DANE TAYLOR  postdoctoral scholar
Department of Mathematics, University of North CarolinaChapel Hill
e: dane. r. taylor [at] gmail. com
w: https://sites.google.com/site/danetaylorresearch/
o: Chapman Hall 442, UNCChapel Hill



Research Highlights

Eigenvectorbased centrality measures for temporal networks D Taylor, SA Myers, A Clauset, MA Porter & PJ Mucha, preprint at arXiv:1507.01266 (2015)
In the study of static networks, numerous "centrality" measures have been developed to quantify the importances of nodes in networks, and one can express many of these measures in terms of the leading eigenvector of a matrix. With the increasing availability of network data that changes in time, it is important to extend eigenvectorbased centrality measures to timedependent networks. In this paper, we introduce a principled generalization that is valid for any eigenvectorbased centrality measure in terms of matrices of size NTxNT, where the components of the dominant eigenvector of such a matrix describes the centralities of N nodes during T time layers. Our approach relies on coupling centrality values between neighboring time layers with a interlayer edge, whose weight controls the extent to which centrality trajectories change over time. By studying the limit of strong coupling between layers, we derive expressions for "timeaveraged centralities," which are given by the zerothorder terms of a singular perturbation expansion. We also study firstorder terms to obtain "firstordermover scores," which concisely describe the magnitude of nodes' centrality changes over time. Importantly, we compute these quantities by solving linear algebraic equations of dimension N. This is much more computationally efficient than computing the dominant eigenvector of the full NTxNT matrix. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among topbilled actors during the Golden Age of Hollywood, and citations of decisions from the Supreme Court of the United States.
 








Topological data analysis of contagion maps for examining spreading processes on networks D Taylor, F Klimm, HA Harrington, M Kramar, K Mischaikow, MA Porter & PJ Mucha, Nature Communications 6, 7723 (2015)
Social and biological contagions are often strongly influenced by the spatial embeddedness of networks. In some cases, such as in the Black Death, they can spread as a wave through space. In many modern contagions, however, longrange edges—e.g., due to airline transportation or communication media—allow clusters of a contagion to arise in distant locations. We study these competing phenomena for the Watts threshold model (WTM) of complex contagions on empirical and synthetic noisy geometric networks, which are networks that are embedded in a manifold and which consist of both shortrange and longrange, "noisy" edges. Our approach involves using the WTM to construct contagion maps that use multiple contagions to map the nodes as a point cloud, which we analyze using tools from highdimensional data analysis and computational homology. For contagions that predominantly exhibit wavefront propagation, we identify a noisy geometric network's underlying manifold in the point cloud, highlighting our approach as a tool for inferring lowdimensional structure. Our approach makes it possible to obtain insights to aid in the modeling, forecast, and control of spreading processes, and it simultaneously leads to a novel technique for manifold learning in noisy geometric networks. 










Dynamics in hybrid complex systems of switches and oscillators
D Taylor, EJ Fertig & JG Restrepo, Chaos 23, 1033144 (2013)
While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finitesize fluctuations.










Social Climber attachment in forming networks produces a phase transition in connectivity
D Taylor & DB Larremore, Physical Review E 86, 031140 (2012)
Formation and fragmentation of networks is typically studied using percolation theory, but most previous research has been restricted to studying a phase transition in cluster size, examining the emergence of a giant component. This approach does not study the effects of evolving network structure on dynamics that occur at the nodes, such as the synchronization of oscillators and the spread of information, epidemics, and neuronal excitations. We introduce and analyze new linkformation rules, called Social Climber (SC) attachment, that may be combined with arbitrary percolation models to produce a previously unstudied phase transition using the largest eigenvalue of the network adjacency matrix as the order parameter. This eigenvalue is significant in the analyses of many networkcoupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important selforganized properties of SC attachment and discuss implications for controlling dynamics on networks.




