arXiv subjects

Andrew Nugent

Publications and source records attributed to Andrew Nugent.

9 recordsLinked to original sources

arXiv2026

Mean-field imitation dynamics on fast assortative networks

The emergence of cooperation in structured populations is fundamental to the success of human societies. Physical and online networks can drive behavioural change by altering who people interact with, thereby modifying social pressures. In this paper, we study imitation dynamics in a population of self-interested agents playing a continuous strategy Prisoner's Dilemma on a dynamically evolving weighted network. In the fast-network regime, we incorporate the edge weights into the strategy evolution before deriving and analysing the large population mean-field limit. Without noise, we establish well-posedness and show the solution collapses to a single Dirac mass. For initially separated clusters, we identify a payoff threshold and sufficient conditions for the overall level of cooperation to increase. We then introduce stochastic strategy updates, and obtain a non-local Fokker-Planck equation in the mean-field limit. We rigorously prove existence and uniqueness of stationary distributions, and show linear stability under sufficient noise. Numerics illustrate that noise can transform the deterministic consensus into stable cooperative stationary behaviour. These findings show that the fast adaptive interactions and stochastic exploration can jointly support the emergence of stable cooperation at a population level.

math.AP↗

arXiv2026

Construction and simulation of a path-valued model of dendrite development

Neurons receive information through their dendrites. During development, when synaptic connections are forming, dendrites grow, retract, and branch. The resulting dendritic tree shapes the structure of the broader neural network. Crucially, retraction and branching make it necessary to track whole dendritic paths rather than only their endpoints. While this is handled implicitly in some existing simulations, here we construct an explicitly path-valued stochastic process for dendrite growth. Combining this with a branching process, using ideas from measure-valued branching particle systems, we show that the model produces the typical tree structures of real dendrites. To complement this analytical work, we also outline several methods for numerical simulation, including time discretisations at different temporal scales and an approximation using a dynamic graph. This provides both a more rigorous mathematical framework and more structured simulation methods for modelling dendrite development.

math.PR↗

arXiv2026

Emergent structures in coupled opinion and network dynamics

This paper investigates a model of opinion formation on an adaptive social network, consisting of a system of coupled ordinary differential equations for individuals' opinions and corresponding network edge weights. A key driver of the system's behaviour is the form of the interaction function, which determines the strength of interactions based on the distance between individuals' opinions and appears in both opinion and network dynamics. Two cases are examined: in the first the interaction function is always positive and in the second case the interaction function is of bounded-confidence type. In both cases there is positive feedback between opinion clustering and the emergence of community structure in the social network. This is confirmed through analytical results on long-term behaviour, extending existing results for a fixed network, as well as through numerical simulations. Transient network dynamics are also examined through a short-time approximation that captures the `typical' early network dynamics. Each approach improves some aspect of our understanding of the interplay between opinion and network evolution.

physics.soc-ph↗

arXiv2026

Speaking of Opinions: Comparing Approaches to Modelling Opinion Manipulation

This review outlines the major approaches to modelling opinion formation and manipulation in mathematics and computer science. Key tools such as ordinary and partial differential equations, stochastic models, control theory, and interaction protocols are introduced and compared as methods for describing manipulation. The review is separated into those models using a continuous opinion space and those using discrete or binary opinions, with the advantages and disadvantages of each discussed. Finally, the authors provide an interdisciplinary perspective on the field of opinion dynamics and its social significance.

math.OC↗

arXiv2026

Cyclists Cardiac Conundrum

Arrhythmia is an abnormality of the heart's rhythm, caused by problems in the conductive system and resulting in irregular heartbeats. There is increasing evidence that undertaking frequent endurance sports training elevates one's risk of arrhythmia. Arrhythmia is diagnosed using an electrocardiogram (ECG) but this is not typically available to athletes while exercising. Previous research by Crickles investigates the usefulness of commonly available heart rate data in detecting signs of arrhythmia. It is hypothesised that a feature termed 'gappiness', defined by jumps in the heart rate while the athlete is under exertion, may be a characteristic of arrhythmia. A correlation was found between the proportion of 'gappy' activities and survey responses about heart rhythm problems. We develop on this measure by exploring various methods to detect spikes in heart rate data, allowing us to describe the extent of irregularity in an activity via the rate of spikes. We first compare the performance of these methods on simulated data, where we find that smoothing using a moving average and setting a constant threshold on the residuals is most effective. This method was then implemented on real data provided by Crickles from 168 athletes, where no significant correlation was found between the spike rates and survey responses. However, when considering only those spikes that occur above a heart rate of 160 beats per minute (bpm) a significant correlation was found. This supports the hypothesis that jumps at only high heart rates are informative of arrhythmia and indicates the need for further research into better measures to characterise features of heart rate data.

stat.AP↗

arXiv2025

Modeling Social Systems: Transparency, Reproducibility, and Responsibility

Mathematical models of complex social systems can enrich social scientific theory, inform interventions, and shape policy. From voting behavior to economic inequality and urban development, such models influence decisions that affect millions of lives. Thus, it is especially important to formulate and present them with transparency, reproducibility, and humility. Modeling in social domains, however, is often uniquely challenging. Unlike in physics or engineering, researchers often lack controlled experiments or abundant, clean data. Observational data is sparse, noisy, partial, and missing in systematic ways. In such an environment, how can we build models that can inform science and decision-making in transparent and responsible ways?

math.HO↗

arXiv2025

Opinion Dynamics with Continuous Age Structure

We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally we establish connections between this and other existing models in various scenarios.

math.AP↗

arXiv2024

Steering opinion dynamics through control of social networks

In this paper we propose a novel control approach for opinion dynamics on evolving networks. The controls modify the strength of connections in the network, rather than influencing opinions directly, with the overall goal of steering the population towards a target opinion. This requires that the social network remains sufficiently connected, the population does not break into separate opinion clusters, and that the target opinion remains accessible. We present several approaches to addressing these challenges, considering questions of controllability, instantaneous control and optimal control. Each of these approaches provides a different view on the complex relationship between opinion and network dynamics and raises interesting questions for future research.

physics.soc-ph↗

arXiv2023

Bridging the gap between agent based models and continuous opinion dynamics

There is a rich literature on microscopic models for opinion dynamics; most of them fall into one of two categories - agent-based models or differential equation models - with a general understanding that the two are connected in certain scaling limits. In this paper we show rigorously this is indeed the case. In particular we show that DEMs can be obtained from ABMs by simultaneously rescaling time and the distance an agent updates their opinion after an interaction. This approach provides a pathway to analyse much more diverse modelling paradigms, for example: the motivation behind several possible multiplicative noise terms in stochastic differential equation models; the connection between selection noise and the mollification of the discontinuous bounded confidence interaction function; and how the method for selecting interacting pairs can determine the normalisation in the corresponding differential equation. Our computational experiments confirm our findings, showing excellent agreement of solutions to the two classes of models in a variety of settings.

math.DS↗