Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and “trustful”, and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.
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机译：用于多主体共识的大多数分布式协议都假定主体是相互合作和“信任的”，因此主体之间的耦合使它们的状态值更接近。但是，由于某些对行为主体之间普遍存在的竞争和不信任，社会群体中的舆论动态需要超越这些常规模型，而这通常以排斥性耦合为特征，并可能导致舆论聚集。 C. Altafini最近提出了一个简单而有见地的，具有吸引力和排斥性耦合的观点动力学模型，他研究了静态签名图上的一阶共识算法。该协议建立了模数共识，其中意见在模数上相同，但可能在符号上有所不同。在本文中，我们将模数共识模型扩展到网络拓扑是任意时变有符号图的情况，并证明在图的均匀连通性的适度充分条件下达到模数共识。对于割平衡图，不仅给出了足够的信息，而且给出了模量共识的必要条件。
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