arXiv subjects

Emilie Hogan

Publications and source records attributed to Emilie Hogan.

7 recordsLinked to original sources

arXiv2015

A Network-of-Networks Model for Electrical Infrastructure Networks

Modeling power transmission networks is an important area of research with applications such as vulnerability analysis, study of cascading failures, and location of measurement devices. Graph-theoretic approaches have been widely used to solve these problems, but are subject to several limitations. One of the limitations is the ability to model a heterogeneous system in a consistent manner using the standard graph-theoretic formulation. In this paper, we propose a {\em network-of-networks} approach for modeling power transmission networks in order to explicitly incorporate heterogeneity in the model. This model distinguishes between different components of the network that operate at different voltage ratings, and also captures the intra and inter-network connectivity patterns. By building the graph in this fashion we present a novel, and fundamentally different, perspective of power transmission networks. Consequently, this novel approach will have a significant impact on the graph-theoretic modeling of power grids that we believe will lead to a better understanding of transmission networks.

physics.soc-ph↗

arXiv2014

Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets

In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this paper by considering conjugate orders which "partition" the space of pairwise comparisons of order elements. For us, these elements are real intervals, and we consider the weak and subset interval orders as (near) conjugates. It is also natural to ask about interval rank functions applied reflexively on whatever poset of intervals we have chosen, and thereby a general iterative strategy for interval ranks. We explore the convergence properties of standard and conjugate interval ranks, and conclude with a discussion of the experimental mathematics needed to support this work.

math.CO↗

arXiv2014

Interval-Valued Rank in Finite Ordered Sets

We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the bottom and from the top. Our rank functions are thus interval-valued, and always exist, even for non-graded posets, providing order homomorphisms to an interval order on the interval-valued ranks. The concept of rank width arises naturally, allowing us to identify the poset region with point-valued width as its longest graded portion (which we call the "spindle"). A standard interval rank function is naturally motivated both in terms of its extremality and on pragmatic grounds. Its properties are examined, including the relationship to traditional grading and rank functions, and methods to assess comparisons of standard interval-valued ranks.

math.CO↗

arXiv2013

Towards a Networks-of-Networks Framework for Cyber Security

Networks-of-networks (NoN) is a graph-theoretic model of interdependent networks that have distinct dynamics at each network (layer). By adding special edges to represent relationships between nodes in different layers, NoN provides a unified mechanism to study interdependent systems intertwined in a complex relationship. While NoN based models have been proposed for cyber-physical systems, in this position paper we build towards a three-layered NoN model for an enterprise cyber system. Each layer captures a different facet of a cyber system. We present in-depth discussion for four major graph- theoretic applications to demonstrate how the three-layered NoN model can be leveraged for continuous system monitoring and mission assurance.

cs.CR↗

arXiv2011

A New Algorithm for Proving Global Asymptotic Stability of Rational Difference Equations

Global asymptotic stability of rational difference equations is an area of research that has been well studied. In contrast to the many current methods for proving global asymptotic stability, we propose an algorithmic approach. The algorithm we summarize here employs the idea of contractions. Given a particular rational difference equation, defined by a function $Q$ which maps the $k+1$ dimensional real numbers to itself, we attempt to find an integer, $K$, for which $Q^K$ shrinks distances to the difference equation's equilibrium point. We state some general results that our algorithm has been able to prove, and also mention the implementation of our algorithm using Maple.

math.DS↗

arXiv2009

Non-linear Recurrences that Quite Unexpectedly Generate Rational Numbers

Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is not linear in the highest order term. In this case we no longer produce a unique sequence, but we sometimes have surprising results. If the highest order term is raised to the $m^{th}$ power we expect answers to have $m^{th}$ roots, but for some specific recurrences it happens that we generate rational numbers ad infinitum. I will give a general example in the case of a first order recurrence with $m=2$, and a more specific example that is order 3 with $m=2$ which comes from a generalized Somos recurrence.

math.CO↗

arXiv2009

A New Family of Somos-like Recurrences

We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that the same sequence is generated by a linear recurrence (with suitable initial conditions). We also give conjectured relations among the three parameters so that the quadratic recurrences generate sequences of integers.

math.CO↗