arXiv subjects

Benjamin Webb

Publications and source records attributed to Benjamin Webb.

At least 19 recordsLinked to original sources

arXiv2026

Structural and Temporal Hallmarks of Genealogical Networks

The rapid growth of the genealogical sector, spanning platforms with billions of records and millions of users, has produced some of the largest and most complex networks available for analysis. Despite substantial advances in genealogical network research, it remains unclear whether human kinship networks exhibit universal structural properties. We address this by developing an integrated approach to genealogical network analysis that combines network-theoretic structure with an inferred notion of time. Using over one hundred datasets from the Kinsources repository, we reinterpret standard network measures in genealogical terms and introduce \emph{pseudogenerations}, a method for extracting temporal structure directly from network topology. Within this framework, we identify common features shared across datasets. We find that genealogical networks exhibit scale-free--like degree and component-size distributions, multiscale family organization, and small-world behavior with respect to genetic and union-based distances. We show that 2-components provide a natural unit of genealogical structure, observe consistent disassortative mixing, and find that recorded unions are strongly biased toward short genetic distances relative to potential pairings. We also document temporal and demographic patterns, including shifts in recorded parental and child information, as well as correlations among recorded unions, parents, and children. These results suggest that diverse genealogical datasets share a common set of structural and temporal characteristics, providing evidence for universal features of human kinship networks and establishing a general framework for their comparative analysis.

cs.SI↗

arXiv2025

Delay-Independent Stability of Nonlinear Delay Differential Equations via Isospectral Reduction

Time delays arise naturally in a wide range of natural and technological systems, yet their influence on the stability remains a challenge to characterize, particularly for nonlinear systems. In this paper, we develop a stability framework that yields a delay-independent criterion for global exponential stability in a broad class of nonlinear, nonautonomous delay differential equations. Our approach is based on a novel method that associates the delayed system with a sequence of finite-dimensional matrices of increasing size, which are analyzed using the graph-theoretic technique of isospectral reduction. In contrast to most existing results for nonlinear delay differential equations, which rely on Lyapunov-based methods, our framework provides a general and computationally efficient alternative. As an application, we apply this criterion to analyze consistency in delayed reservoir computing systems, illustrating how the proposed approach can be used to assess stability properties relevant to prediction tasks.

math.DS↗

arXiv2023

Minimal Specialization: Coevolution of Network Structure and Dynamics

The changing topology of a network is driven by the need to maintain or optimize network function. As this function is often related to moving quantities such as traffic, information, etc. efficiently through the network the structure of the network and the dynamics on the network directly depend on the other. To model this interplay of network structure and dynamics we use the dynamics on the network, or the dynamical processes the network models, to influence the dynamics of the network structure, i.e., to determine where and when to modify the network structure. We model the dynamics on the network using Jackson network dynamics and the dynamics of the network structure using minimal specialization, a variant of the more general network growth model known as specialization. The resulting model, which we refer to as the integrated specialization model, coevolves both the structure and the dynamics of the network. We show this model produces networks with real-world properties, such as right-skewed degree distributions, sparsity, the small-world property, and non-trivial equitable partitions. Additionally, when compared to other growth models, the integrated specialization model creates networks with small diameter, minimizing distances across the network. Along with producing these structural features, this model also sequentially removes the network's largest bottlenecks. The result are networks that have both dynamic and structural features that allow quantities to more efficiently move through the network.

physics.soc-ph↗

arXiv2020

Stability of Stochastically Switched and Stochastically Time-Delayed Systems

In this paper we introduce the notion of a patient first-mean stable system. Such systems are switched systems that are first-mean stable meaning that they converge to a globally attracting fixed point on average. They are also patient so that they do not lose their first-mean stability when time-delays are introduced into the system. As time-delays are, in general, a source of instability and poor performance patient first-mean stability is a much stronger condition than first-mean stability. This notion of patient stability allows one to design systems that cannot be destabilized via time-delays. It also significantly reduces the difficulty of modeling such systems since in patient systems time-delays can, to a large extent, be safely ignored. The paper's main focus is on giving a sufficient criteria under which a system is patient first-mean stable and we give a number of examples that demonstrate the simplicity of this criteria.

math.DS↗

arXiv2019

Link Prediction in Networks Using Effective Transitions

We introduce a new method for predicting the formation of links in real-world networks, which we refer to as the method of effective transitions. This method relies on the theory of isospectral matrix reductions to compute the probability of eventually transitioning from one vertex to another in a (biased) random walk on the network. Unlike the large majority of link prediction techniques, this method can be used to predict links in networks that are directed or undirected which are either weighted or unweighted. We apply this method to a number of social, technological, and natural networks and show that it is competitive with other link predictors often outperforming them. We also provide a method of approximating our effective transition method and show that aside from having much lower temporal complexity, this approximation often provides more accurate predictions than the original effective transition method. We also, prove a number of mathematical results regarding our effective transition algorithm and its approximation.

cs.SI↗

arXiv2019

Multiparticle Dynamics on the Triangular Lattice in Interacting Media

We study the motion of $N$ particles moving on a two-dimensional triangular lattice, whose sites are occupied by either left or right rotators. These rotators deterministically scatter the particles to the left (right), changing orientation from left to right (right to left) after scattering a particle. This interplay between the scatterers and the particle's motion causes a single particle to propagate in one direction away from its initial position. For multiple particles we show that the particles' dynamics can be vastly different. Specifically, we show that a particle can become entangled with another particle potentially causing the particle's trajectory to become periodic and that this can happen when the particles have the same or differing speeds. We describe two classes of periodic orbits based on the particles' initial velocities. We also describe how a particle with an unbounded past trajectory implies that some, possibly other, particle(s) has an unbounded future trajectory in this and other related multiparticle models.

cond-mat.stat-mech↗

arXiv2019

Spectral and Dynamic Consequences of Network Specialization

One of the hallmarks of real networks is their ability to perform increasingly complex tasks as their topology evolves. To explain this, it has been observed that as a network grows certain subsets of the network begin to specialize the function(s) they perform. A recent model of network growth based on this notion of specialization has been able to reproduce some of the most well-known topological features found in real-world networks including right-skewed degree distributions, the small world property, modular as well as hierarchical topology, etc. Here we describe how specialization under this model also effects the spectral properties of a network. This allows us to give conditions under which a network is able to maintain its dynamics as its topology evolves. Specifically, we show that if a network is intrinsically stable, which is a stronger version of the standard notion of global stability, then the network maintains this type of dynamics as the network evolves. This is one of the first steps toward unifying the rigorous study of the two types of dynamics exhibited by networks. These are the \emph{dynamics of} a network, which is the study of the topological evolution of the network's structure, modeled here by the process of network specialization, and the \emph{dynamics on} a network, which is the changing state of the network elements, where the type of dynamics we consider is global stability. The main examples we apply our results to are recurrent neural networks, which are the basis of certain types of machine learning algorithms.

nlin.AO↗

arXiv2019

Characterizing cospectral vertices via isospectral reduction

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices.

math.CO↗

arXiv2019

Intrinsic Stability: Global Stability of Dynamical Networks and Switched Systems Resilient to any Type of Time-Delays

In real-world networks the interactions between network elements are inherently time-delayed. These time-delays can not only slow the network but can have a destabilizing effect on the network's dynamics leading to poor performance. The same is true in computational networks used for machine learning etc. where time-delays increase the network's memory but can degrade the network's ability to be trained. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network or high-dimensional dynamical system is intrinsically stable, which is a stronger form of the standard notion of global stability, then it maintains its stability when constant time-delays are introduced into the system. Here we show that intrinsically stable systems, including intrinsically stable networks and a broad class of switched systems, i.e. systems whose mapping is time-dependent, remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known stability results. We also show that the asymptotic state of an intrinsically stable switched system is exponentially independent of the system's initial conditions.

math.OC↗

arXiv2018

General Equitable Decompositions for Graphs with Symmetries

Using the theory of equitable decompositions it is possible to decompose a matrix $M$ appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix $M$ is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix $M$.

math.CO↗

arXiv2018

Hidden Symmetries in Real and Theoretical Networks

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called \emph{latent symmetry}, which is an extension of the standard notion of symmetry. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a \emph{size}, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erd{\H o}s-R\'enyi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that \emph{latent symmetries} are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

physics.soc-ph↗

arXiv2017

Fast link prediction for large networks using spectral embedding

Many link prediction algorithms require the computation of a similarity metric on each vertex pair, which is quadratic in the number of vertices and infeasible for large networks. We develop a class of link prediction algorithms based on a spectral embedding and the k closest pairs algorithm that are scalable to very large networks. We compare the prediction accuracy and runtime of these methods to existing algorithms on several large link prediction tasks. Our methods achieve comparable accuracy to standard algorithms but are significantly faster.

cs.SI↗

arXiv2016

Mechanisms for Network Growth that Preserve Spectral and Local Structure

We introduce a method that can be used to evolve the topology of a network in a way that preserves both the network's spectral as well as local structure. This method is quite versatile in the sense that it can be used to evolve a network's topology over any collection of the network's elements. This evolution preserves both the eigenvector centrality of these elements as well as the eigenvalues of the original network. Although this method is introduced as a tool to model network growth, we show it can also be used to compare the topology of different networks where two networks are considered similar if their evolved topologies are the same. Because this method preserves the spectral structure of a network, which is related to the network's dynamics, it can also be used to study the interplay of network growth and function. We show that if a network's dynamics is intrinsically stable, which is a stronger version of the standard notion of stability, then the network remains intrinsically stable as the network's topology evolves. This is of interest since the growth of a network can have a destabilizing effect on the network's dynamics, in general. In this sense the methods developed here can be used as a tool for designing mechanisms of network growth that ensure a network remains stabile as it grows.

nlin.AO↗

arXiv2015

Subdiffusion, Anomalous Diffusion and Propagation of a Particle Moving in Random and Periodic Media

We investigate the motion of a single particle moving on a two-dimensional square lattice whose sites are occupied by right and left rotators. These left and right rotators deterministically rotate the particle's velocity to the right or left, respectively and \emph{flip} orientation from right to left or from left to right after scattering the particle. We study three types of configurations of left and right rotators, which we think of as types of media, through with the particle moves. These are completely random (CR), random periodic (RP), and completely periodic (CP) configurations. For CR configurations the particle's dynamics depends on the ratio $r$ of right to left scatterers in the following way. For small $r\simeq0$, when the configuration is nearly homogeneous, the particle subdiffuses with an exponent of 2/3, similar to the diffusion of a macromolecule in a crowded environment. Also, the particle's trajectory has a fractal dimension of $d_f\simeq4/3$, comparable to that of a self-avoiding walk. As the ratio increases to $r\simeq 1$, the particle's dynamics transitions from subdiffusion to anomalous diffusion with a fractal dimension of $d_f\simeq 7/4$, similar to that of a percolating cluster in 2-d. In RP configurations, which are more structured than CR configurations but also randomly generated, we find that the particle has the same statistic as in the CR case. In contrast, CP configurations, which are highly structured, typically will cause the particle to go through a transient stage of subdiffusion, which then abruptly changes to propagation. Interestingly, the subdiffusive stage has an exponent of approximately 2/3 and a fractal dimension of $d_f\simeq4/3$, similar to the case of CR and RP configurations for small $r$.

cond-mat.stat-mech↗

arXiv2015

Self-Limiting Trajectories of a Particle Moving Deterministically in a Random Medium

We study the motion of a particle moving on a two-dimensional honeycomb lattice, whose sites are randomly occupied by either right or left rotators, which rotate the particle's velocity to its right or left, according to deterministic rules. In the model we consider, the scatterers are each initially oriented to the right with probability $p\in[0,1]$. This is done independently, so that the initial configuration of scatterers, which forms the medium through which the particle moves, are both independent and identically distributed. For $p\in(0,1)$, we show that as the particle moves through the lattice, it creates a number of reflecting structures. These structures ultimately \emph{limit} the particle's motion, causing it to have a periodic trajectory. As $p$ approaches either 0 or 1, and the medium becomes increasingly homogenous, the particle's dynamics undergoes a discontinuous transition from this self-limiting, periodic motion to a self-avoiding motion, where the particle's trajectory, away from its initial position, is a self-avoiding walk. Additionally, we show that the periodic dynamics observed for $p\in(0,1)$ can persist, even if the initial configuration of scatterers are not identically distributed. Furthermore, we show that if these orientations are not chosen independently, this can drastically change the particle's motion causing it to have a behavior that is nonperiodic.

math-ph↗

arXiv2014

Self-Avoiding Modes of Motion in a Deterministic Lorentz Lattice Gas

We study the motion of a particle on the two-dimensional honeycomb lattice, whose sites are occupied by either flipping rotators or flipping mirrors, which scatter the particle according to a deterministic rule. For both types of scatterers we find a new type of motion that has not been observed in a Lorentz Lattice gas, where the particle's trajectory is a self-avoiding walk between returns to its initial position. We show that this behavior is a consequence of the deterministic scattering rule and the particular class of initial scatterer configurations we consider. Since self-avoiding walks are one of the main tools used to model the growth of crystals and polymers, the particle's motion in this class of systems is potentially important for the study of these processes.

math-ph↗

arXiv2012

Improved Estimates of Survival Probabilities via Isospectral Transformations

We consider open systems generated from one-dimensional maps that admit a finite Markov partition and use the recently developed theory of isospectral graph transformations to estimate a system's survival probabilities. We show that these estimates are better than those obtained through a more direct approach.

math.DS↗

arXiv2012

Isospectral Compression and Other Useful Isospectral Transformations of Dynamical Networks

It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network. This procedure allows for the introduction of new equivalence relations between networks, where two networks are spectrally equivalent if they can be reduced to the same network. Additionally, using this theory it is possible to establish whether a network, modeled as a dynamical system, has a globally attracting fixed point (is strongly synchronizing). In this paper we further develop this theory of isospectral network transformations and demonstrate that our procedures are applicable to families of parameterized networks and networks of arbitrary size.

math.DS↗