Analysis of threshold models for collective actions in social networks

Abstract

In this paper, we study the asymptotic behaviors of threshold models used to describe the formation of collective actions in social networks. At each time instant, each agent of the network makes a choice between two possible actions. The decision is made on the basis of the actions chosen by its neighbors and the value of a dynamically updated threshold. The main novelty of the proposed model is the introduction of a parameter accounting for the level of self-confidence of the agents, which affects the dynamic evolution of the threshold and in turn the way the agents make their decision. The objective is to identify which are the possible limiting behaviors of the network and under which conditions each of them occurs. Three different network topologies are considered. In the case of complete graph, the asymptotic behaviors are analytically characterized, whereas for star and ring graphs an extensive numerical analysis is presented

Asymptotic behaviours of a class of threshold models for collective action in social networks

“…A wide variety of collective limiting behaviors is observed and its relationship with the self-confidence parameter and the chosen decision mechanism is highlighted. A preliminary version of this work has been presented in [28]. This paper extends previous results to the case of star and ring graphs.…”

“…Corollary 6. System (5), (6), with F and G given by (28) and (29), respectively, and initial condition (8) As for the complete graph, also for the star graph it can be observed that the WAL scenario shows a wider variety of asymptotic behaviors with respect to the UAL scenario. In particular, in the latter the initial activity pattern a(0) is never maintained indefinitely and the persistent oscillations for β < 1 occur only under the singular condition τ = µ(β) (while in the WAL scenario they show up for the entire range of τ values).…”

“…The following technical results are instrumental to the asymptotic analysis of the star graph interconnection. (5), with matrix F given by (28).…”

“…Proof: Let us first observe that, due to the structure of F in (28), one has that system (5) with the initial condition θ(0) = [0, τ, . .…”

“…Theorem 4. System (5), (6), with F and G given by (28) and (29), respectively, and initial condition (8), exhibits the following asymptotic behaviors.…”