arXiv subjects

Nathan Albin

Publications and source records attributed to Nathan Albin.

33 records · Page 2Linked to original sources

arXiv2019

Fourier multipliers for nonlocal Laplace operators

Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The Fourier multipliers are given through an integral representation. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function $_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is utilized to identify the asymptotic behavior of the Fourier multipliers $m(ν)$ as $\|ν\|\rightarrow \infty$. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel, and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the dimension $n$, the integral kernel, and the peridynamic Laplacian nonlocality. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation and, moreover, show that its solution converges to the solution of the classical Poisson equation.

math.CA↗

arXiv2019

Fourier spectral methods for nonlocal models

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in a previous work. It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in $n$ dimensions reduces to the computation of a 1D smooth function given in terms of $_2F_3$. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are compared against a finite-difference solver.

math.NA↗

arXiv2019

Spanning tree modulus for secure broadcast games

The theory of $p$-modulus provides a general framework for quantifying the richness of a family of objects on a graph. When applied to the family of spanning trees, $p$-modulus has an interesting probabilistic interpretation. In particular, the $2$-modulus problem in this case has been shown to be equivalent to the problem of finding a probability distribution on spanning trees that utilizes the edges of the graph as evenly as possible. In the present work, we use this fact to produce a game-theoretic interpretation of modulus by employing modulus to solve a secure broadcast game.

math.CO↗

arXiv2018

Generalization of Effective Conductance Centrality for Egonetworks

We study the popular centrality measure known as effective conductance or in some circles as information centrality. This is an important notion of centrality for undirected networks, with many applications, e.g., for random walks, electrical resistor networks, epidemic spreading, etc. In this paper, we first reinterpret this measure in terms of modulus (energy) of families of walks on the network. This modulus centrality measure coincides with the effective conductance measure on simple undirected networks, and extends it to much more general situations, e.g., directed networks as well. Secondly, we study a variation of this modulus approach in the egocentric network paradigm. Egonetworks are networks formed around a focal node (ego) with a specific order of neighborhoods. We propose efficient analytical and approximate methods for computing these measures on both undirected and directed networks. Finally, we describe a simple method inspired by the modulus point-of-view, called shell degree, which proved to be a useful tool for network science.

physics.data-an↗

arXiv2018

Fairest edge usage and minimum expected overlap for random spanning trees

Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or "law"), $μ$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $μ$ in order to utilize the edges as "fairly" as possible. This turns out to be equivalent to minimizing, with respect to $μ$, the expected overlap of two independent random spanning trees sampled with law $μ$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probability. The main result is a deflation process that identifies a hierarchical structure of arbitrary graphs in terms of homogeneous subgraphs, which we call homogeneous cores. A key tool in the analysis is the spanning tree modulus, for which there exists an algorithm based on minimum spanning tree algorithms, such as Kruskal's or Prim's.

math.CO↗

arXiv2018

Modulus metrics on networks

The concept of $p$-modulus gives a way to measure the richness of a family of objects on a graph. In this paper, we investigate the families of connecting walks between two fixed nodes and show how to use $p$-modulus to form a parametrized family of graph metrics that generalize several well-known and widely-used metrics. We also investigate a characteristic of metrics called the "antisnowflaking exponent" and present some numerical findings supporting a conjecture about the new metrics. We end with explicit computations of the new metrics on some selected graphs.

math.MG↗

arXiv2018

Infinity modulus and the essential metric

We study $\infty$-modulus on general metric spaces and establish its relation to shortest lengths of paths. This connection was already known for modulus on graphs, but the formulation in metric measure spaces requires more attention to exceptional families. We use this to define a metric that we call the essential metric, and show how this recovers a metric that had already been advanced in the literature by De Cecco and Palmieri.

math.MG↗

arXiv2017

Blocking duality for $p$-modulus on networks and applications

This paper explores the implications of blocking duality---pioneered by Fulkerson et al.---in the context of $p$-modulus on networks. Fulkerson's blocking duality is an analogue on networks to the method of conjugate families of curves in the plane. The technique presented here leads to a general framework for studying families of objects on networks; each such family has a corresponding dual family whose $p$-modulus is essentially the reciprocal of the original family's. As an application, we give a modulus-based proof for the fact that effective resistance is a metric on graphs. This proof immediately generalizes to yield a family of graph metrics, depending on the parameter $p$, that continuously interpolates among the shortest-path metric, the effective resistance metric, and the mincut ultrametric. In a second application, we establish a connection between Fulkerson's blocking duality and the probabilistic interpretation of modulus. This connection, in turn, provides a straightforward proof of several monotonicity properties of modulus that generalize known monotonicity properties of effective resistance. Finally, we use this framework to expand on a result of Lovász in the context of randomly weighted graphs.

math.CO↗

arXiv2017

Modulus of families of walks on graphs

We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide.

math.CO↗

arXiv2016

Network clustering and community detection using modulus of families of loops

We study the structure of loops in networks using the notion of modulus of loop families. We introduce a new measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link-usage optimally. We propose weighting networks using these expected link-usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks.

cs.SI↗

arXiv2016

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies have at most $|E|$ elements and that these elements carry a weight related to their "importance" in relation to the corresponding $p$-modulus problem. When $p=2$, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.

math.OC↗

arXiv2015

Numerical Investigation of Metrics for Epidemic Processes on Graphs

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.

physics.soc-ph↗

arXiv2015

Modulus on graphs as a generalization of standard graph theoretic quantities

This paper presents new results for the modulus of families of walks on a graph---a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut.

math.CO↗

arXiv2015

Maximizing Algebraic Connectivity in Interconnected Networks

Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with inter-layer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these inter-layer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one inter-layer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be non-uniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically.

physics.soc-ph↗

arXiv2015

An algorithmic exploration of the existence of high-order summation by parts operators with diagonal norm

This paper explores a common class of diagonal-norm summation by parts (SBP) operators found in the literature, which can be parameterized by an integer triple $(s,t,r)$ representing the interior order of accuracy ($2s)$, the boundary order of accuracy ($t$), and the dimension of the boundary closure ($r$). There is no simple formula for determining whether or not an SBP operator exists for a given triple of parameters. Instead, one must check that certain compatibility conditions are met: namely that a particular linear system of equations has a positive solution. Partly because of the complexity involved, not much is known about diagonal-norm SBP operators with $2s>10$. By utilizing a new algorithm for answering the question "Does an SBP operator exist for the parameters $(s,t,r)$?", it is possible to explore the existence of SBP operators with high order accuracy, and previously unknown SBP operators with interior order of accuracy as large as $2s=30$ are found. Additionally, a method for optimizing the spectral radius of the SBP derivative is introduced, and the effectiveness of this method is explored through numerical experiment.

math.NA↗