Opinion Dynamics and the Evolution of Social Power in Social Networks

A fundamental aspect of society is the exchange and discussion of opinions between individuals, occurring in mediums and situations as varied as company boardrooms, elementary school classrooms and online social media. This thesis studies several mathematical models of how an individual’s opinion(s) evolves via interaction with others in a social network, developed to reflect and capture different socio-psychological processes that occur during the interactions. In the first part, and inspired by Solomon E. Asch’s seminal experiments on conformity, a novel discrete-time model of opinion dynamics is proposed, with each individual having both an expressed and a private opinion on the same topic. Crucially, an individual’s expressed opinion is altered from the individual’s private opinion due to pressures to conform to the majority opinion of the social network. Exponential convergence of the opinion dynamical system to a unique configuration is established for general networks. Several conclusions are established, including how differences between an individual’s expressed and private opinions arise, and how to estimate disagreement among the private opinions at equilibrium. Asch’s experiments are revisited and re-examined, and then it is 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 of individuals! The second part builds on the recently proposed discrete-time DeGroot–Friedkin model, which describes the evolution of an individual’s self-confidence (termed social power) in his/her opinion over the discussion of a sequence of issues. Using nonlinear contraction analysis, exponential convergence to a unique equilibrium is established for networks with constant topology. Networks with issue-varying topology (which remain constant for any given issue) are then studied; exponential convergence to a unique limiting trajectory is established. In a social context, this means that each individual forgets his/her initial social power exponentially fast; in the limit, his/her social power for a given issue depends only on the previously occurring sequence of dynamic topology. Two further related works are considered; a network modification problem, and a different convergence proof based on Lefschetz Fixed Point Theory. In the final part, a continuous-time model is proposed to capture simultaneous discussion of logically interdependent topics; the interdependence is captured by a “logic matrix”. When no individual remains attached to his/her initial opinion, a necessary and sufficient condition for the network to reach a consensus of opinions is provided. This condition depends on the interplay between the network interactions and the logic matrix; if the network interactions are too strong when compared to the logical couplings, instability can result. Last, when some individuals remain attached to their initial opinions, sufficient conditions are given for opinions to converge to a state of persistent disagreement

Event-Triggered Algorithms for Leader-Follower Consensus of Networked Euler-Lagrange Agents

This paper proposes three different distributed event-triggered control algorithms to achieve leader-follower consensus for a network of Euler-Lagrange agents. We firstly propose two model-independent algorithms for a subclass of Euler-Lagrange agents without the vector of gravitational potential forces. By model-independent, we mean that each agent can execute its algorithm with no knowledge of the agent self-dynamics. A variable-gain algorithm is employed when the sensing graph is undirected; algorithm parameters are selected in a fully distributed manner with much greater flexibility compared to all previous work concerning event-triggered consensus problems. When the sensing graph is directed, a constant-gain algorithm is employed. The control gains must be centrally designed to exceed several lower bounding inequalities which require limited knowledge of bounds on the matrices describing the agent dynamics, bounds on network topology information and bounds on the initial conditions. When the Euler-Lagrange agents have dynamics which include the vector of gravitational potential forces, an adaptive algorithm is proposed which requires more information about the agent dynamics but can estimate uncertain agent parameters. For each algorithm, a trigger function is proposed to govern the event update times. At each event, the controller is updated, which ensures that the control input is piecewise constant and saves energy resources. We analyse each controllers and trigger function and exclude Zeno behaviour. Extensive simulations show 1) the advantages of our proposed trigger function as compared to those in existing literature, and 2) the effectiveness of our proposed controllers.Comment: Extended manuscript of journal submission, containing omitted proofs and simulation

On incentivizing innovation diffusion in a network of coordinating agents

Innovation diffusion is fundamental for societal growth and development, and understanding how to unlock it is key toward devising policies encouraging the adoption of new practices, e.g., sustainable innovations. Here, we propose a mathematical model to investigate such a problem. Specifically, we consider a coordination game —which is a standard game-theoretic model used to study innovation diffusion—and we embed it on an activity-driven network. Within this model, we integrate three policies to incentivize the adoption of the innovation: i) providing a direct advantage for adopting it, ii) making people sensitive to emerging trends at the population level, and iii) increasing the visibility of adopters of the innovation, respectively. We provide analytical insights to shed light on the effect of the joint use of these three policies on unlocking innovation diffusion, supported by numerical simulations

Facilitating innovation diffusion in social networks using dynamic norms

Dynamic norms have recently emerged as a powerful method to encourage individuals to adopt an innovation by highlighting a growing trend in its uptake. However, there have been no concrete attempts to understand how this individual-level mechanism might shape the collective population behavior. Here, we develop a framework to examine this by encapsulating dynamic norms within a game-theoretic mathematical model for innovation diffusion. Specifically, we extend a network coordination game by incorporating a probabilistic mechanism where an individual adopts the action with growing popularity, instead of the standard best-response update rule; the probability of such an event captures the population’s “sensitivity” to dynamic norms. Theoretical analysis reveals that sensitivity to dynamic norms is key to facilitating social diffusion. Small increases in sensitivity reduces the advantage of the innovation over status quo or the number of initial innovators required to unlock diffusion, while a sufficiently large sensitivity alone guarantees diffusion

On modeling social diffusion under the impact of dynamic norms

We develop and analyze a collective decision-making model concerning the adoption and diffusion of a novel product, convention, or behavior within a population. Motivated by the growing social psychology literature on dynamic norms, under which an individual is influenced by changing trends in the population, we propose a stochastic model for the decision-making process encompassing two behavioral mechanisms. The first is social influence, which drives coordination among individuals. Consistent with the literature on social diffusion modeling, we capture such a mechanism through an evolutionary game-theoretic framework for a network of interacting individuals. The second, which is the main novelty of our model, represents the impact of dynamic norms, capturing the tendency of individuals to be attracted to products or behaviors with growing popularity. We analytically determine sufficient conditions under which a novel alternative spreads to the majority of the population. Our findings provide insights into the unique and nontrivial role of human sensitivity to dynamic norms in facilitating social diffusion

A Coevolutionary Model for Actions and Opinions in Social Networks

© 2020 IEEE. In complex social networks, the decision-making mechanisms behind human actions and the cognitive processes that shape opinion formation processes are often intertwined, leading to complex and varied collective emergent behavior. In this paper, we propose a mathematical model that captures such a coevolution of actions and opinions. Following a discrete-time process, each individual decides between binary actions, aiming to coordinate with the actions of other members observed on a network of interactions and taking into account their own opinion. At the same time, the opinion of each individual evolves due to the opinions shared by other members, the actions observed on the network, and, possibly, an external influence source. We provide a global convergence result for a special case of the coupled dynamics. Steady state configurations in which all the individuals take the same action are then studied, elucidating the role of the model parameters and the network structure on the collective behavior of the system