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In models of opinion dynamics, the opinions of individual agents evolve with time. 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 those opinions are sufficiently similar. In studies of BCMs, it is typically assumed that interactions between agents occur at deterministic times. This assumption neglects an inherent element of randomness in social systems. 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 interactions, which can follow arbitrary waiting-time distributions (WTDs). We establish connections between these random-time-interaction BCMs and deterministic-time-interaction BCMs. We find that BCMs with Markovian WTDs have consistent statistical properties on different networks but that the statistical properties of BCMs with non-Markovian WTDs depend on network structure., Comment: 9 pages, 5 figures

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., Comment: 95-page review article

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 can possess a collective opinion that may differ from the opinions of the 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., Comment: 15 pages, 9 figures, 1 table

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 $\GL^{W_n}_\ep$ 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 $\Gamma$-converges to a continuous and nonlocal functional that we call the \emph{graphon GL functional}. We also investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total variation. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the variational problem in the large-graph limit. Finally, to develop intuition about the graphon GL functional, we compute the GL minimizer for several example families of graphons., Comment: 37 pages

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), in which we incorporate 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 optimum minimizing exit time {\tau} . We substantiate this finding by using mean-field theory on annealed uniform hypergraphs with N nodes, for which {\tau} scales as A ln N, where the prefactor 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., Comment: 6 pages, 5 figures

Each year, the United States Congress considers {thousands of legislative proposals to select bills} to present to the US President to sign into law. Naturally, the decision processes of members of Congress are subject to peer influence. In this paper, we examine the effect on bill passage of accrued influence between US Congress members in the US House of Representatives. We explore how the influence of a bill's cosponsors affects the bill's outcome (specifically, whether or not it passes in the House). We define a notion of influence by analyzing the structure of a network that we construct {using} cosponsorship dynamics. We award `influence' between a pair of Congress members when they cosponsor a bill that achieves some amount of legislative success. We find that properties of the bill cosponsorship network can be a useful signal to examine influence in Congress; they help explain why some bills pass and others fail. We compare our measure of influence to off-the-shelf centrality measures and conclude that our influence measure is more indicative of bill passage., Comment: submitted

Social interactions influence people's opinions. In some situations, these interactions result in a consensus opinion; in others, they result in opinion fragmentation and the formation of different opinion groups in the form of "echo chambers". Consider a social network of individuals, who hold continuous-valued scalar opinions and change their opinions when they interact with each other. In such an opinion model, it is common for an opinion-update rule to depend on the mean opinion of interacting individuals. However, we consider an alternative update rule - which may be more realistic in some situations - that instead depends on a weighted median opinion of interacting individuals. Through numerical simulations of our opinion model, we investigate how the limit opinion distribution depends on network structure. For configuration-model networks, we also derive a mean-field approximation for the asymptotic dynamics of the opinion distribution when there are infinitely many individuals in a network., Comment: 30 pages, 13 figures, Submitted to SIAM Journal on Applied Dynamical Systems

The leading eigenvalue $\lambda$ 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 $\lambda$ and its associated eigenvector. We study a previously derived measure of edge importance known as ``dynamical importance'', which estimates how much $\lambda$ 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 $\lambda$ 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., Comment: 12 pages; revised version

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-spread model introduces a twist into BCMs by using bounded confidence for the content spread itself. To study the spread of content, 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 will die off or propagate widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination using 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 large expected mean degree or large receptiveness to content. When the amount of content spread only slightly exceeds the critical opinion reproduction number (i.e., the infodemic threshold), there can be longer dissemination trees than when the expected mean degree or receptiveness is larger, even though the total number of content shares is smaller., Comment: 14 pages; submitted to \emph{Chaos} for the special issue for the 80th birthday of David Campbell. (Happy birthday, David!)

As people's opinions change, their social networks typically coevolve with them. People are often more susceptible to influence by people with similar opinions than by people with dissimilar opinions. In a bounded-confidence model (BCM) of opinion dynamics, interacting individuals influence each other through dyadic influence if and only if their opinions are sufficiently similar to each other. We introduce `neighborhood BCMs' (NBCMs) that include both the usual dyadic influence and a transitive influence, which models the effect of friends of a friend when determining whether or not an interaction with a friend influences an individual. 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 their own opinion. We formulate neighborhood Deffuant--Weisbuch (NDW) and neighborhood Hegselmann--Krause (NHK) BCMs. We simulate our NDW model on time-independent networks and observe interesting opinion states that cannot occur in an associated baseline DW model. We also simulate our NDW model on adaptive networks that coevolve with opinions by changing its structure through `transitive homophily'. An individual that breaks a tie to one of its neighbors and then rewires that tie to a new individual, with a preference for individuals with a mean neighbor opinion that is closer to that individual's opinion. We explore how the qualitative opinion dynamics and network properties of our time-independent and adaptive NDWM models change as we adjust the relative proportions of dyadic and transitive influence. Finally, we study a two-layer opinion--disease model in which we couple our NDW model with disease spread through a shared adaptive network that can change both on the opinion layer and on the disease layer and we examine how the opinion dynamics affect disease spread., Comment: 21 pages, 6 figures; the abstract on the arXiv page is abbreviated in parts because of character-count limitations

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 Schr\"odinger 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., Comment: 33 pages, 12 figures

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 \textit{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 electro-mechanical oscillators, and cooperation in cattle herds., Comment: 63 pages, 26 figures

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., Comment: This article is written for teenagers and preteens. A newly revised version, based on copy-editor comments and feedback and our subsequent revisions to address them

Information sharing is a ubiquitous and consequential behavior that has been proposed to play 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 behavioral studies and found (1) that people were particularly likely to share content about which they believed others in their social circles would share their viewpoints and (2) that this relationship is causal. 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., Comment: 44 pages, including main text + supplementary information

The computational investigation of Fermi, Pasta, Ulam, and Tsingou of arrays of nonlinearly coupled oscillators has led to a wealth of studies in nonlinear dynamics. Most studies of oscillator arrays have considered homogeneous oscillators, even though there are inherent heterogeneities between {individual} oscillators in real-world arrays. Well-known FPUT phenomena, such as energy recurrence, can break down in such heterogeneous systems. In this paper, we present an approach -- the use of structured heterogeneities -- to recover recurrence in FPUT systems in the presence of oscillator heterogeneities. We examine oscillator variabilities in FPUT systems with cubic nonlinearities, and we demonstrate that centrosymmetry in oscillator arrays may be an important source of recurrence., Comment: revised version

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., Comment: a few minor edits for clarity; 45 pages

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., Comment: 20 pages, 3 figures

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., Comment: 17 pages, 7 figures, 1 table

Media bias can significantly impact the formation and development of opinions and sentiments in a population. It is thus important to study the emergence and development of partisan media and political polarization. However, it is challenging to quantitatively infer the ideological positions of media outlets. In this paper, we present a quantitative framework to infer both political bias and content quality of media outlets from text, and we illustrate this framework with empirical experiments with real-world data. We apply a bidirectional long short-term memory (LSTM) neural network to a data set of more than 1 million tweets to generate a two-dimensional ideological-bias and content-quality measurement for each tweet. We then infer a ``media-bias chart'' of (bias, quality) coordinates for the media outlets by integrating the (bias, quality) measurements of the tweets of the media outlets. We also apply a variety of baseline machine-learning methods, such as a naive-Bayes method and a support-vector machine (SVM), to infer the bias and quality values for each tweet. All of these baseline approaches are based on a bag-of-words approach. We find that the LSTM-network approach has the best performance of the examined methods. Our results illustrate the importance of leveraging word order into machine-learning methods in text analysis., Comment: 21 pages, 7 figures, 4 tables

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 corpus of text, one can attempt to describe and discern the polarity of that text. 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, whereas 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 dimension and its semantic dimensions. Our experimental results demonstrate that our PEM model can successfully learn polarity-aware embeddings that perform well classification tasks. We examine a variety of applications and we thereby demonstrate the effectiveness of our PEM model. We also discuss important limitations of our work and encourage caution when applying the it to real-world scenarios., Comment: Submitted to EPJ -- Data Science, under review

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 $\gamma$, 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 $\gamma = 0$, 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 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., Comment: 29 pages, 7 figures

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 are have non-Markovian dynamics. We derive an opinion model that is governed by an arbitrary waiting-time distribution (WTD) and 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 to a steady state more slowly than those with light tails (or with compact support) and that entities with larger waiting times exert a larger influence on the mean opinion at steady state., Comment: 24 pages, 7 figures

We investigate a majority-vote model on two-layer multiplex networks with community structure. In our majority-vote model, the edges on each layer encode one type of social relationship and an individual changes their opinion based on the majority opinions of their neighbors in each layer. To capture the fact that different relationships often have different levels of importance, we introduce a layer-preference parameter, which determines the probability of a node to adopt an opinion when the node's neighborhoods on the two layers have different majority opinions. We construct our networks so that each node is a member of one community on each layer, and we consider situations in which nodes tend to have more connections with nodes from the same community than with nodes from different communities. We study the influence of the layer-preference parameter, the intralayer communities, and interlayer membership correlation on the steady-state behavior of our model using both direct numerical simulations and a mean-field approximation. We find three different types of steady-state behavior: a fully-mixed state, consensus states, and polarized states. We demonstrate that a stronger interlayer community correlation makes polarized steady states reachable for wider ranges of the other model parameters. We also show that different values of the layer-preference parameter result in qualitatively different phase diagrams for the mean opinions at steady states.

Agent-based models of opinion dynamics allow one to examine the spread of opinions between entities and to study phenomena such as consensus, polarization, and fragmentation. By studying a model of opinion dynamics on a social network, one can explore the effects of network structure on these phenomena. In social networks, some individuals share their ideas and opinions more frequently than others. These disparities can arise from heterogeneous sociabilities, heterogeneous activity levels, different prevalences to share opinions when engaging in a social-media platform, or something else. To examine the impact of such heterogeneities on opinion dynamics, we generalize the Deffuant--Weisbuch (DW) bounded-confidence model (BCM) of opinion dynamics by incorporating node weights. The node weights allow us to model agents with different probabilities of interacting. Using numerical simulations, we systematically investigate (using a variety of network structures and node-weight distributions) the effects of node weights, which we assign uniformly at random to the nodes. We demonstrate that introducing heterogeneous node weights results in longer convergence times and more opinion fragmentation than in a baseline DW model. The node weights in our BCM allow one to consider a variety of sociological scenarios in which agents have heterogeneous probabilities of interacting with other agents., Comment: revised version, 11 figures

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., Comment: revised version

Social interactions often occur between three or more agents simultaneously. Examining opinion dynamics on hypergraphs allows one to study the effect of such polyadic interactions on the opinions of agents. In this paper, we consider a bounded-confidence model (BCM), in which opinions take continuous values and interacting agents comprise their opinions if they are close enough to each other. We study a density description of a Deffuant--Weisbuch BCM on hypergraphs. We derive a rate equation for the mean-field opinion density as the number of agents becomes infinite, and we prove that this rate equation yields a probability density that converges to noninteracting opinion clusters. Using numerical simulations, we examine bifurcations of the density-based BCM's steady-state opinion clusters and demonstrate that the agent-based BCM converges to the density description of the BCM as the number of agents becomes infinite., Comment: 19 pages, 4 figures

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., Comment: new Lyapunov analysis (including 2 new figures)

We consider generalizations of nonlinear Schr\"odinger equations, which we call "Karpman equations", that include additional linear higher-order derivatives. Singularly-perturbed Karpman equations produce generalized solitary waves (GSWs) in the form of solitary waves with exponentially small oscillatory tails. Nanoptera are a special case of GSWs in which these oscillatory tails do not decay. Previous research on continuous third-order and fourth-order Karpman equations has shown that nanoptera occur in specific settings. We use exponential asymptotic techniques to identify traveling nanoptera in singularly-perturbed continuous Karpman equations. We then study the effect of discretization on nanoptera by applying a finite-difference discretization to continuous Karpman equations and studying traveling-wave solutions. The finite-difference discretization turns a continuous Karpman equation into an advance--delay equation, which we study using exponential asymptotic analysis. By comparing nanoptera in these discrete Karpman equations with nanoptera in their continuous counterparts, we show that the oscillation amplitudes and periods in the nanoptera tails differ in the continuous and discretized equations. We also show that the parameter values at which there is a bifurcation between nanopteron and decaying oscillatory solutions depends on the choice of discretization. Finally, by comparing different higher-order discretizations of the fourth-order Karpman equation, we show that the bifurcation value tends to a nonzero constant for large orders, rather than to $0$ as in the associated continuous Karpman equation., Comment: 14 figures, 42 pages, 4 tables

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., Comment: A few more small tweaks. This is a book chapter. It is associated with my data-ethics lecture at the 2021 AMS Short Course on Mathematical and Computational Methods for Complex Social Systems

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.

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 paper, 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 neighboring nodes and (2) break connections with neighbors 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 importance 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., Comment: accepted by Journal of Complex Networks; further cosmetic polishing from the last version

We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a data set and quantify the persistence of such voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (which are an approach for visualizing PH), to track how the locations of the anomalies change with time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code at a single point in time. Second, we study a year-long data set of COVID-19 case rates in neighborhoods of the city of Los Angeles., Comment: revised version

During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we construct a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. On one layer, we consider the concurrent evolution of two opinions -- pro-physical-distancing and anti-physical-distancing -- that compete with each other and have mutual immunity to each other. The disease evolves on the other layer, and individuals are less likely (respectively, more likely) to become infected when they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. Through numerical simulations, we illustrate the influence of opinion dynamics on the spread of the disease from complex interactions both between the two conflicting opinions and between the opinions and the disease. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

Loneliness is detrimental to well-being and is often accompanied by self-reported feelings of not being understood by others. What contributes to such feelings in lonely people? We used functional magnetic resonance imaging (fMRI) of 66 participants to unobtrusively measure the relative alignment of people's mental processing of naturalistic stimuli and tested whether or not lonely people actually process the world in idiosyncratic ways. We found evidence for such idiosyncrasy: lonely individuals' neural responses were dissimilar to their peers, particularly in regions of the default-mode network in which similar responses have been associated with shared perspectives and subjective understanding. These relationships persisted when controlling for demographic similarities, objective social isolation, and participants' friendships with each other. Our findings suggest the possibility that being surrounded by people who see the world differently from oneself, even if one is friends with them, may be a risk factor for loneliness., Comment: revised version. arXiv admin note: text overlap with arXiv:2106.02726

People differ in how they attend to, interpret, and respond to their surroundings. Convergent processing of the world may be one factor that contributes to social connections between individuals. We used neuroimaging and network analysis to investigate whether the most central individuals in their communities (as measured by in-degree centrality, a notion of popularity) process the world in a particularly normative way. We found that more central individuals had exceptionally similar neural responses to their peers and especially to each other in brain regions that are associated with high-level interpretations and social cognition (e.g., in the default-mode network), whereas less-central individuals exhibited more idiosyncratic responses. Self-reported enjoyment of and interest in stimuli followed a similar pattern, but accounting for these data did not change our main results. These findings suggest that highly-central individuals process the world in exceptionally similar ways, whereas less-central individuals process the world in idiosyncratic ways., Comment: revised version; title changed. arXiv admin note: text overlap with arXiv:2107.01312

In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review the recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two cases studies: street networks in Shanghai and hotspots of COVID-19 infections. We then summarize our results and provide an outlook on TDA in spatial systems., Comment: draft of book chapter

Dynamic-mode decomposition (DMD) is a versatile framework for model-free analysis of time series that are generated by dynamical systems. We develop a DMD-based algorithm to investigate the formation of "functional communities" in networks of coupled, heterogeneous Kuramoto oscillators. In these functional communities, the oscillators in the network have similar dynamics. We consider two common random-graph models (Watts--Strogatz networks and Barab\'asi--Albert networks) with different amounts of heterogeneities among the oscillators. In our computations, we find that membership in a community reflects the extent to which there is establishment and sustainment of locking between oscillators. We construct forest graphs that illustrate the complex ways in which the heterogeneous oscillators associate and disassociate with each other.

We study ``nanoptera'', which are non-localized solitary waves with exponentially small but non-decaying oscillations, in two singularly-perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from Stokes phenomena and appear as special curves, called Stokes curves, are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling waves in a singularly perturbed woodpile chain have a single Stokes curve, across which oscillations appear. Comparing these asymptotic predictions with numerical simulations reveals that this accurately describes the non-decaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, that the nanpteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing asymptotic and numerical results in a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions., Comment: 29 pages, 9 figures

It is common to use networks to encode the architecture of interactions between entities in complex systems in the physical, biological, social, and information sciences. 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. We present a new approach for describing low-rank mesoscale structures in networks, and we illustrate our approach using several synthetic network models and empirical friendship, collaboration, and protein--protein interaction (PPI) networks. We find that these networks possess a relatively small number of `latent motifs' that together can successfully approximate most subgraphs of a network at a fixed mesoscale. We use an algorithm for `network dictionary learning' (NDL), which combines a network-sampling method and nonnegative matrix factorization, to learn the latent motifs of a given network. The ability to encode a network using a set of latent motifs has a wide variety of applications to network-analysis tasks, such as comparison, denoising, and edge inference. Additionally, using a new network denoising and reconstruction (NDR) algorithm, we demonstrate how to denoise a corrupted network by using only the latent motifs that one learns directly from the corrupted network., Comment: 82 pages, 25 figures, 2 tables

People's opinions evolve over time as they interact with their friends, family, colleagues, and others. In the study of opinion dynamics on networks, one often encodes interactions between people in the form of dyadic relationships, but many social interactions in real life are polyadic (i.e., they involve three or more people). In this paper, we extend an asynchronous bounded-confidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to an asynchronous BCM on hypergraphs, in which arbitrarily many nodes can be connected by a single hyperedge. We show that our hypergraph BCM converges to consensus under a wide range of initial conditions for the opinions of the nodes, including for non-uniform and asymmetric initial opinion distributions. We also show that, under suitable conditions, echo chambers can form on hypergraphs with community structure. We demonstrate that the opinions of individuals can sometimes jump from one opinion cluster to another in a single time step, a phenomenon (which we call ``opinion jumping'') that is not possible in standard dyadic BCMs. Additionally, we observe that there is a phase transition in the convergence time on {a complete hypergraph} when the variance $\sigma^2$ of the initial opinion distribution equals the confidence bound $c$. We prove that the convergence time grows at least exponentially fast with the number of nodes when $\sigma^2 > c$ and the initial opinions are normally distributed. Therefore, to determine the convergence properties of our hypergraph BCM when the variance and the number of hyperedges are both large, it is necessary to use analytical methods instead of relying only on Monte Carlo simulations., Comment: revised version

A major strategy to prevent the spread of COVID-19 is the limiting of in-person contacts. However, this is impractical or impossible for the many disabled people who do not live in care facilities, but still require caregivers. We seek to determine which interventions can prevent infections among disabled people and their caregivers. We simulate transmission with a model that includes susceptible, exposed, asymptomatic, symptomatically ill, hospitalized, and removed individuals. The networks on which we simulate disease spread incorporate heterogeneity in the risks of different types of interactions, time-dependent lockdown and reopening measures, and contact distributions for four different groups (caregivers, disabled people, essential workers, and the general population). We find the probability of becoming infected is largest for caregivers and second largest for disabled people. Our analysis of network structure illustrates that caregivers have the largest modal eigenvector centrality. We find that two interventions -- contact-limiting by all groups and mask-wearing by disabled people and caregivers -- most reduce the cases among disabled people and caregivers. We also test which group spreads COVID-19 most readily by seeding infections in a subset of each group. We find caregivers are the most potent spreaders of COVID-19, particularly to other caregivers and to disabled people. We test where to use limited vaccine doses most effectively and find (1) vaccinating caregivers better protects disabled people than vaccinating the general population or essential workers and (2) vaccinating caregivers protects disabled people about as much as vaccinating disabled people themselves. Our results highlight the potential effectiveness of mask-wearing, contact-limiting throughout society, and strategic vaccination for limiting the exposure of disabled people and their caregivers to COVID-19., Comment: 44 pages, 14 figures

Multilayer network analysis is a useful approach for studying the structural properties of entities with diverse, multitudinous relations. Classifying the importance of nodes and node-layer tuples is an important aspect of the study of multilayer networks. To do this, it is common to calculate various centrality measures, which allow one to rank nodes and node-layers according to a variety of structural features. In this paper, we formulate occupation, PageRank, betweenness, and closeness centralities in terms of node-occupation properties of different types of continuous-time classical and quantum random walks on multilayer networks. We apply our framework to a variety of synthetic and real-world multilayer networks, and we identify marked differences between classical and quantum centrality measures. Our computations also give insights into the correlations between certain random-walk-based and geodesic-path-based centralities., Comment: 21 pages, 8 figures

Traveling to different destinations is a big part of our lives. We visit a variety of locations both during our daily lives and when we're on vacation. How can we find the best way to navigate from one place to another? Perhaps we can test all of the different ways of traveling between two places, but another method is to use mathematics and computation to find a shortest path. We discuss how to construct a shortest path and introduce Dijkstra's algorithm to minimize the total cost of a path, where the cost may be the travel distance, travel time, or some other measurement. We also discuss how to use shortest paths in the real world to save time and increase traveling efficiency., Comment: This is an article for teens and pre-teens. Let us know if you have any comments

In social networks, interaction patterns typically change over time. We study opinion dynamics on tie-decay networks in which tie strength increases instantaneously when there is an interaction and decays exponentially between interactions. Specifically, we formulate continuous-time Laplacian dynamics and a discrete-time DeGroot model of opinion dynamics on these tie-decay networks, and we carry out numerical computations for the continuous-time Laplacian dynamics. We examine the speed of convergence by studying the spectral gaps of combinatorial Laplacian matrices of tie-decay networks. First, we compare the spectral gaps of the Laplacian matrices of tie-decay networks that we construct from empirical data with the spectral gaps for corresponding randomized and aggregate networks. We find that the spectral gaps for the empirical networks tend to be smaller than those for the randomized and aggregate networks. Second, we study the spectral gap as a function of the tie-decay rate and time. Intuitively, we expect small tie-decay rates to lead to fast convergence because the influence of each interaction between two nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interactions occur, we expect eventual convergence. However, we demonstrate that the spectral gap need not decrease monotonically with respect to the decay rate or increase monotonically with respect to time. Our results highlight the importance of the interplay between the times that edges strengthen and decay in temporal networks., Comment: 15 pages, 8 figures, 2 tables

In the study of infectious diseases on networks, researchers calculate epidemic thresholds to help forecast whether a disease will eventually infect a large fraction of a population. Because network structure typically changes in time, which fundamentally influences the dynamics of spreading processes on them and in turn affects epidemic thresholds for disease propagation, it is important to examine epidemic thresholds in temporal networks. Most existing studies of epidemic thresholds in temporal networks have focused on models in discrete time, but most real-world networked systems evolve continuously in time. In our work, we encode the continuous time-dependence of networks into the evaluation of the epidemic threshold of a susceptible--infected--susceptible (SIS) process by studying an SIS model on tie-decay networks. We derive the epidemic-threshold condition of this model, and we perform numerical experiments to verify it. We also examine how different factors---the decay coefficients of the tie strengths in a network, the frequency of interactions between nodes, and the sparsity of the underlying social network in which interactions occur---lead to decreases or increases of the critical values of the threshold and hence contribute to facilitating or impeding the spread of a disease. We thereby demonstrate how the features of tie-decay networks alter the outcome of disease spread., Comment: revised version, 24 pages