Mathematical modelling of political opinions and societal change has been used as a tool to simulate populations and address questions concerned with the polarization of society and other phenomena of concern. However, the development of opinion dynamics has typically been in separation to that of social sciences, dealing with artificial agents governed by pairwise interaction and abstractions of reality that are inspired by complex physical systems rather than social cognition and perception, which presents us with an important area of research to connect the fields.

Here we present work that addresses the challenge of the empirical gap within opinion dynamics, or how do we link these models to reality? For data, we use multi-dimensional political positions that are inferred from a large number of social media users, by leveraging behavioural data traces (e.g., what politicians users follow) on platforms. With traditional agent-based modelling there is then a problem of how to overcome the computational complexity incurred by simulating large populations which requires significant resource. In order to overcome this obstacle, we employ mean-field theory to approximate behaviour and model distributions - rather than agents - with the finite volume method.

The result is an approach to understanding how population-level multi-dimensional distributions of opinion might develop, and what modelling conditions maintain populations in states of polarization (low or high) that are encountered in the real world. The multi-dimensional aspect of modelling also points towards analysis of the dimensionality of opinion space and relevance to European contexts.

We consider a non-cooperative model of network formations where agents decide on whom

to form costly links to. Links are unilaterally formed and payoff flows one way to the active

side. We study discontinuous information flows where agents only receive benefits from other agents that are at a distance of two in the network. For the static game, we show that the set of strict Nash equilibria encompasses a multiplicity of core-periphery network structures. We further study a noisy best response process to obtain long-run predictions. Doing so, we find that the set of stochastically stable states retains a multiplicity of network structures, many of which are not efficient. In addition, our simulation results from MatLab suggest that when there is a small probability that agents make mistakes, core-periphery networks are uniquely stochastically stable in the perturbed evolution.