dane taylor research

DANE TAYLOR - postdoctoral scholar

Department of Mathematics, University of North Carolina-Chapel Hill, NC
Statistical and Applied Mathematical Sciences Institute (SAMSI)Research Triangle Park, NC

email: dane. r. taylor [at] gmail. com    
office: Chapman Hall 442, UNC-Chapel Hill

Scientific Interests
  • Network Science (biological, technological, and social networks)
  • High-dimensional data analysis (clustering and manifold learning)
  • Dynamical and complex systems (synchronization, diffusion and contagion)
  • Spectral graph theory (perturbation, evolution, and optimization of spectra)
  • Topological data analysis (persistent homology of networks, dynamics on simplicial complexes
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Research Highlights

 
Complex contagions for topological data analysis of networks
D Taylor, F Klimm, HA Harrington, M Kramar, K Mischaikow, MA Porter & PJ Mucha, in revision for Nature Communications (2014)

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, long-range 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 short-range and long-range, "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 high-dimensional 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 low-dimensional 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. 

   
 

Several Watts threshold model (WTM) contagions on a small world network map the nodes as a point cloud. When contagions are dominated by wavefront propagation then the network's underlying manifold (i.e., a ring) appears in this high dimensional point cloud.

 
     
 
Dynamics on hybrid 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 finite-size 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 link-formation 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 network-coupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important self-organized properties of SC attachment and discuss implications for controlling dynamics on networks.
     



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