Daryl R. DeFord

A new framework for dynamical models on multiplex networks

Many complex systems have natural representations as multi-layer networks. While these formulations retain more information than standard single-layer network models, there is not yet a fully developed theory for computing network metrics and statistics on these objects. We introduce a family of models of multiplex processes motivated by dynamical applications and investigate the properties of their spectra both theoretically and computationally. We study special cases of multiplex diffusion and Markov dynamics, using the spectral results to compute their rates of convergence. We use our framework to define a version of multiplex eigenvector centrality, which generalizes some existing notions in the literature. Last, we compare our operator to structurally-derived models on synthetic and real-world networks, helping delineate the contexts in which the different frameworks are appropriate.

Empirical Analysis of Space-Filling Curves for Scientific Computing Applications

Space-Filling Curves are frequently used in parallel processing applications to order and distribute inputs while preserving proximity. Several different metrics have been proposed for analyzing and comparing the efficiency of different space-filling curves, particularly in database settings. In this paper, we introduce a general new metric, called Average Communicated Distance, that models the average pair wise communication cost expected to be incurred by an algorithm that makes use of an arbitrary space-filling curve. For the purpose of empirical evaluation of this metric, we modeled the communications structure of the Fast Multipole Method for n-body problems. Using this model, we empirically address a number of interesting questions pertaining to the effectiveness of space-filling curves in reducing communication, under different combinations of network topology and input distribution settings. We consider these problems from the perspective of ordering the input data, as well as using space-filling curves to assign ranks to the processors. Our results for these varied scenarios point towards a list of recommendations based on specific knowledge about the input data. In addition, we present some new empirical results, relating to proximity preservation under the average nearest neighbor stretch metric, that are application independent.

Multiplex Dynamics on the World Trade Web

The network of international trade, where countries are represented with nodes and trade relations are represented by directed, weighted edges is an important economic model. In this paper, we consider a multiplex version of this system, wherein nations are connected by multiple edges, representing different commodities, based on a new approach for multilayer network models. The central idea behind this method is to use a network dynamics interpretation of the commodity flows to construct a single operator representing the entire system. This allows us to study the global structure of the trade network as a single object instead of as a disjoint collection of layers. We analyze centralities and communities determined by this model and show that studying the multiplex as a whole allows us to uncover structure not evident in the individual layers or the aggregate network.

Random Walk Null Models for Time Series Data

Permutation entropy has become a standard tool for time series analysis that exploits the temporal and ordinal relationships within data. Motivated by a Kullback–Leibler divergence interpretation of permutation entropy as divergence from white noise, we extend pattern-based methods to the setting of random walk data. We analyze random walk null models for correlated time series and describe a method for determining the corresponding ordinal pattern distributions. These null models more accurately reflect the observed pattern distributions in some economic data. This leads us to define a measure of complexity using the deviation of a time series from an associated random walk null model. We demonstrate the applicability of our methods using empirical data drawn from a variety of fields, including to a variety of stock market closing prices.