The Effect of Topology on the Dynamical Behavior of Oscillator Networks

Open Access
Author:
Valente, Daniel Paul
Graduate Program:
Acoustics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
February 14, 2006
Committee Members:
  • David Carl Swanson, Committee Chair
  • Thomas B Gabrielson, Committee Member
  • Julian Decatur Maynard Jr., Committee Member
  • Anthony A Atchley, Committee Member
Keywords:
  • oscillator networks
  • complex networks
  • genetic algorithm
  • optimization
  • network topology
Abstract:
Recent years have seen an upsurge of interest in the topological properties of real networks. The underlying motivation for much of this research is that an understanding of network structure will elucidate the dynamical behaviors of the system. A common hypothesis is that the topological characteristics of these networks have evolved to facilitate some dynamical process. Working with that hypothesis, this dissertation explores the role of network topology on the dynamical behavior of a collection of nonlinear oscillators. Specifically, we address the question of whether there are ideal topologies for certain functions performed by the network; that is, given a required dynamical response of the system, can the set of connections be chosen to elicit this behavior? If so, are there any common topological properties of the networks that accomplish this task? Of particular interest to oscillator networks is the minimization of vibration at target locations in the structure. In this dissertation, a genetic algorithm (GA) is used to optimize the topology of the network so that the vibration of target oscillators is minimized. It is demonstrated that reduction of target energy is nearly always possible by a change of connection topology. The components of the GA are then discussed in detail. Using this method, target minimization is observed to be a largely local phenomenon; the ideal topologies found by the GA contain the target as a hub and otherwise have a large number of homogenously distributed edges. Hub creation is observed as the primary mechanism for minimization; the addition of edges to the graph is secondary, as is illustrated by imposing an edge penalty on the system. The reduction in amplitude of the target oscillator is attributed to a balance between a minimization of the local mean field of the target and a maximization of the target degree.