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BIRS Workshop Lecture Videos

Clustering in Markov Chains with Subdominant Eigenvalues Close to One Breen, Jane


Finite, discrete, time-homogeneous Markov chains are frequently used as a simple mathematical model of real-world dynamical systems. In many such applications, an analysis of clustering behaviour is desirable, and it is well-known that the eigendecomposition of the transition matrix $T$ of the chain can provide such insight. In a recent paper (see [1]), a method is presented for determining clusters from a subdominant real eigenvalue $\lambda$ of $T$ which is close to the spectral radius 1. In this talk, we extend the method to include an analysis for complex eigenvalues of $T$ which are close to 1. Since a real spectrum is not guaranteed in most applications, this is a valuable result in the area of spectral clustering in Markov chains. This is joint work with Emanuele Crisostomi, Mahsa Faizrahnemoon, Steve Kirkland, and Robert Shorten. [1] Emanuele Crisostomi, Stephen Kirkland, and Robert Shorten. A Google-like model of road network dynamics and its application to regulation and control. $\textit{International Journal of Control}$, 84(3):633--651, 2011.

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