Multi-agreements in social networks based on LaSalle invariance principle and positive definiteness

The paper considers multi-agreements that may exist in social networks that do not have connectivity across the network, and where each agent can decide on how an opinion from an influencing agent is accepted or not. The opinion dynamics considered in this paper are therefore nonlinear. Because of this nonlinearity, and because general directed graphs are considered, the first contribution is using a decomposition of the directed graph, which identifies the key influencers (with strong connectivity) and the pure followers in the graph, to partition the dynamics into two cascaded subsystems. Based on a LaSalle invariance principle and a max–min Lyapunov function, global asymptotic stability of “agreements” can then be established under very weak, nearly necessary conditions. The result generalizes a number of results for consensus networks with first-order multi-agent systems, as well as three important models, namely, stubborn extremists, stubborn positives, and stubborn neutrals. A Gaussian elimination based algorithm is introduced to simultaneously identify the key influencers and a particular basis of the left kernel space of the Laplacian matrix. To compute a special basis of the right kernel space of the Laplacian matrix, an effective algorithm is then proposed based on the Cramer’s rule. These algorithms are closely related to the prediction of final agreement values. Simulation studies complement the theoretical results, hinting toward agreement design.