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Mason A. Porter

Publications and source records attributed to Mason A. Porter.

At least 19 recordsLinked to original sources
arXiv2026

Analog and hybrid computing in the 21st century

Analog computers solve problems through physical representations of mathematical variables and relations, often using the dynamics of electronic circuits and their associated signals. Many analog computers directly represent nonlinear dynamical systems; they encode their solutions in the signals that they generate. Although most modern scientific computing relies on digital architectures with inherently discrete-valued representations and algorithmic approaches, the design constraints of such architectures increasingly limit both performance and energy efficiency. Analog systems can circumvent some of these constraints; they execute specific computations very efficiently by mapping them directly onto physical processes using smaller and simpler computing elements than the ones in digital architectures. In this article, we provide an interdisciplinary perspective on modern analog and hybrid (i.e., analog-digital) computing. We discuss how modern scientific computing workflows can leverage analog approaches in various applications, such as in simulating dynamical systems, performing optimization, and enabling more energy-efficient implementations of artificial intelligence.

math.DS↗
arXiv2026

Inference of Hierarchical Core-Periphery Structure in Temporal Networks

Networks can have various types of mesoscale structures. One such structure is core--periphery structure, which consists of sets of densely-connected core nodes and sets of sparsely-connected peripheral nodes. Core nodes are connected densely to other core nodes and are potentially connected densely to 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 \cite{Polanco23} 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 group as a strict subset of the other group), 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 hierarchical core--periphery structure, we use a Markov-chain Monte Carlo (MCMC) approach. We illustrate our method for detecting hierarchical core--periphery structure in one synthetic temporal network and two real-world temporal networks, and we discuss the structures that we identify in these networks.

cs.SI↗
arXiv2026

DeGroot Dynamics on Multiplex Networks with Directed and Acyclic Interlayer Connectivities

In models of opinion dynamics, researchers typically examine how an individual's opinion on one topic is influenced by the opinions of their neighbors in a network. However, people's opinions on a topic are also influenced by their opinions and their neighbors' opinions on related topics. In an election, for example, the opinions of a voter on a variety of issues inform their opinions of electoral candidates. To consider the influence between people's opinions on multiple distinct topics, we study a multiplex extension of the traditional DeGroot consensus model. In our multiplex DeGroot model, different network layers represent distinct topics and directed interlayer edges (which are not allowed to form cycles) encode relationships between distinct topics. In our model, an individual's opinion on one topic is influenced both by the opinions of their neighbors on that topic and by their own opinions on related topics. We prove under mild conditions that our multiplex DeGroot model converges to a limit state. We also prove that strong connectedness in ``source layers'' (which represent fundamental beliefs) eventually yields a consensus opinion in each layer. In particular, we show that consensus on source layers can lead to consensus on other layers even when those layers otherwise would be polarized. By numerically simulating our multiplex DeGroot model on a two-layer network, we examine the effects on opinion dynamics of (1) different network structures and (2) different types of initial opinion data.

cs.SI↗
arXiv2026

Local Geometric and Transport Properties of Networks that are Generated from Hyperuniform Point Patterns

Hyperuniformity, which is a type of 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 in many natural and engineered phenomena. In this paper, we harness hyperuniform point patterns to generate disordered, spatially-embedded networks that are distinct from both perfectly ordered lattices and uniformly random spatial networks. We refer to this class of disordered networks as \emph{hyperuniform-point-pattern-induced (HuPPI) networks}, and we compare them to counterpart \emph{Poisson-point-pattern-induced (PoPPI) networks} and networks that we generate from random-sequential-adsorption (RSA) packings (which have local order but are not hyperuniform). We examine HuPPI, PoPPI, and RSA networks that we generate using Gabriel, Delaunay, Delaunay-centroidal, and Voronoi network-generation methods. We show that HuPPI networks have smaller mean normalized total effective resistances (TERs) than PoPPI networks in Gabriel, Delaunay-centroidal, and Voronoi networks and have shorter mean random-walk mixing times than PoPPI networks in Gabriel, Delaunay, and Delaunay-centroidal networks. For all four network-generation methods and for both HuPPI and PoPPI networks, we find that the Ollivier--Ricci curvature (ORC) of an edge is negatively correlated with its TER edge contribution (i.e., the increase in TER that results from removing that edge). Collectively, our results (1) illustrate how the local network geometry that ORC measures is related to transport and network connectivity and (2) demonstrate that the network-generation method provides an additional design lever for spatial networks beyond the choice of their underlying point pattern. [This abstract is abridged significantly from the full abstract.]

cond-mat.dis-nn↗
arXiv2026

An Approximate-Master-Equation Formulation of the Watts Threshold Model on Hypergraphs

In traditional models of behavioral and opinion dynamics on networks, researchers suppose that interactions occur between pairs of individuals. However, in reality, social interactions also occur in sets of three or more individuals. A common way to incorporate such polyadic interactions is to study dynamical processes on hypergraphs, in which interactions can occur between any number of the individuals. A major current effort is to generalize popular dynamical processes from graphs to hypergraphs. One important model is the Watts threshold model (WTM), which describes a simplistic social spreading process and was extended recently by Chen et al. from dyadic networks (i.e., ordinary graphs) to polyadic networks (i.e., hypergraphs). In the present paper, we extend their discrete-time model to continuous time by using approximate master equations (AMEs). By employing AMEs, we are able to very accurately model the mean behavior of the continuous-time system. We reduce the high-dimensional AME system to a system of three coupled ordinary differential equations without any detectable loss of accuracy. This much lower-dimensional system is computationally faster to solve numerically, and it is also easier to interpret. We linearize the reduced AME system and derive a cascade condition, which allows us to determine when a large spreading event occurs. We then examine our reduced AME system on a contact hypergraph of a French primary school and on a hypergraph of computer-science coauthorships. We find that our full and reduced AME systems give reasonably good descriptions of the mean time-dependent fraction of active nodes in the polyadic WTM dynamics on these empirical networks.

physics.soc-ph↗
arXiv2026

Using Persistent Homology to Analyze Access to Heterogeneous-Quality Resources and Heterogeneous-Severity Nuisances

We develop a framework to use multiparameter persistent homology (PH) to examine access to heterogeneous-quality resources and exposure to heterogeneous-severity nuisances in a geographic region. Persistent homology, which is a type of topological data analysis {(TDA)}, has been employed previously to examine resource coverage. Unlike prior approaches, which used one-parameter PH to study resource coverage and nuisance exposure, our method accounts for heterogeneous-quality resources. Our framework, which employs a computationally-efficient approximation of multiparameter PH, allows one to study access to any resource ({or} exposure of any nuisance) using any notion of quality (or severity). Using the city of Chicago as an example region, we employ our framework to detect clusters of poor access to public parks, overexposure to landfills, and both underexposure and overexposure to pubs and bars.

cs.CG↗
arXiv2026

Interacting Hosts with Microbiome Exchange: An Extension of Metacommunity Theory for Discrete Interactions between Hosts

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.

math.DS↗
arXiv2026

A Network-Based Measure of Cosponsorship Influence on Bill Passing in the United States House of Representatives

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.

cs.SI↗
arXiv2026

Drift Behavior in a Bounded-Confidence Opinion Model with Media Influence

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.

physics.soc-ph↗
arXiv2026

Generalizing Perron--Frobenius theory and eigenvector-based centralities to networks with complex edge weights

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.

math.SP↗
arXiv2026

A Bounded-Confidence Model of Opinion Dynamics with Adaptive Interaction Probabilities

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.

physics.soc-ph↗
arXiv2026

Community-Size Biases in Statistical Inference of Communities in Temporal Networks

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.

cs.SI↗
arXiv2026

Graph energy as a measure of community detectability in 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.

cs.SI↗
arXiv2026

A disease-spread model on hypergraphs with distinct droplet and aerosol transmission modes

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.

physics.soc-ph↗
arXiv2025

Ginzburg--Landau Functionals in the Large-Graph Limit

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 $Γ$-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 (TV) functional. 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 determine the GL minimizer for several example families of graphons.

math.FA↗
arXiv2025

Ising dynamics on multilayer networks with heterogeneous layers

Multilayer networks provide a framework to study complex systems with multiple types of interactions, multiple dynamical processes, and/or multiple subsystems. When studying a dynamical process on a multilayer network, it is important to consider how both layer structure and heterogeneity across layers impacts the overall dynamics. As a concrete example, we study Ising dynamics on multilayer networks and investigate how network structure affects its qualitative features. We focus primarily on multiplex networks, which are multilayer networks in which interlayer edges occur only between manifestations of the same entity on different layers, although we also consider one empirical example with a more general multilayer structure. We use numerical simulations and a mean-field approximation to examine the steady-state behavior of the Ising dynamics as a function of temperature (which is a key model parameter) for a variety of two-layer multilayer networks from both models and empirical data. We examine both the steady-state behavior and a metastable state in which the two layers are anti-aligned, and we explore the effects of interlayer coupling strength and structural heterogeneity. In synthetic multilayer networks with core--periphery structure, we show that interlayer edges that involve peripheral nodes can exert more influence than interlayer edges that involve only core nodes. Finally, we consider empirical multilayer networks from biological and social systems. Our work illustrates how heterogeneity across the layers of a multilayer network influences dynamics on the whole network.

physics.soc-ph↗
arXiv2025

Comparing dragonfly wings to jars of marbles through the lens of hyperuniformity

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.

math.HO↗