Discrete-Time Polar Opinion Dynamics with Heterogeneous Individuals

Abstract

This paper considers a discrete-time opinion dynamics model in which each individual's susceptibility to being influenced by others is dependent on her current opinion. We assume that the social network has time-varying topology and that the opinions are scalars on a continuous interval. We first propose a general opinion dynamics model based on the DeGroot model, with a general function to describe the functional dependence of each individual's susceptibility on her own opinion, and show that this general model is analogous to the Friedkin-Johnsen model, which assumes a constant susceptibility for each individual. We then consider two specific functions in which the individual's susceptibility depends on the polarity of her opinion, and provide motivating social examples. First, we consider stubborn positives, who have reduced susceptibility if their opinions are at one end of the interval and increased susceptibility if their opinions are at the opposite end. A court jury is used as a motivating example. Second, we consider stubborn neutrals, who have reduced susceptibility when their opinions are in the middle of the spectrum, and our motivating examples are social networks discussing established social norms or institutionalized behavior. For each specific susceptibility model, we establish the initial and graph topology conditions in which consensus is reached, and develop necessary and sufficient conditions on the initial conditions for the final consensus value to be at either extreme of the opinion interval. Simulations are provided to show the effects of the susceptibility function when compared to the DeGroot model.

Asynchrony rescues statistically optimal group decisions from information cascades through emergent leaders

“…Therefore, through these decentralized algorithms, each agent on a sparse graph can converge on the same average confidence value. The Belief Consensus algorithm [24] uses a linear function, while other averaging algorithms employ nonlinear [51][52][53][54] or heterogenous functions [55]. The advantage of consensus averaging algorithms is a guarantee of convergence in a relatively small number of time steps.…”

Computing Different Realizations of Linear Dynamical Systems with Embedding Eigenvalue Assignment

“…Discrete time linear compartmental models -having many applications in modeling biological systems -also satisfy the above non-negativity condition [14,13]. Social networks provide an important application field of modeling discrete time dynamical systems defined on networks [26,28,29,21]. The DeGroot and Friedkin-Johnsen models are well-known discrete time linear models of opinion dynamics and information spreading in networks where the off-diagonal entries of state transition matrices are also constrained to be non-negative [6,11].…”

Asynchrony rescues statistically-optimal group decisions from information cascades through emergent leaders

“…Therefore, through these decentralised algorithms, each agent on a sparse graph can converge on the same average confidence value. The Belief Consensus algorithm [24] uses a linear function, while other averaging algorithms employ nonlinear [51, 52, 53, 54] or heterogenous functions [55]. The advantage of consensus averaging algorithms is a guarantee of convergence in a relatively small number of timesteps.…”