Ranking Problems Arising from ODE Models on Networks

• Author / Creator

  Biglands, Adrian J.

• The use of ordinary differential equations modelled on networks has become
  an increasingly important technique in many areas of research. The local
  behaviour of the system is modelled with differential equations and interactions
  between members or nodes are described using weighted digraphs. For
  instance, in public health nodes can represent different groups of people affected
  by an infectious disease, while edges in the network represent the cross-infection
  between the groups. The local behaviours of the disease in each group
  are described with ODE dynamics. In ecology, the spatial dispersal of one or
  more species considers the habitation patches as nodes and the edges between
  nodes describe the movement of the species between patches.
  This thesis develops a method of ranking the nodes of an ODE network
  at a positive equilibrium x^* = (x1^* , . . . , xn^)T . Such a ranking is called an
  equilibrium ranking. More specifically, assuming an ODE system modelled on
  a network (G,B) has a positive equilibrium x^, we associate xi^* to node i of
  the network. These positive equilibrium values xi^* are used to rate, and hence
  rank, the individual nodes of the network. Such an equilibrium ranking reflects
  both the graph structure and the local ODE parameters of the model.
  In my dissertation I investigate equilibrium ranking for several ODE networks
  including SIR epidemiology models with n different groups or spatial
  regions, single or multiple species ecological models, and coupled oscillator
  models from engineering. For an SIR model the equilibrium ranking can
  be obtained using the equilibrium values of the disease prevalence vector
  I^* = (I1^* , . . . , In^)T . This will indicate which of the n groups or patches has
  the highest number of infected individuals per capita. In the single species ecology model an equilibrium ranking vector comes from the species density
  vector x^ = (x1^* , . . . , xn^*)T , and reflects overpopulation or extinction on different
  patches or in different groups.
  The dependence of equilibrium ranking on both graph structure and local
  parameters is also investigated. In particular, the dependence of the equilibrium
  ranking is considered for several digraph structures including rooted
  trees, loop digraphs, unicyclic and multi-cyclic digraphs. This will allow researchers
  to fix the network structure (G,B) of the system and focus on how
  the dynamics play a role in the importance of nodes in a network.

• Subjects / Keywords

  • Equilibrium Ranking
  • Eigenvector Centrality
  • ODE Network

• Graduation date

  Fall 2015

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R34X54T73

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.