Age Structure 

We incorporate age structure in a classical model of continuous opinion formation, studying both a stochastic differential equation (SDE) model and the corresponding mean-field partial differential equation (PDE). We prove the existence of stationary states of the PDE model, but also see non-uniqueness of stationary states and periodic behaviour, an example of which is shown on the left. 

Dynamic Networks 

This model introduces an adaptive network, capturing the changing relationships between individuals over time. We consider a network that responds to the strength of individuals' interactions, as well as network dynamics driven by a control (with an example shown in the video on the right). We focus on the impact of edge dynamics on the opinion formation process: when does the dynamic network encourage consensus and when does it reinforce polarisation?

Comparing Modelling Approaches

Most models of opinion dynamics that track individuals are either agent-based models or differential equation models. We show how the latter can be obtained from the former by simultaneously rescaling the time-step and strength of interactions. The video shows realisations of the ABM in solid lines and the limiting ODE in dashed lines.

This connection helps answer questions in both settings, like the motivation of multiplicative noise terms in SDE models or the link between selection noise and mollification of interaction functions.

Early Warning Signals

Critical slowing down states that systems display increasing relaxation times prior to a critical transition, an effect that can be observed in timeseries statistics to give an early warning of the transition. However, in epidemiological models there is frequent disagreement with this general theory, and the alternative theory of critical speeding up predicts contradictory behaviour of early warning signals. We describe the behaviour of common early warning signals in terms of a system’s potential surface and noise around a quasi-steady state, then describe an equation-free method to obtain these key features from timeseries, using a version of the SIS model as a case study. The figure shows example reconstructed potential surfaces.