Both interpersonal communication and media contact are important information sources and play a significant role in shaping public opinions of large populations. In this article, we investigate how the opinion-forming process evolves over social networks under media influence. In addition to being affected by the opinions of their connected peers, the media cooperate and/or compete mutually with each other. In this endeavor, a novel mathematical model of opinion dynamics is introduced, which captures the information diffusion process under consideration, makes use of the community-based network structure, and takes into account personalized biases among individuals in social networks. By employing port-Hamiltonian system theory to analyze the modeled opinion dynamics, we predict how public opinions evolve in the long run through social entities and find applications in political strategy science. A key technical observation is that as a result of the port-Hamiltonian formulation, the mathematical passivity property of individuals' self-dynamics facilitates the convergence analysis of opinion evolution. We show that public opinions become polarized or neutralized, and investigate how an autocratic media coalition might emerge that ignores the public opinions. We also assess the role of interpersonal communication and media exposure, which in itself is an important topic in mathematical sociology.

Opinion dynamics on social networks with coopetitive (cooperative-competitive) interactions may result in polarity, consensus or neutrality under different opinion protocols. The antecedent of protocol design is to study the accessibility problem: whether or not there exist admissible control rules to polarize, consensus, or neutralize individual opinions in a large population. From an operational perspective, this technical note is aimed at the investigation of polarizability, consensusability, and neutralizability of opinion dynamics in question. Particular emphasis is on the joint impact of the dynamical properties of individuals and the interaction topology among them on polarizability, consensusability, and neutralizability, respectively. Sufficient and/or necessary conditions for those accessibility problems are provided by using powerful tools from spectral analysis and algebraic graph theory. To characterize the individual diversity in real life, we further investigate the solvability of opinion formation problems in heterogeneous systems with nonidentical dynamics. Accordingly, sufficient and/or necessary criteria for heterogeneous network polarizability, consensusability, and neutralizability are derived.

This paper considers the problem of network structure manipulation in the absence of a central coordinator that collects network information and makes decisions. The primary focus is on the spectral radius minimization problem by removing/rewiring links to control epidemic spreading over networks. The resulting optimizations are generally combinatorial and NP-hard. The lack of the central base prevents us from solving such problems in a centralized fashion, thus driving the need for distributed computation and collaborative decision-making to support effective topology modification. In this paper, a distributed estimation scheme involving event-based communication and parallel algorithms is developed to enhance network capability against epidemics. Based on this low-complexity estimation algorithm, fully distributed strategies are proposed to enable individuals to sequentially discern the critical in-network contact, and to implement the desired link operation solely. We further extend the acquisitions to a more flexible framework based on a “coevolutionary networks” picture and discuss the optimality of solutions from an algebraic and topological perspective. Extensive simulation examples are presented to demonstrate the effectiveness of the proposed strategies.

The process of enhancing the ability of a complex network against various malicious attacks through link addition/rewiring has been the subject of extensive interest and research. The performance of existing methods often highly depends on full knowledge about the network topology. In this study, the authors devote ourselves to developing new distributed strategies to perform link manipulation sequentially using only local accessible topology information. This strategy is concerned with a matrix‐perturbation‐based approximation of the network‐based optimisation problems and a distributed algorithm to compute eigenvectors and eigenvalues of graph matrices. In addition, the development of a distributed stopping criterion, which provides the desired accuracy on the distributed estimation algorithm, enables us to solve the link‐operation problem in a finite‐time manner. Finally, all results are illustrated and validated using numerical demonstrations and examples.