"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.

What Happens in Boston Stays in Boston

I am off to Boston to speak in the satellite conference on Physical Networks at the 2026 NetSci conference. (It's my triumphant return to NetSci!) As usual, I also helped organize the satellite conference on Network Science in Education (NetSciEd 2026).

"Bounded-Confidence Opinion Models with Random-Time Interactions"

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

Title: Bounded-Confidence Opinion Models with Random-Time Interactions

Authors: Weiqi Chu and Mason A. Porter

Abstract: In models of opinion dynamics, agents interact with each other and can change their opinions as a result of those interactions. One type of opinion model is a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents compromise their opinions with each other if their opinions are sufficiently similar. In studies of BCMs, researchers typically assume that interactions between agents occur at deterministic times. This assumption neglects an inherent element of randomness in social interactions, and it is desirable to account for it. In this paper, we study BCMs on networks and allow agents to interact at random times. To incorporate random-time interactions, we use renewal processes to determine social-interaction event times, which can follow arbitrary interevent-time distributions (ITDs). We establish connections between these random-time-interaction BCMs and deterministic-time-interaction BCMs. We analyze the quantitative impact of ITDs on the transient dynamics of BCMs and derive approximate governing equations for the time-dependent expectations of the BCM dynamics. We find that BCMs with Markovian ITDs have consistent statistical properties (in particular, they have the same expected time-dependent opinions) when the ITDs have the same mean but that the statistical properties of BCMs with non-Markovian ITDs depend on the type of ITD even when the ITDs have the same mean. We numerically examine the transient and steady-state dynamics of our BCMs with various ITDs on different networks, and we compare their expected order-parameter values and expected convergence times.

"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

"Oscillatory and Excitable Dynamics in an Opinion Model with Group Opinions"

The published version of one of my papers just came out today. Here are some details.

Title: Oscillatory and Excitable Dynamics in an Opinion Model with Group Opinions

Authors: Corbit R. Sampson, Juan G. Restrepo, and Mason A. Porter

Abstract:

"Competition Between Group Interactions and Nonlinearity in Voter Dynamics on Hypergraphs"

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

Title: Competition Between Group Interactions and Nonlinearity in Voter Dynamics on Hypergraphs

Authors: Jihye Kim, Deok-Sun Lee, Byungjoon Min, Mason A. Porter, Maxi San Miguel, and Kwang-Il Goh

"Dynamical Importance and Network Perturbations"

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

Title: Dynamical Importance and Network Perturbations

Authors: Ethan Young and Mason A. Porter

Abstract: 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 importance of network structures to λ and its associated eigenvectors. 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 several undirected network structures and compare it to the relative 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.

"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.

What Happens in Seoul Stays in Seoul (2024 Edition)

I am off to Seoul to speak in a workshop on Theoretical Challenges in Network Science! I really enjoy visiting Seoul, and I am very happy to have another chance to visit.

"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.

"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.

What Happens in Warsaw Stays in Warsaw

I am heading to Warsaw to participate a couple of days in the WAW 2024 conference. This is my first trip to Poland in several years, and unfortunately it's going to be very brief.

"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.

"Complex Networks with Complex Weights"

The published version of one of my papers came out today. Its title is one of my favorites among all of the papers that I've ever written. Here are some details about the paper.

Title: Complex Networks with Complex Weights

Authors: Lucas Böttcher and Mason A. Porter

Abstract: 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.

"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.

"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.

"Minimizing Congestion in Single-Source, Single-Sink Queuing Networks"

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

Title: Minimizing Congestion in Single-Source, Single-Sink Queuing Networks

Authors: Fabian Ying, Alisdair O. G. Wallis, Mason A. Porter, Sam D. Howison, and Mariano Beguerisse-Díaz

Abstract: Motivated by the modeling of customer mobility and congestion in supermarkets, we study queueing networks with a single source and a single sink. We assume that walkers traverse a network according to an unbiased random walk, and we analyze how network topology affects the total mean queue size Q, which we use to measure congestion. We examine network topologies that minimize Q and provide proofs of optimality for some cases and numerical evidence of optimality for others. Finally, we present greedy algorithms that add edges to and delete edges from a network to reduce Q, and we apply these algorithms to a network that we construct using a supermarket store layout. We find that these greedy algorithms, which typically tend to add edges to the sink node, are able to significantly reduce Q. Our work helps improve understanding of how to design networks with low congestion and how to amend networks to reduce congestion.

"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.

"Detecting Political Biases of Named Entities and Hashtags on Twitter"

One of my papers came out in final form earlier today. Here are some details. (This is in collaboration with computer scientists, and stylistically it is rather different from much of my work. However, you'll still notice my hand in it. :P)

Title: Detecting Political Biases of Named Entities and Hashtags on Twitter

Authors: Zhiping Xiao, Jeffrey Zhu, Yining Wang, Pei Zhou, Wen Hong Lam, Mason A. Porter, and Yizhou Sun

Abstract: Ideological divisions in the United States have become increasingly prominent in daily communication. Accordingly, there has been much research on political polarization, including many recent efforts that take a computational perspective. By detecting political biases in a text document, one can attempt to discern and describe its polarity. Intuitively, the named entities (i.e., the nouns and the phrases that act as nouns) and hashtags in text often carry information about political views. For example, people who use the term “pro-choice” are likely to be liberal and people who use the term “pro-life” are likely to be conservative. In this paper, we seek to reveal political polarities in social-media text data and to quantify these polarities by explicitly assigning a polarity score to entities and hashtags. Although this idea is straightforward, it is difficult to perform such inference in a trustworthy quantitative way. Key challenges include the small number of known labels, the continuous spectrum of political views, and the preservation of both a polarity score and a polarity-neutral semantic meaning in an embedding vector of words. To attempt to overcome these challenges, we propose the Polarity-aware Embedding Multi-task learning (PEM) model. This model consists of (1) a self-supervised context-preservation task, (2) an attention-based tweet-level polarity-inference task, and (3) an adversarial learning task that promotes independence between an embedding’s polarity component and its semantic component. Our experimental results demonstrate that our PEM model can successfully learn polarity-aware embeddings that perform well at tweet-level and account-level classification tasks. We examine a variety of applications—including a study of spatial and temporal distributions of polarities and a comparison between tweets from Twitter and posts from Parler—and we thereby demonstrate the effectiveness of our PEM model. We also discuss important limitations of our work and encourage caution when applying the PEM model to real-world scenarios.

"An Adaptive Bounded-Confidence Model of Opinion Dynamics on Networks "

An article of mine just appeared in final form a couple of days ago. Here are some details.

Title: An Adaptive Bounded-Confidence Model of Opinion Dynamics on Networks

Authors: Unchitta Kan, Michelle Feng, and Mason A. Porter

Abstract: Individuals who interact with each other in social networks often exchange ideas and influence each other’s opinions. A popular approach to study the spread of opinions on networks is by examining bounded-confidence models (BCMs), in which the nodes of a network have continuous-valued states that encode their opinions and are receptive to other nodes’ opinions when they lie within some confidence bound of their own opinion. In this article, we extend the Deffuant–Weisbuch (DW) model, which is a well-known BCM, by examining the spread of opinions that coevolve with network structure. We propose an adaptive variant of the DW model in which the nodes of a network can (1) alter their opinions when they interact with neighbouring nodes and (2) break connections with neighbours based on an opinion tolerance threshold and then form new connections following the principle of homophily. This opinion tolerance threshold determines whether or not the opinions of adjacent nodes are sufficiently different to be viewed as ‘discordant’. Using numerical simulations, we find that our adaptive DW model requires a larger confidence bound than a baseline DW model for the nodes of a network to achieve a consensus opinion. In one region of parameter space, we observe ‘pseudo-consensus’ steady states, in which there exist multiple subclusters of an opinion cluster with opinions that differ from each other by a small amount. In our simulations, we also examine the roles of early-time dynamics and nodes with initially moderate opinions for achieving consensus. Additionally, we explore the effects of coevolution on the convergence time of our BCM.