈ R n×n be a symmetric nonnegative matrix, then: i) D − A is positive definite if and only if there exists a strictly positive vector…”