Models of opinion dynamics aim to capture how individuals' opinions change when they interact with each other. One well-known model of opinion dynamics is the Deffuant–Weisbuch (DW) model, which is a type of bounded-confidence model (BCM). In the DW model, agents have pairwise interactions, and they are receptive to other agents' opinions when their opinions are sufficiently close to each other. In this paper, we extend the DW model by studying it on networks with heterogeneous and adaptive edge weights between pairs of agents. These edge weights govern the interaction probabilities between the agents and thereby encode the idea that people are more likely to communicate with individuals with whom they have previously compromised or had other positive interactions. We prove theoretical guarantees of our adaptive edge-weighted DW model's convergence properties, the long-time dynamics of its edge weights, and the model's associated “effective graph", which is a time-dependent subgraph that includes edges only between agents that are receptive to each other's opinions. We support our theoretical results with numerical simulations of our adaptive edge-weighted DW model on a variety of networks and find that including adaptive edge weights yields different qualitative dynamics for different types of networks. In particular, for small confidence bounds, we observe that incorporating adaptive edge weights decreases the convergence time for dense networks but increases the convergence time for sparse networks.

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.

In the study of time-dependent (i.e., temporal) networks, researchers often examine the evolution of communities, which are sets of densely connected sets of nodes that are connected sparsely to other nodes. An increasingly prominent approach to studying community structure in temporal networks is statistical inference. In the present paper, we study the performance of a class of statistical-inference methods for community detection in temporal networks. We represent temporal networks as multilayer networks, with each layer encoding a time step, and we illustrate that statistical-inference models that generate community assignments via either a uniform distribution on community assignments or discrete-time Markov processes are biased against generating communities with large or small numbers of nodes. In particular, we demonstrate that statistical-inference methods that use such generative models tend to poorly identify community structure in networks with large or small communities. To rectify this issue, we introduce a novel statistical model that generates the community assignments of the nodes in given layer (i.e., at a given time) using all of the community assignments in the previous layer. We prove results that guarantee that our approach greatly mitigates the bias against large and small communities, so using our generative model is beneficial for studying community structure in networks with large or small communities. Our code is available at https://github.com/tfaust0196/TemporalCommunityComparison.

We examine the spread of an infectious disease, such as one that is caused by a respiratory virus, with two distinct modes of transmission. To do this, we consider a susceptible–infected–susceptible (SIS) disease on a hypergraph, which allows us to incorporate the effects of both dyadic (i.e., pairwise) and polyadic (i.e., group) interactions on disease propagation. This disease can spread either via large droplets through direct social contacts, which we associate with edges (i.e., hyperedges of size 2), or via infected aerosols in the environment through hyperedges of size at least 3 (i.e., polyadic interactions). We derive mean-field approximations of our model for two types of hypergraphs, and we obtain threshold conditions that characterize whether the disease dies out or becomes endemic. Additionally, we numerically simulate our model and a mean-field approximation of it to examine the impact of various factors, such as hyperedge size (when the size is uniform), hyperedge-size distribution (when the sizes are nonuniform), and hyperedge-recovery rates (when the sizes are nonuniform) on the disease dynamics.

People's opinions can change both from their interactions with each other and from their interactions with media sources. Bounded-confidence models (BCMs) of opinion dynamics provide one framework to study such dynamics. In a BCM, the nodes of a network are agents with continuous-valued opinions, and these agents interact with each other via the edges of the network. In this paper, we extend the original Deffuant–Weisbuch (DW) BCM by incorporating influence from two media sources – one with a positive value and one with a negative value – to capture the effects of a polarized media landscape. We show both numerically and analytically that our extended DW model exhibits drifting behavior in which a large cluster of opinions shifts toward one of the media agents. We analyze how the drift trajectory and speed depend on the model parameters, and we identify conditions in which drift is promoted or suppressed. Our results provide insight into how competing media sources can influence collective opinion formation in social systems.

A fundamental concept in linear algebra and its applications to network analysis is the Perron–Frobenius (PF) theorem, which underpins eigenvector-based centrality measures such as eigenvector centrality, PageRank, and hubs and authorities. By invoking the PF theorem, we know for strongly connected networks with positive edge weights that the eigenvector corresponding to the largest eigenvalue of the weight matrix yields a well-defined centrality measure (namely, eigenvector centrality). Traditional formulations of the PF theorem and associated centrality measures assume that networks have real-valued weights. However, many networks in areas such as quantum information, quantum chemistry, electrodynamics, and machine learning have complex-valued edge weights. In this paper, we study generalizations of the PF theorem to complex-valued matrices, establish connections between these generalizations, and propose generalized eigenvector-based centrality measures to analyzing node importances in networks with complex edge weights. We also prove results about the existence of complex-weighted networks that satisfy generalized PF properties and calculate associated centrality measures for several examples, which we draw from application areas such as electron transport, circuit analysis, mathematical chemistry, and communication networks.

A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called “detectability limit”, no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős–Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

Quantum walks on networks are a paradigmatic model in quantum information theory. Quantum-walk algorithms have been developed for various applications, including spatial-search problems, element-distinctness problems, and node-centrality analysis. Unlike their classical counterparts, the evolution of quantum walks is unitary, so they do not converge to a stationary distribution. However, for many applications, it is important to understand the long-time behavior of quantum walks and the impact of network structure on their evolution. In the present paper, we study localization of continuous-time quantum walks on networks. We derive an analytical expression for the long-time inverse participation ratio that depends on combinations of eigenvectors of the quantum-walk Hamiltonian. We then demonstrate how localization emerges in highly clustered networks that we construct by recursively attaching triangles. Building on the insights from this example, we then show that localization also occurs in Kleinberg navigable small-world networks and Holme-Kim power-law cluster networks. Our results illustrate that local clustering, which is a key structural feature of networks, can induce localization of quantum walks.

Networks can have various types of mesoscale structures. One type of mesoscale structure in networks is core-periphery structure, which consists of densely-connected core nodes and sparsely-connected peripheral nodes. The core nodes are connected densely to each other and can be connected to the peripheral nodes, which are connected sparsely to other nodes. There has been much research on core-periphery structure in time-independent networks, but few core-periphery detection methods have been developed for time-dependent (i.e., “temporal") networks. Using a multilayer-network representation of temporal networks and an inference approach that employs stochastic block models, we generalize a recent method for detecting hierarchical core-periphery structure from time-independent networks to temporal networks. In contrast to “onion-like” nested core-periphery structures (where each node is assigned to a group according to how deeply it is nested in a network's core), hierarchical core-periphery structures encompass networks with nested structures, tree-like structures (where any two groups must either be disjoint or have one as a strict subset of the other), and general non-nested mesoscale structures (where the group assignments of nodes do not have to be nested in any way). To perform statistical inference and thereby identify core-periphery structure, we use a Markov-chain Monte Carlo (MCMC) approach. We illustrate our method for detecting hierarchical core-periphery structure in two real-world temporal networks, and we briefly discuss the structures that we identify in these networks.

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

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 only by neighboring nodes whose opinions are within their 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) 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 can currently influence each other. For a wide range of parameters that control the increase and decrease of confidence bounds, we demonstrate for a variety of networks 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 pairs of adjacent nodes that converge to the same opinion but are not able to influence each other. This qualitative behavior does not occur in the associated baseline BCMs.

Social dynamics are often driven by both pairwise (i.e., dyadic) relationships and higher-order (i.e., polyadic) group relationships, which one can describe using hypergraphs. To gain insight into the impact of polyadic relationships on dynamical processes on networks, we formulate and study a polyadic voter process, which we call the group-driven voter model (GVM), that incorporates the effects of group dynamics through nonlinear interactions that are subject to a group (i.e., hyperedge) constraint. By examining the competition between nonlinearity and group sizes, we show that the GVM achieves consensus faster than standard voter-model dynamics, with an optimal minimizing exit time. We substantiate this finding by using mean-field theory on annealed uniform hypergraphs with N nodes, for which the exit time scales as AlnN, 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.

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. Our content-spreading model, which one can also interpret as an independent-cascade model, introduces a twist into BCMs by using bounded confidence for the content spread itself. We define an analog of the basic reproduction number from disease dynamics that we call an opinion reproduction number. A critical value of the opinion reproduction number indicates whether or not there is an "infodemic" (i.e., a large content-spreading cascade) of content that reflects a particular opinion. By determining this critical value, one can determine whether or not an opinion dies off or propagates widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination by calculating a variety of summary statistics, and we illustrate how network structure and spreading-model parameters affect these statistics. We find that content spreads most widely when agents have a large expected mean degree or a large receptiveness to content. When the spreading process slightly exceeds the infodemic threshold, there can be longer dissemination trees than for larger expected mean degrees or receptiveness (which both promote content sharing and hence help push content spread past the infodemic threshold), even though the total number of content shares is smaller.

When we look at the world around us, we see both organized (also called ordered) and disorganized (also called disordered) arrangements of things. Carefully-tiled floors and brick walls have organized and repeating patterns, but the stars in the sky and the trees in a forest look like they're arranged in a disordered way. We also see objects, like jars of marbles and the lacy wings of insects, that lie between ordered and disordered extremes. Although the marbles in a jar don't sit on a regular grid like carefully-arranged tiles, the collection of marbles does have some consistent features, such as the typical size and spacing between them. However, the positions of the marbles are much less random than the positions of the stars in the sky. To help understand and classify these patterns, mathematicians and physicists use the term hyperuniform to help them describe the situations of being perfectly organized or being disorganized in an organized way. In this article, we discuss various fascinating properties of hyperuniform patterns. We explore where they occur in the natural world and how engineers are using them to build new structures.

People's opinions evolve through social interactions, and their social networks evolve as their opinions change. In this paper, we generalize bounded-confidence models of opinion dynamics by incorporating neighborhood effects. In our neighborhood bounded-confidence models (NBCMs), interacting agents are influenced both by each other's opinions and by the opinions of the agents in each other's neighborhoods (through so-called "transitive influence"). We extend our NBCMs to adaptive network models that incorporate "transitive homophily", with agents rewiring to connect preferentially to agents whose mean neighbor opinions are close to their own opinions. We numerically simulate an asynchronously updating adaptive NBCM on a variety of networks, and we explore how its opinion dynamics and structural properties change as we adjust the relative importances of dyadic and transitive influence. We observe that the transitive influence of neighborhoods can play an important role in shaping the qualitative features of opinion dynamics.

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 that one can describe using coupled ordinary differential equations, difference equations, stochastic processes, or agent-based models. 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 a network's 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 insightful for analyzing the interplay between structure and dynamics in metric networks. We start by considering a time-independent Schro"\dinger 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 PDEs on metric networks. In our numerical experiments, we consider metric networks with up to about 104 nodes and 104 edges. Additionally, using our spectral approach, we are able to resolve degenerate eigenmo des. 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.

Hyperuniformity, which is a type of long-range order that is characterized by the suppression of long-range density fluctuations in comparison to the fluctuations in standard disordered systems, has emerged as a powerful concept to aid in the understanding of diverse natural and engineered phenomena. In the present paper, we harness hyperuniform point patterns to generate a class of disordered, spatially embedded networks that are distinct from both perfectly ordered lattices and uniformly random geometric graphs. We refer to these networks as hyperuniform-point-pattern-induced (HuPPI) networks, and we compare them to their counterpart Poisson-point-pattern-induced (PoPPI) networks. By computing the local geometric and transport properties of HuPPI networks, we demonstrate how hyperuniformity imparts advantages in both transport efficiency and robustness. Specifically, we show that HuPPI networks have systematically smaller total effective resistances, slightly faster random-walk mixing times, and fewer extreme-curvature edges than PoPPI networks. Counterintuitively, we also find that HuPPI networks simultaneously have more negative mean Ollivier–Ricci curvatures and smaller global resistances than PoPPI networks, indicating that edges with moderately negative curvatures need not create severe bottlenecks to transport. We also demonstrate that the network-generation method strongly influences these properties and in particular that it often overshadows differences that arise from underlying point patterns. These results collectively demonstrate potential advantages of hyperuniformity in network design and motivate further theoretical and experimental exploration of HuPPI networks.

Microbiomes, which are collections of interacting microbes in an environment, often substantially impact the environmental patches or living hosts that they occupy. In microbiome models, it is important to consider both the local dynamics within an environment and exchanges of microbiomes between environments. One way to incorporate these and other interactions across multiple scales is to employ metacommunity theory. Metacommunity models commonly assume continuous microbiome dispersal between the environments in which local microbiome dynamics occur. Under this assumption, a single parameter between each pair of environments controls the dispersal rate between those environments. This metacommunity framework is well-suited to abiotic environmental patches, but it fails to capture an essential aspect of the microbiomes of living hosts. Living hosts generally do not interact continuously with each other. Instead, living hosts interact with each other in discrete time intervals. In this paper, we develop a modeling framework that encodes such discrete interactions and uses two parameters to separately control the interaction frequencies between hosts and the amount of microbiome exchange during each interaction. We derive analytical approximations of models in our framework in three parameter regions and prove that they are accurate in these regions. We compare these approximations to numerical simulations for an illustrative model, and we demonstrate that both parameters in our modeling framework are necessary to determine microbiome dynamics. Key features of the dynamics, such as microbiome convergence across hosts, depend sensitively on the interplay between interaction frequency and strength.

In traditional models of behavioral or opinion dynamics on social networks, researchers suppose that all interactions occur between pairs of individuals. However, in reality, social interactions also occur in groups of three or more individuals. A common way to incorporate such polyadic interactions is to study dynamical processes on hypergraphs. In a hypergraph, interactions can occur between any number of the individuals in a network. The Watts threshold model (WTM) is a well-known model of a simplistic social spreading process. Very recently, Chen et al. extended the WTM from dyadic networks (i.e., graphs) to polyadic networks (i.e., hypergraphs). In the present paper, we extend their discrete-time model to continuous time using approximate master equations (AMEs). By using AMEs, we are able to model the system with very high accuracy. We then reduce the high-dimensional AME system to a system of three coupled differential equations without any detectable loss of accuracy. This much lower-dimensional system is more computationally efficient to solve numerically and is also easier to interpret. We linearize the reduced AME system and calculate a cascade condition, which allows us to determine when a large spreading event occurs. We then apply our model to a social contact network of a French primary school and to a hypergraph of computer-science coauthorships. We find that the AME system is accurate in modelling the polyadic WTM on these empirical networks; however, we expect that future work that incorporates structural correlations between nearby nodes and groups into the model for the dynamics will lead to more accurate theory for real-world networks.

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.