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Random Propagation in Complex Systems: Nonlinear Matrix Recursions and Epidemic Spread


Ahn, Hyoung Jun (2014) Random Propagation in Complex Systems: Nonlinear Matrix Recursions and Epidemic Spread. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MC7M-EE22.


This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Nonlinear dynamics, Mixing time, Markov chain, Nonlinear matrix equation
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Hassibi, Babak
Thesis Committee:
  • Hassibi, Babak (chair)
  • Ligett, Katrina A.
  • Owhadi, Houman
  • Tropp, Joel A.
Defense Date:21 May 2014
Record Number:CaltechTHESIS:05232014-172754261
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8391
Deposited By: Hyoung Jun Ahn
Deposited On:29 May 2014 21:31
Last Modified:04 Oct 2019 00:05

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