Analytic treatment of tipping points for social consensus in large random networks

Abstract

We introduce a homogeneous pair approximation to the naming game (NG) model by deriving a six-dimensional Open Dynamics Engine (ODE) for the two-word naming game. Our ODE reveals the change in dynamical behavior of the naming game as a function of the average degree {k} of an uncorrelated network. This result is in good agreement with the numerical results. We also analyze the extended NG model that allows for presence of committed nodes and show that there is a shift of the tipping point for social consensus in sparse networks.

Role of committed minorities in times of crisis

“…This result was shown to be largely independent of the structure of the model interactions within society but can be determined by as much as 10% to as little as 4% for a sparse network10. The percentage at which the tipping point (critical point of a phase transition) occurs is clearly model dependent and can vary from 4 to 15%1112.…”

Social opinion dynamics is not chaotic

“…For a general social system of the above type, we can obtain a similar ODE system using exactly the same approach as in Ref. 6. The type of a link in this ODEs is ultimately given by the opinions or node-spins of its two ends (γ i − γ j ) regardless of the order but it is more convenient to work with link-based macrostates; a transformation between the link-based macrostate and the nodebased macrostate n is given in Ref.…”

“…The type of a link in this ODEs is ultimately given by the opinions or node-spins of its two ends (γ i − γ j ) regardless of the order but it is more convenient to work with link-based macrostates; a transformation between the link-based macrostate and the nodebased macrostate n is given in Ref. 6. Changes of l come from two parts: the direct change and the related change.…”

“…For naming game, R is given explicitly in Ref. 6. For general opinion dynamics, R is given in detail later.…”

“…4 However, when it comes to the case of social opinion dynamics, the direct application of previous monotonicity results, is difficult because the corresponding systems are highly nonlinear. [5][6][7] Our main aim in this paper is to extend the applications of monotonicity to a large class of social opinion dynamics by providing an easy-to-verify criterion of monotonicity for this type of systems.…”