Matrix analysis of a Markov chain small-world model

Motivated by Higham's work, we show that the Markov chain of the small-world model can be investigated more completely by direct matrix-theoretic methods which produces exact results for all N and for all ϵ. Our results therefore allow a fuller examination of the behavior of the small-world phenomenon in the Markov chain model for small to moderate values of N and for all α > 1.