Continuous-time opinion dynamics on multiple interdependent topics

Abstract

In this paper, and inspired by the recent discrete-time model in [1,2], we study two continuous-time opinion dynamics models (Model 1 and Model 2) where the individuals discuss opinions on multiple logically interdependent topics. The logical interdependence between the different topics is captured by a "logic" matrix, which is distinct from the Laplacian matrix capturing interactions between individuals. For each of Model 1 and Model 2, we obtain a necessary and sufficient condition for the network to reach to a consensus on each separate topic. The condition on Model 1 involves a combination of the eigenvalues of the logic matrix and Laplacian matrix, whereas the condition on Model 2 requires only separate conditions on the logic matrix and Laplacian matrix. Further investigations of Model 1 yields two sufficient conditions for consensus, and allow us to conclude that one way to guarantee a consensus is to reduce the rate of interaction between individuals exchanging opinions. By placing further restrictions on the logic matrix, we also establish a set of Laplacian matrices which guarantee consensus for Model 1. The two models are also expanded to include stubborn individuals, who remain attached to their initial opinions. Sufficient conditions are obtained for guaranteeing convergence of the opinion dynamics system, with the final opinions generally being at a persistent disagreement. Simulations are provided to illustrate the results.

On the Controllability of Matrix-Weighted Networks

“…In the meantime, existing works on network controllability are mainly concerned with networks with scalar weighted edges; such network models are restrictive in characterizing interdependence amongst subsets of the underlying node states [19]. Matrix-weighted networks are a natural extension of scalar-valued networks; they have been examined in scenarios such as graph effective resistance (motivated by distributed estimation and control) [20], [21], logical inter-dependency of multiple topics in opinion evolution [22], [23], bearing-based formation control [24], as well as the array of coupled LC oscillators [25]. More recently, consensus and synchronization problems on matrix-weighted networks have been examined in [26], [27], [28], [29].…”

Bipartite Consensus on Matrix-Valued Weighted Networks

“…Recently, a broader category of networks referred to as matrix-weighted networks has been introduced to address such interdependencies [5], [6]. In fact, matrix-weighted networks arise in scenarios such as graph effective resistance examined in the context of distributed control and estimation [7], [8], logical inter-dependencies amongst topics in opinion evolution [9], [10], bearing-based formation control [11], dynamics of an array of coupled LC oscillators [12], as well as consensus and synchronization on matrix-weighted networks [5], [13], [14].…”

Recent Advances in the Modelling and Analysis of Opinion Dynamics on Influence Networks

“…Phenomena and behaviour related to pluralistic ignorance are of particular interest in studying how misinformation spreads through high profile media figures, or hostile bot accounts on social media (such as the Twitter bots of a foreign country commenting on political matters). We also expect this to be closely linked with the concept of the spiral of silence [80,84] , which is a social phenomenon where an individual is more likely to stop expressing his or her opinion if that individual perceives (rightly or wrongly) that everyone else is moving away from his or her opinion. We hope that study and development of these models (including incorporation of a spiral of silence mechanism, perhaps by inclusion of an eventbased communication component) will reveal to us deeper insight about the role extremists play in creating a divergence in the private and expressed opinions of the general population, and provide some guidance for developing effective countermeasures.…”

“…where and are the same as that defined in Section 2.2. Here is called the public opinion as consistent with [80] and is termed the resilience, i.e., the ability for individual to withstand group pressure. It is instructive to observe that if , then the individual is fully resilient to pressure and the expressed and private opinions coincide, while if , the individual's own opinion is totally overwhelmed by the network's average opinion.x…”