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Exponential Convergence of the Discrete- and Continuous-Time Altafini Models

机译：离散时间和连续时间Altafini模型的指数收敛

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This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [3], exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that 1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; 2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is provided. The results are also extended to the continuous-time Altafini model.
机译：本文考虑了Altafini模型的离散时间版本的意见动态，其中一组代理之间的交互由时变有符号图描述。由文献[3]提出的想法，使用图形方法研究了系统的指数收敛性。提供了关于每种可能类型的极限状态的指数收敛的充要条件。具体而言，在反复共同强连接的假设下，表明：1）仅当有符号有向图的序列反复共同结构平衡时，对于几乎所有初始条件，特定类型的两群聚都将以指数方式快速达到对应于这种两种类型的集群； 2）当且仅当带符号的有向图的序列在结构上反复联合不平衡时，系统才会对所有初始条件快速收敛到零。提供了收敛速度的上限。结果也扩展到连续时间Altafini模型。

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