Belief Dynamics in Social Networks: A Fluid-Based Analysis

Abstract

The advent and proliferation of social media have led to the development of mathematical models describing the evolution of beliefs/opinions in an ecosystem composed of socially interacting users. The goal is to gain insights into collective dominant social beliefs and into the impact of different components of the system, such as users' interactions, while being able to predict users' opinions. Following this thread, in this paper we consider a fairly general dynamical model of social interactions, which captures all the main features exhibited by a social system. For such model, by embracing a mean-field approach, we derive a diffusion differential equation that represents asymptotic belief dynamics, as the number of users grows large. We then analyze the steady-state behavior as well as the time dependent (transient) behavior of the system. In particular, for the steady-state distribution, we obtain simple closed-form expressions for a relevant class of systems, while we propose efficient semi-analytical techniques in the most general cases. At last, we develop an efficient semi-analytical method to analyze the dynamics of the users' belief over time, which can be applied to a remarkably large class of systems.Comment: submitted to IEEE TNS

Macroscopic Noisy Bounded Confidence Models With Distributed Radical Opinions

“…Setting p = 1 in (31) gives the sufficient condition σ 2 > 16R/ ln 3 for uniqueness of stationary solution. This result corresponds to the sufficient condition provided in[52, Theorem 2].…”

“…One of the important directions in analysis of bounded confidence models is examination of their asymptotic properties as the number of social actors becomes very large N → ∞ and their individual opinions are replaced by infinitesimal "elements". The arising macroscopic approximations of agent-based models describe the evolution of the distribution of opinion (usually supposed to have a density) and are referred to as density-based [46], continuum-agent [47], [48], Eulerian [49], [50], kinetic [51], hydrodynamical [28] or mean-field [43], [52] models of opinion formation. In the continuous-time situation, the density obeys a nonlinear Fokker-Planck (FP) equation.…”

Opinion Dynamics on Correlated Subjects in Social Networks

“…We now extend the above model to the continuous case by using the mean-field theory. We leverage on the procedure presented in [18] and apply it to the multi-subject scenario. More in detail, we define the empirical probability measure, ρ (U ) ( dp, dx, t) over Y at time t, as: ρ (U ) ( dp, dx, t) = 1 U i∈U δ (Pi,xi(t)) ( dp, dx) .…”

“…Note that in (4) agents are seen as particles in the continuous space Y, moving along the opinion axis x. As shown in [18], by applying the mean-field theory [31], [32], as U → ∞, ρ (U ) ( dp, dx, t) converges in law to the asymptotic distribution ρ(p, x, t), provided that ρ (U ) ( dp, dx, 0) converges in law to ρ(p, x, 0). Moreover, ρ(p, x, t) can be obtained from the following non-linear Fokker-Planck (FP) equation [31], [32]:…”

“…• Continuous models, in which the opinion of individuals on a particular subject is described by means of a continuous variable, whose value is adapted as a result of social interactions with individuals having different opinions [11], [12], [13], [14], [15], [16], [17], [18]. Also in this case, social interactions between individuals are typically modeled by using (static or dynamic) graphs, which reflect the structure of the society and describe how individuals interact.…”