A fundamental aspect of society is the exchange and discussion of opinions between individuals, occurring in situations as varied as company boardrooms, elementary school classrooms and online social media. After a very brief introduction to the established results of the most fundamental opinion dynamics models, which seek to mathematically capture observed social phenomena, a brief discussion follows on several recent themes pursued by the authors building on the fundamental ideas. In the first theme, we study the way an individual′s self-confidence can develop through contributing to discussions on a sequence of topics, reaching a consensus in each case, where the consensus value to some degree reflects the contribution of that individual to the conclusion. During this process, the individuals in the network and the way they interact can change. The second theme introduces a novel discrete-time model of opinion dynamics to study how discrepancies between an individual′s expressed and private opinions can arise due to stubbornness and a pressure to conform to a social norm. It is also shown that a few extremists can create “pluralistic ignorance”, where people believe there is majority support for a position but in fact the position is privately rejected by the majority. Last, we consider a group of individuals discussing a collection of logically related topics. In particular, we identify that for topics whose logical interdependencies take on a cascade structure, disagreement in opinions can occur if individuals have competing and/or heterogeneous views on how the topics are related, i.e., the logical interdependence structure varies between individuals.

This paper studies bearing-based formation control of a group of autonomous agents with the leader-first follower (LFF) structure in an arbitrary dimensional space. Firstly, the bearing-based Henneberg construction and some properties of the LFF formation are introduced. Then, we propose and analyze bearing-only control laws which almost globally stabilize LFF formations to a target formation. Further strategies to rotate and to rescale the target formation are also discussed. Finally, simulation results are provided to support the analysis.

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The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. The equation for $x_i$ includes additive terms weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the $i$-th and $j$-th agents are adjacent. The results are extended to the case where $A$ is not square but has full row rank, and bounds are given on the convergence rate.

The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that the initial (perceived) social power of each individual is exponentially forgotten. Specifically, individual social power is dependent only on the dynamic network topology, and initial social power is forgotten as a result of sequential opinion discussion. Finally, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results.