"Long-Time and Short-Time Dynamics in a Weighted-Median Opinion Model on Networks"

One of my papers was published in final form late last month. Here are some details.

Title: Long-Time and Short-Time Dynamics in a Weighted-Median Opinion Model on Networks

Authors: Lasse Mohr, Poul G. Hjorth, and Mason A. Porter

Abstract: Social interactions influence people's opinions. In some situations, these interactions eventually yield a consensus opinion; in others, they can lead to opinion fragmentation and the formation of different opinion groups in the form of "echo chambers". Consider a social network of individuals with continuous-valued scalar opinions, and suppose that they can change their opinions when they interact with each other. In many models of the opinion dynamics of individuals in a network, it is common for opinion updates to depend on the mean opinion of interacting individuals. As an alternative, which may be more realistic in some situations, we study an opinion model with an opinion-update rule that depends on the weighted median of the opinions of interacting individuals. Through numerical simulations of our median-update opinion model, we investigate how the final opinion distribution depends on network structure. For configuration-model networks, we derive a mean-field approximation of the asymptotic dynamics of the opinion distribution when there are infinitely many individuals. We numerically investigate its accuracy for short-time opinion dynamics on various networks.

"Ginzburg–Landau Functionals in the Large-Graph Limit"

Another of my papers has appeared in final form. Here are some details.

Title: Ginzburg–Landau Functionals in the Large-Graph Limit

Authors: Edith J. Zhang, James Scott, Qiang Du, and Mason A. Porter

Abstract: Ginzburg–Landau (GL) functionals on graphs, which are relaxations of graph-cut functionals on graphs, have yielded a variety of insights in image segmentation and graph clustering. In this paper, we study large-graph limits of GL functionals by taking a functional-analytic view of graphs as nonlocal kernels. For a graph W_n with n nodes, the corresponding graph GL functional GL_ϵ^{W_n} is an energy for functions on W_n. We minimize GL functionals on sequences of growing graphs that converge to functions called graphons. For such sequences of graphs, we show that the graph GL functional Γ-converges to a continuous and nonlocal functional that we call the graphon GL functional. We investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total-variation (TV) functional. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the minimization problem in the large-graph limit. Finally, to develop intuition about graphon GL functionals, we determine the GL minimizer for several example families of graphons.

RIP Erik Bollt (1967–2025)

My collaborator Erik Bollt died suddenly and unexpectedly last Sunday (the 7th). Erik is a well-known applied mathematician, and he was especially in the applied dynamical-systems community. Erik and I wrote two papers together, including our paper on mathematically modeling cow synchronization.

You can read about Erik's research on his web page and his Google Scholar page.

There will be a memorial article at some point.

"Dynamical Processes on Metric Networks"

A paper of mine was just published in final form. Here are some details.

Title: Dynamical Processes on Metric Networks

Authors: Lucas Böttcher and Mason A. Porter

What Happens in Montreal Stays in Montreal (2025 Edition)

I am heading to Montreal for the first time in several years to attend the 2025 SIAM Annual Meeting, which is being held jointly with the CAIMS Annual Meeting.

What Happens in "Snowbird" Stays in "Snowbird" (2025 Edition)

I'm off to the 2025 "Snowbird" SIAM dynamical-systems conference. I love this biennial migration to my favorite conference series!

"A Non-Expert’s Introduction to Data Ethics for Mathematicians"

My chapter in our new book is also out. Here are some details about it.

Title: A Non-Expert’s Introduction to Data Ethics for Mathematicians

Author: Mason A. Porter

Abstract: 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, the tension between privacy and open data and code, and a few controversial studies and reactions to studies. I also 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.

"Mathematical and Computational Methods for Complex Social Systems"

Our edited book is out! You can find it on this website. Here are some details.

Title: Mathematical and Computational Methods for Complex Social Systems

Authors: Heather Z. Brooks, Michelle Feng, Mason A. Porter, and Alexandria Volkening

What Happens in Denver Stays in Denver (2025 Edition)

I am flying to Denver for the US Dynamics Days 2025 conference. I really love this conference series! This year, the conference hotel is actually about half a mile from the 2025 Snowbird-conference hotel.

I am really looking forward to this year's Dynamics Daze conference!

"Oscillatory Networks: Insights from Piecewise-Linear Modeling"

Another of my papers just appeared in final form. Here are some details.

Title: Oscillatory Networks: Insights from Piecewise-Linear Modeling

Authors: Stephen Coombes, Mustafa Şayli, Rüdiger Thul, Rachel Nicks, Mason A. Porter, and Yi Ming Lai

Dedication: We dedicate this paper to the memory of our dear friend and colleague Yi Ming Lai. Although he began with us on the journey to write this paper, which in part reviews some of his research activity in recent years, sadly he did not end that journey with us. RIP Yi Ming Lai 1988–2022.

Abstract: There is enormous interest—both mathematically and in diverse applications—in understanding the dynamics of coupled-oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled-oscillator networks. We discuss phase reductions, phase–amplitude reductions, and the master stability function for smooth dynamical systems. We then focus, in particular, on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use saltation operators, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.

"Using Mathematics to Study how People Influence Each Other’s Opinions"

Our article for teenagers and preteens about mathematical modeling of opinion dynamics has just been published in final form. Here are some details.

Title: Using Mathematics to Study how People Influence Each Other’s Opinions

Authors: Grace J. Li, Jiajie (Jerry) Luo, Kaiyan Peng, and Mason A. Porter

Abstract: People sometimes change their opinions when they discuss things with each other. Researchers can use mathematics to study opinion changes in simplifications of real-life situations. These simplified scenarios, 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 opinion models can tell us and how researchers try to make them more realistic.

What Happens in Boston Stays in Boston

I'm on my way to Boston for the workshop to celebrate David Campbell's 80th birthday! It will surely be chaotic. :)

"Adapting InfoMap to Absorbing Random Walks Using Absorption-Scaled Graphs"

One of my papers just came out in final published form. Here are some details.

Title: Adapting InfoMap to Absorbing Random Walks Using Absorption-Scaled Graphs

Authors: Esteban Vargas Bernal, Mason A. Porter, and Joseph H. Tien

Abstract: 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.

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

One of my paper was published in final form last week. Here are some details.

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

Authors: Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter

Abstract: 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 availability and coverage of polling sites. The information from persistent homology allows us to infer holes in a 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.

"Emergence of Polarization in a Sigmoidal Bounded-Confidence Model of Opinion Dynamics"

A paper of mine was just published in final form. Here are zome details.

Title: Emergence of Polarization in a Sigmoidal Bounded-Confidence Model of Opinion Dynamics

Authors: Heather Z. Brooks, Philip S. Chodrow, and Mason A. Porter

Abstract: 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 nonnegative scalar \gamma, which controls the steepness of a smooth influence function. This influence function encodes the relative weights that individuals place on the opinions of other individuals. When \gamma = 0, this influence function recovers Taylor's averaging model; when \gamma \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 \gamma. We show that the set of steady states of our SBCM is qualitatively similar to that of the Taylor model when \gamma is small and that the set of steady states approaches a subset of the set of steady states of a modified HK model as \gamma \rightarrow \infty. For certain 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 graph topology and the stability of polarized states in a simple special case that models 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.

"Inference of Interaction Kernels in Mean-Field Models of Opinion Dynamics"

A paper of mine just came out in final form. Here are some details about it.

Title: Inference of Interaction Kernels in Mean-Field Models of Opinion Dynamics

Authors: Weiqi Chu, Qin Li, and Mason A. Porter

Abstract: 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.

"Learning Low-Rank Latent Mesoscale Structures in Networks"

One of my papers came out in final form today. Here are some details.

Title: Learning Low-Rank Latent Mesoscale Structures in Networks

Authors: Hanbaek Lyu, Yacoub H. Kureh, Joshua Vendrow, and Mason A. Porter

Abstract: Researchers in many fields use networks to represent interactions between entities in complex systems. To study the large-scale behavior of complex systems, it is useful to examine mesoscale structures in networks as building blocks that influence such behavior. In this paper, we present an approach to describe low-rank mesoscale structures in networks. We find that many real-world networks possess a small set of latent motifs that effectively approximate most subgraphs at a fixed mesoscale. Such low-rank mesoscale structures allow one to reconstruct networks by approximating subgraphs of a network using combinations of latent motifs. Employing subgraph sampling and nonnegative matrix factorization enables the discovery of these latent motifs. The ability to encode and reconstruct networks using a small set of latent motifs has many applications in network analysis, including network comparison, network denoising, and edge inference.

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

A paper of mine just came out in final form. Here are some details.

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

Authors: Tachin Ruangkriengsin and Mason A. Porter

Abstract: 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 examine the original high-dimensional system in a situation in which we draw the intrinsic frequencies of the oscillators from Gaussian distributions with different variances.

"Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling" (Second Edition)

The unnecessary second edition of the book Temporal Network Theory is now out. It includes a second edition of a chapter that I coauthored. Here are a few details.

Title: Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

Authors: Dane Taylor, Mason A. Porter, and Peter J. Mucha

Abstract: We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change with time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a supracentrality matrix C(ω), where ω controls the extent to which centrality trajectories change with time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of C(ω) represent joint centralities, which simultaneously quantify the importances of every node in every time layer. Inspired by probability theory, we also compute marginal and conditional centralities. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes’ centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large ω.

"Non-Markovian Models of Opinion Dynamics on Temporal Networks"

One of my papers was published in final form today. Here are some details.

Title: Non-Markovian Models of Opinion Dynamics on Temporal Networks

Authors: Weiqi Chu and Mason A. Porter

Abstract: Traditional models of opinion dynamics, in which the nodes of a network change their opinions based on their interactions with neighboring nodes, consider how opinions evolve either on time-independent networks or on temporal networks with edges that follow Poisson statistics. Most such models are Markovian. However, in many real-life networks, interactions between individuals (and hence the edges of a network) follow non-Poisson processes and thus yield dynamics with memory-dependent effects. In this paper, we model opinion dynamics in which the entities of a temporal network interact and change their opinions via random social interactions. When the edges have non-Poisson interevent statistics, the corresponding opinion models have non-Markovian dynamics. We derive a family of opinion models that are induced by arbitrary waiting-time distributions (WTDs), and we illustrate a variety of induced opinion models from common WTDs (including Dirac delta distributions, exponential distributions, and heavy-tailed distributions). We analyze the convergence to consensus of these models and prove that homogeneous memory-dependent models of opinion dynamics in our framework always converge to the same steady state regardless of the WTD. We also conduct a numerical investigation of the effects of waiting-time distributions on both transient dynamics and steady states. We observe that models that are induced by heavy-tailed WTDs converge more slowly to a steady state than models that are induced by WTDs with light tails (or with compact support) and that entities with longer waiting times exert more influence on the mean opinion at steady state.