The Spacey Random Walk: a Stochastic Process for Higher-order Data

Random walks are a fundamental model in applied mathematics and are a common example of a Markov chain. The limiting stationary distribution of the Markov chain represents the fraction of the time spent in each state during the stochastic process. A standard way to compute this distribution for a random walk on a finite set of states is to compute the Perron vector of the associated transition matrix. There are algebraic analogues of this Perron vector in terms of probability transition tensors of higher-order Markov chains. These vectors are nonnegative, have dimension equal to the dimension of the state space, and sum to one. These were derived by making an algebraic substitution in the equation for the joint-stationary distribution of a higher-order Markov chains. Here, we present the spacey random walk, a non-Markovian stochastic process whose stationary distribution is given by the tensor eigenvector. The process itself is a vertex-reinforced random walk, and its discrete dynamic...

• Type: Journal
• Year: 2016
• Where: CORR
• Authors: Austin R. Benson, David F. Gleich, Lek-Heng Lim
• Keywords: CORR 2016, Education