"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.

"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.

"Clustering-Induced Localization of Quantum Walks on Networks"

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

Title: Clustering-Induced Localization of Quantum Walks on Networks

Authors: Lucas Böttcher and Mason A. Porter

"Bounded-Confidence Models of Opinion Dynamics with Neighborhood Effects"

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

"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:

"Perceived Community Alignment Increases Information Sharing"

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

Title: Perceived Community Alignment Increases Information Sharing

Authors: Elisa C. Baek, Ryan Hyon, Karina López, Mason A. Porter, and Carolyn Parkinson

Abstract: 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 test 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 data collected while people who live in the same residential community viewed brief film clips, we find that more similar neural responses across participants is associated with a greater likelihood to share content. We then test this relationship using two behavioral studies and find (1) that people are particularly likely to share content that they believe others in their social circles will 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.

"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

"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

"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.

"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.

"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.

"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.

"Human-Network Regions as Effective Geographic Units for Disease Mitigation"

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

Title: "Human-Network Regions as Effective Geographic Units for Disease Mitigation"

Authors: Clio Andris, Caglar Koylu, and Mason A. Porter

Abstract: Susceptibility to infectious diseases such as COVID-19 depends on how those diseases spread. Many studies have examined the decrease in COVID-19 spread due to reduction in travel. However, less is known about how much functional geographic regions, which capture natural movements and social interactions, limit the spread of COVID-19. To determine boundaries between functional regions, we apply community-detection algorithms to large networks of mobility and social-media connections to construct geographic regions that reflect natural human movement and relationships at the county level in the coterminous United States. We measure COVID-19 case counts, case rates, and case-rate variations across adjacent counties and examine how often COVID-19 crosses the boundaries of these functional regions. We find that regions that we construct using GPS-trace networks and especially commute networks have the lowest COVID-19 case rates along the boundaries, so these regions may reflect natural partitions in COVID-19 transmission. Conversely, regions that we construct from geolocated Facebook friendships and Twitter connections yield less effective partitions. Our analysis reveals that regions that are derived from movement flows are more appropriate geographic units than states for making policy decisions about opening areas for activity, assessing vulnerability of populations, and allocating resources. Our insights are also relevant for policy decisions and public messaging in future emergency situations.

"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.