On the Analysis of the DeGroot-Friedkin Model with Dynamic Relative Interaction Matrices

This paper analyses the DeGroot-Friedkin model for evolution of the individuals’ social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices). The DeGroot-Friedkin model describes how individual social power (self-appraisal, self-weight) evolves as a network of individuals discuss opinions on a sequence of issues. We seek to study dynamically changing relative interactions because interactions may change depending on the issue being discussed. Specifically, we study relative interaction matrices which vary periodically with respect to the issues. This may reflect a group of individuals, e.g. a government cabinet, that meet regularly to discuss a set of issues sequentially. It is shown that individuals’ social powers admit a periodic solution. Initially, we study a social network which varies periodically between two relative interaction matrices, and then generalise to an arbitrary number of relative interaction matrices.This work was supported by the Australian Research Council (ARC) under the ARC grants DP-130103610 and DP-160104500, by the National Natural Science Foundation of China (grant 61375072), and by Data61-CSIRO (formerly NICTA). The work of Liu and Basar was supported in part by Office of Naval Research (ONR) MURI Grant N00014-16-1-2710, and in part by NSF under grant CCF 11-11342

Evolution of Social Power for Opinion Dynamics Networks

This article studies the evolution of opinions and interpersonal influence structures in a group of agents as they discuss a sequence of issues, each of which follows an opinion dynamics model. In this work, we propose a general opinion dynamics model and an evolution of interpersonal influence structures based on the model of reflected appraisals proposed by Friedkin. Our contributions can be summarized as follows: (i) we introduce a model of opinion dynamics and evolution of interpersonal influence structures between issues viewed as a best response cost minimization to the neighbor's actions, (ii) we show that DeGroot's and Friedkin-Johnsen's models of opinion dynamics and their evolution of interpersonal influence structures are particular cases of our proposed model, and (iii) we prove the existence of an equilibrium. This work is a step towards providing a solid formulation of the evolution of opinions and interpersonal influence structures over a sequence of issues

Evolution of Social Power in Social Networks with Dynamic Topology

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 each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, 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.The work of Mengbin Ye, Brian D. O. Anderson, and Changbin Yu was supported by the Australian Research Council under Grant DP-130103610 and Grant DP-160104500, by 111-Project D17019, by NSFC Projects 61385702 and 61761136005, and by Data61-CSIRO. The work of Mengbin Ye was supported by an Australian Government Research Training Program Scholarship. The work of Ji Liu and Tamer Bas¸ar was supported by the Office of Naval Research MURI Grant N00014-16-1-2710, and by NSF under Grant CCF 11-11342. Recommended by Associate Editor C. M. Kellett

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

Learning Hidden Influences in Large-Scale Dynamical Social Networks: A Data-Driven Sparsity-Based Approach, in Memory of Roberto Tempo

The processes of information diffusion across social networks (for example, the spread of opinions and the formation of beliefs) are attracting substantial interest in disciplines ranging from behavioral sciences to mathematics and engineering (see "Summary"). Since the opinions and behaviors of each individual are infl uenced by interactions with others, understanding the structure of interpersonal infl uences is a key ingredient to predict, analyze, and, possibly, control information and decisions [1]. With the rapid proliferation of social media platforms that provide instant messaging, blogging, and other networking services (see "Online Social Networks") people can easily share news, opinions, and preferences. Information can reach a broad audience much faster than before, and opinion mining and sentiment analysis are becoming key challenges in modern society [2]. The first anecdotal evidence of this fact is probably the use that the Obama campaign made of social networks during the 2008 U.S. presidential election [3]. More recently, several news outlets stated that Facebook users played a major role in spreading fake news that might have infl uenced the outcome of the 2016 U.S. presidential election [4]. This can be explained by the phenomena of homophily and biased assimilation [5]-[7] in social networks, which correspond to the tendency of people to follow the behaviors of their friends and establish relationships with like-minded individuals

Modelling of individual behaviour in the degroot-friedkin self-appraisal dynamics on social networks

© 2019 EUCA. The DeGroot-Friedkin model describes how an individual's self-confidence in his or her own opinion evolves as that individual participates in a group discussing a sequence of topics; as the individual has more impact or less impact (termed social power) on a given topic discussion, his or her self-confidence increases or decreases due to the process of reflected self-appraisal. This paper proposes a broad generalisation of the DeGroot-Friedkin model by allowing each individual's self-appraisal process to be distorted by behavioural characteristics such as humility. We establish the generalised dynamical model for the evolution of individuals' social power (a measure of an individual's contribution to each topic discussion). For some types of individuals, whom we term 'humble' and 'unreactive', results are provided on the existence of equilibria, whether such equilibria are unique, and convergence to said equilibria. Simulations are used to illustrate that networks of 'emotional' individuals, who at times act like humble individuals and at other times like arrogant individuals, can have at least two attractive equilibria