Ordering dynamics and aging in the Symmetrical Threshold model

TL;DR

This paper explores the ordering dynamics and aging effects in a symmetric complex contagion model, revealing new phases and dynamical behaviors influenced by network structure and agent resistance over time.

Contribution

It introduces a symmetric version of the Granovetter-Watts model with aging, identifying new phases and providing a theoretical description via an Approximate Master Equation.

Findings

Three phases: mixed, ordered, and heterogeneous frozen.

Aging replaces the mixed phase with a slow ordering phase in sparse networks.

Theoretical predictions align well with numerical simulations.

Abstract

The so-called Granovetter-Watts model was introduced to capture a situation in which the adoption of new ideas or technologies requires a certain redundancy in the social environment of each agent to take effect. This model has become a paradigm for complex contagion. Here we investigate a symmetric version of the model: agents may be in two states that can spread equally through the system via complex contagion. We find three possible phases: a mixed one (dynamically active disordered state), an ordered one, and a heterogeneous frozen phase. These phases exist for several configurations of the contact network. Then we consider the effect of introducing aging as a non-Markovian mechanism in the model, where agents become increasingly resistant to change their state the longer they remain in it. We show that when aging is present, the mixed phase is replaced, for sparse networks, by a new phase with different dynamical properties. This new phase is characterized by an initial disordering stage followed by a slow ordering process towards a fully ordered absorbing state. In the ordered phase, aging modifies the dynamical properties. For random contact networks, we develop a theoretical description based on an Approximate Master Equation that describes with good accuracy the results of numerical simulations for the model with and without aging.