arXiv subjects

Mason A. Porter

Publications and source records attributed to Mason A. Porter.

At least 37 records · Page 2Linked to original sources

arXiv2025

Bounded-Confidence Models of Opinion Dynamics with Neighborhood Effects

We generalize bounded-confidence models (BCMs) of opinion dynamics by incorporating neighborhood effects. In a BCM, interacting agents influence each other through dyadic influence if their opinions are sufficiently similar to each other. In our "neighborhood BCMs" (NBCMs), interacting agents are influenced both by each other's opinions and by the opinions of the agents in each other's neighborhoods. Our NBCMs thus include both the usual dyadic influence between agents and a "transitive influence", which encodes the influence of an agent's neighbors, when determining whether or not an interaction changes the opinions of agents. In this transitive influence, an individual's opinion is influenced by a neighbor when, on average, the opinions of the neighbor's neighbors are sufficiently similar to its own opinion. We formulate both neighborhood Deffuant--Weisbuch (NDW) and neighborhood Hegselmann--Krause (NHK) BCMs. We build further on our NBCMs by introducing a neighborhood-based network adaptation in which a network coevolves with agent opinions by changing its structure through "transitive homophily". In this network evolution, an agent breaks a tie to one of its neighbors and then rewires that tie to a new agent, with a preference for agents with a mean neighbor opinion that is closer to its own opinion. Using numerical simulations on a variety of types of networks, we explore how the qualitative opinion dynamics and network properties of our adaptive NDW model change as we adjust the relative proportions of dyadic and transitive influence. In our numerical experiments, we find that incorporating neighborhood effects into the opinion dynamics and the network-adaptation rewiring strategy tends to reduce the spectral gap and degree assortativity of networks. (This is a shortened version of the paper's abstract.)

physics.soc-ph↗

arXiv2025

Oscillatory and Excitable Dynamics in an Opinion Model with Group Opinions

In traditional models of opinion dynamics, each agent in a network has an opinion and changes in opinions arise from pairwise (i.e., dyadic) interactions between agents. However, in many situations, groups of individuals possess a collective opinion that can differ from the opinions of its constituent individuals. In this paper, we study the effects of group opinions on opinion dynamics. We formulate a hypergraph model in which both individual agents and groups of 3 agents have opinions, and we examine how opinions evolve through both dyadic interactions and group memberships. In some parameter regimes, we find that the presence of group opinions can lead to oscillatory and excitable opinion dynamics. In the oscillatory regime, the mean opinion of the agents in a network has self-sustained oscillations. In the excitable regime, finite-size effects create large but short-lived opinion swings (as in social fads). We develop a mean-field approximation of our model and obtain good agreement with direct numerical simulations. We also show -- both numerically and via our mean-field description -- that oscillatory dynamics occur only when the number of dyadic and polyadic interactions per agent are not completely correlated. Our results illustrate how polyadic structures, such as groups of agents, can have important effects on collective opinion dynamics.

physics.soc-ph↗

arXiv2025

A Social Dynamical System for Twitter Analysis

Understanding the evolution of public opinion is crucial for informed decision-making in various domains, particularly public affairs. The rapid growth of social networks, such as Twitter (now rebranded as X), provides an unprecedented opportunity to analyze public opinion at scale without relying on traditional surveys. With the rise of deep learning, Graph Neural Networks (GNNs) have shown great promise in modeling online opinion dynamics. Notably, classical opinion dynamics models, such as DeGroot, can be reformulated within a GNN framework. We introduce Latent Social Dynamical System (LSDS), a novel framework for modeling the latent dynamics of social media users' opinions based on textual content. Since expressed opinions may not fully reflect underlying beliefs, LSDS first encodes post content into latent representations. It then leverages a GraphODE framework, using a GNN-based ODE function to predict future opinions. A decoder subsequently utilizes these predicted latent opinions to perform downstream tasks, such as interaction prediction, which serve as benchmarks for model evaluation. Our framework is highly flexible, supporting various opinion dynamic models as ODE functions, provided they can be adapted into a GNN-based form. It also accommodates different encoder architectures and is compatible with diverse downstream tasks. To validate our approach, we constructed dynamic datasets from Twitter data. Experimental results demonstrate the effectiveness of LSDS, highlighting its potential for future applications. We plan to publicly release our dataset and code upon the publication of this paper.

cs.SI↗

arXiv2024

Perceived community alignment increases information sharing

It has been proposed that information sharing, which is a ubiquitous and consequential behavior, plays a critical role in cultivating and maintaining a sense of shared reality. Across three studies, we tested this theory by investigating whether or not people are especially likely to share information that they believe will be interpreted similarly by others in their social circles. Using neuroimaging while members of the same community viewed brief film clips, we found that more similar neural responding of participants was associated with a greater likelihood to share content. We then tested this relationship using two behavioral studies and found (1) that people were particularly likely to share content that they believed others in their social circles would interpret similarly and (2) that perceived similarity with others leads to increased sharing likelihood. In concert, our findings support the idea that people are driven to share information to create and reinforce shared understanding, which is critical to social connection.

q-bio.NC↗

arXiv2024

Competition between group interactions and nonlinearity in voter dynamics on hypergraphs

Social dynamics are often driven by both pairwise (i.e., dyadic) relationships and higher-order (i.e., polyadic) group relationships, which one can describe using hypergraphs. To gain insight into the impact of polyadic relationships on dynamical processes on networks, we formulate and study a polyadic voter process, which we call the group-driven voter model (GVM), that incorporates the effect of group interactions by nonlinear interactions that are subject to a group (i.e., hyperedge) constraint. By examining the competition between nonlinearity and group sizes, we show that the GVM achieves consensus faster than standard voter-model dynamics, with an optimal minimizing exit time. We substantiate this finding by using mean-field theory on annealed uniform hypergraphs with $N$ nodes, for which the exit time scales as ${\cal A}\ln N$, where the prefactor ${\cal A}$ depends both on the nonlinearity and on group-constraint factors. Our results reveal how competition between group interactions and nonlinearity shapes GVM dynamics. We thereby highlight the importance of such competing effects in complex systems with polyadic interactions.

physics.soc-ph↗

arXiv2024

An "Opinion Reproduction Number" for Infodemics in a Bounded-Confidence Content-Spreading Process on Networks

We study the spreading dynamics of content on networks. To do this, we use a model in which content spreads through a bounded-confidence mechanism. In a bounded-confidence model (BCM) of opinion dynamics, the agents of a network have continuous-valued opinions, which they adjust when they interact with agents whose opinions are sufficiently close to theirs. The employed content-spreading model introduces a twist into BCMs by using bounded confidence for the content spread itself. We define an analogue of the basic reproduction number from disease dynamics that we call an \emph{opinion reproduction number}. A critical value of the opinion reproduction number indicates whether or not there is an ``infodemic'' (i.e., a large content-spreading cascade) of content that reflects a particular opinion. By determining this critical value, one can determine whether or not an opinion dies off or propagates widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination by calculating a variety of summary statistics, and we illustrate how network structure and spreading-model parameters affect these statistics. We find that content spreads most widely when the agents have a large expected mean degree or a large receptiveness to content. When the spreading process slightly exceeds the infodemic threshold, there can be longer dissemination trees than when the expected mean degree or receptiveness are larger, even though the total number of content shares is smaller.

physics.soc-ph↗

arXiv2024

Structural Robustness and Vulnerability of Networks

Networks are useful descriptions of the structure of many complex systems. Unsurprisingly, it is thus important to analyze the robustness of networks in many scientific disciplines. In applications in communication, logistics, finance, ecology, biomedicine, and many other fields, researchers have studied the robustness of networks to the removal of nodes, edges, or other subnetworks to identify and characterize robust network structures. A major challenge in the study of network robustness is that researchers have reported that different and seemingly contradictory network properties are correlated with a network's robustness. Using a framework by Alderson and Doyle~\cite{Alderson2010}, we categorize several notions of network robustness and we examine these ostensible contradictions. We survey studies of network robustness with a focus on (1)~identifying robustness specifications in common use, (2)~understanding when these specifications are appropriate, and (3)~understanding the conditions under which one can expect different notions of robustness to yield similar results. With this review, we aim to give researchers an overview of the large, interdisciplinary body of work on network robustness and develop practical guidance for the design of computational experiments to study a network's robustness.

physics.soc-ph↗

arXiv2024

Dynamical importance and network perturbations

The leading eigenvalue $λ$ of the adjacency matrix of a graph exerts much influence on the behavior of dynamical processes on that graph. It is thus relevant to relate notions of the importance (specifically, centrality measures) of network structures to $λ$ and its associated eigenvector. We study a previously derived measure of edge importance known as ``dynamical importance'', which estimates how much $λ$ changes when one removes an edge from a graph or adds an edge to it. We examine the accuracy of this estimate for different network structures and compare it to the true change in $λ$ after an edge removal or edge addition. We then derive a first-order approximation of the change in the leading eigenvector. We also consider the effects of edge additions on Kuramoto dynamics on networks, and we express the Kuramoto order parameter in terms of dynamical importance. Through our analysis and computational experiments, we find that studying dynamical importance can improve understanding of the relationship between network perturbations and dynamical processes on networks.

cs.SI↗

arXiv2024

Using mathematics to study how people influence each other's opinions

People sometimes change their opinions when they discuss things with other people. Researchers can use mathematics to study opinion changes in simplifications of real-life situations. These simplified settings, which are examples of mathematical models, help researchers explore how people influence each other through their social interactions. In today's digital world, these models can help us learn how to promote the spread of accurate information and reduce the spread of inaccurate information. In this article, we discuss a simple mathematical model of opinion changes that arise from social interactions. We briefly describe what such opinion models can tell us and how researchers try to make them more realistic.

physics.soc-ph↗

arXiv2024

Bounded-Confidence Models of Opinion Dynamics with Adaptive Confidence Bounds

People's opinions change with time as they interact with each other. In a bounded-confidence model (BCM) of opinion dynamics, individuals (which are represented by the nodes of a network) have continuous-valued opinions and are influenced by neighboring nodes whose opinions are sufficiently similar to theirs (i.e., are within a confidence bound). In this paper, we formulate and analyze discrete-time BCMs with heterogeneous and adaptive confidence bounds. We introduce two new models: (1) a BCM with synchronous opinion updates that generalizes the Hegselmann--Krause (HK) model and (2) a BCM with asynchronous opinion updates that generalizes the Deffuant--Weisbuch (DW) model. We analytically and numerically explore our adaptive BCMs' limiting behaviors, including the confidence-bound dynamics, the formation of clusters of nodes with similar opinions, and the time evolution of an "effective graph", which is a time-dependent subgraph of a network with edges between nodes that {are currently receptive to each other.} For a variety of networks and a wide range of values of the parameters that control the increase and decrease of confidence bounds, we demonstrate numerically that our adaptive BCMs result in fewer major opinion clusters and longer convergence times than the baseline (i.e., nonadaptive) BCMs. We also show that our adaptive BCMs can have adjacent nodes that converge to the same opinion but are not {receptive to each other.} This qualitative behavior does not occur in the associated baseline BCMs.

cs.SI↗

arXiv2024

A Non-Expert's Introduction to Data Ethics for Mathematicians

I give a short introduction to data ethics. I begin with some background information and societal context for data ethics. I then discuss data ethics in mathematical-science education and indicate some available course material. I briefly highlight a few efforts -- at my home institution and elsewhere -- on data ethics, society, and social good. I then discuss open data in research, research replicability and some other ethical issues in research, and the tension between privacy and open data and code, and a few controversial studies and reactions to studies. I then discuss ethical principles, institutional review boards, and a few other considerations in the scientific use of human data. I then briefly survey a variety of research and lay articles that are relevant to data ethics and data privacy. I conclude with a brief summary and some closing remarks. My focal audience is mathematicians, but I hope that this chapter will also be useful to others. I am not an expert about data ethics, and this chapter provides only a starting point on this wide-ranging topic. I encourage you to examine the resources that I discuss and to reflect carefully on data ethics, its role in mathematics education, and the societal implications of data and data analysis. As data and technology continue to evolve, I hope that such careful reflection will continue throughout your life.

math.HO↗

arXiv2024

An adaptation of InfoMap to absorbing random walks using absorption-scaled graphs

InfoMap is a popular approach to detect densely connected "communities" of nodes in networks. To detect such communities, InfoMap uses random walks and ideas from information theory. Motivated by the dynamics of disease spread on networks, whose nodes can have heterogeneous disease-removal rates, we adapt InfoMap to absorbing random walks. To do this, we use absorption-scaled graphs (in which edge weights are scaled according to absorption rates) and Markov time sweeping. One of our adaptations of InfoMap converges to the standard version of InfoMap in the limit in which the node-absorption rates approach $0$. We demonstrate that the community structure that one obtains using our adaptations of InfoMap can differ markedly from the community structure that one detects using methods that do not account for node-absorption rates. We also illustrate that the community structure that is induced by heterogeneous absorption rates can have important implications for susceptible-infected-recovered (SIR) dynamics on ring-lattice networks. For example, in some situations, the outbreak duration is maximized when a moderate number of nodes have large node-absorption rates.

cs.SI↗

arXiv2024

Dynamical processes on metric networks

The structure of a network has a major effect on dynamical processes on that network. Many studies of the interplay between network structure and dynamics have focused on models of phenomena such as disease spread, opinion formation and changes, coupled oscillators, and random walks. In parallel to these developments, there have been many studies of wave propagation and other spatially extended processes on networks. These latter studies consider metric networks, in which the edges are associated with real intervals. Metric networks give a mathematical framework to describe dynamical processes that include both temporal and spatial evolution of some quantity of interest -- such as the concentration of a diffusing substance or the amplitude of a wave -- by using edge-specific intervals that quantify distance information between nodes. Dynamical processes on metric networks often take the form of partial differential equations (PDEs). In this paper, we present a collection of techniques and paradigmatic linear PDEs that are useful to investigate the interplay between structure and dynamics in metric networks. We start by considering a time-independent Schrödinger equation. We then use both finite-difference and spectral approaches to study the Poisson, heat, and wave equations as paradigmatic examples of elliptic, parabolic, and hyperbolic PDE problems on metric networks. Our spectral approach is able to account for degenerate eigenmodes. In our numerical experiments, we consider metric networks with up to about $10^4$ nodes and about $10^4$ edges. A key contribution of our paper is to increase the accessibility of studying PDEs on metric networks. Software that implements our numerical approaches is available at https://gitlab.com/ComputationalScience/metric-networks.

math.DS↗

arXiv2023

Inference of interaction kernels in mean-field models of opinion dynamics

In models of opinion dynamics, many parameters -- either in the form of constants or in the form of functions -- play a critical role in describing, calibrating, and forecasting how opinions change with time. When examining a model of opinion dynamics, it is beneficial to infer its parameters using empirical data. In this paper, we study an example of such an inference problem. We consider a mean-field bounded-confidence model with an unknown interaction kernel between individuals. This interaction kernel encodes how individuals with different opinions interact and affect each other's opinions. Because it is often difficult to quantitatively measure opinions as empirical data from observations or experiments, we assume that the available data takes the form of partial observations of a cumulative distribution function of opinions. We prove that certain measurements guarantee a precise and unique inference of the interaction kernel and propose a numerical method to reconstruct an interaction kernel from a limited number of data points. Our numerical results suggest that the error of the inferred interaction kernel decays exponentially as we strategically enlarge the data set.

cs.SI↗

arXiv2023

Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites

It is important to choose the geographical distributions of public resources in a fair and equitable manner. However, it is complicated to quantify the equity of such a distribution; important factors include distances to resource sites, availability of transportation, and ease of travel. We use persistent homology, which is a tool from topological data analysis, to study the effective availability and coverage of polling sites. The information from persistent homology allows us to infer holes in the distribution of polling sites. We analyze and compare the coverage of polling sites in Los Angeles County and five cities (Atlanta, Chicago, Jacksonville, New York City, and Salt Lake City), and we conclude that computation of persistent homology appears to be a reasonable approach to analyzing resource coverage.

cs.CG↗

arXiv2023

Emergence of polarization in a sigmoidal bounded-confidence model of opinion dynamics

We study a nonlinear bounded-confidence model (BCM) of continuous-time opinion dynamics on networks with both persuadable individuals and zealots. The model is parameterized by a scalar $γ$, which controls the steepness of a smooth influence function. This influence function encodes the relative weights that nodes place on the opinions of other nodes. When $γ= 0$, this influence function recovers Taylor's averaging model; when $γ\rightarrow \infty$, the influence function converges to that of a modified Hegselmann--Krause (HK) BCM. Unlike the classical HK model, however, our sigmoidal bounded-confidence model (SBCM) is smooth for any finite $γ$. We show that the set of steady states of our SBCM is qualitatively similar to that of the Taylor model when $γ$ is small and that the set of steady states approaches a subset of the set of steady states of a modified HK model as $γ\rightarrow \infty$. For several special graph topologies, we give analytical descriptions of important features of the space of steady states. A notable result is a closed-form relationship between the stability of a polarized state and the graph topology in a simple model of echo chambers in social networks. Because the influence function of our BCM is smooth, we are able to study it with linear stability analysis, which is difficult to employ with the usual discontinuous influence functions in BCMs.

math.DS↗

arXiv2023

Low-Dimensional Behavior of a Kuramoto Model with Inertia and Hebbian Learning

We study low-dimensional dynamics in a Kuramoto model with inertia and Hebbian learning. In this model, the coupling strength between oscillators depends on the phase differences between the oscillators and changes according to a Hebbian learning rule. We analyze the special case of two coupled oscillators, which yields a five-dimensional dynamical system that decouples into a two-dimensional longitudinal system and a three-dimensional transverse system. We readily write an exact solution of the longitudinal system, and we then focus our attention on the transverse system. We classify the stability of the transverse system's equilibrium points using linear stability analysis. We show that the transverse system is dissipative and that all of its trajectories are eventually confined to a bounded region. We compute Lyapunov exponents to infer the transverse system's possible limiting behaviors, and we demarcate the parameter regions of three qualitatively different behaviors. Using insights from our analysis of the low-dimensional dynamics, we study the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.

math.DS↗

arXiv2023

Complex networks with complex weights

In many studies, it is common to use binary (i.e., unweighted) edges to examine networks of entities that are either adjacent or not adjacent. Researchers have generalized such binary networks to incorporate edge weights, which allow one to encode node--node interactions with heterogeneous intensities or frequencies (e.g., in transportation networks, supply chains, and social networks). Most such studies have considered real-valued weights, despite the fact that networks with complex weights arise in fields as diverse as quantum information, quantum chemistry, electrodynamics, rheology, and machine learning. Many of the standard network-science approaches in the study of classical systems rely on the real-valued nature of edge weights, so it is necessary to generalize them if one seeks to use them to analyze networks with complex edge weights. In this paper, we examine how standard network-analysis methods fail to capture structural features of networks with complex edge weights. We then generalize several network measures to the complex domain and show that random-walk centralities provide a useful approach to examine node importances in networks with complex weights.

physics.soc-ph↗