A stochastic uncoupling process for graphs

A discrete stochastic uncoupling process for nite spaces is introduced, called the Markov Cluster Process. The process takes a stochastic matrix as input, and then alternates ow expansion and ow innation, each step deening a stochastic matrix in terms of the previous one. Flow expansion corresponds with taking the k th power of a stochastic matrix, where k 2 IN. Flow innation corresponds with a parametrized operator ?r, r 0, which maps the set of (column) stochastic matrices onto itself. The image ?rM is obtained by raising each entry in M to the r th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit which is idempotent under both matrix multiplication and innation, with quadratic convergence around the limit points. The limit is in general extremely sparse and the number of components of its associated graph may be larger than the number associated with the input matrix. This uncoupling is a desired eeect as it reveals structure in the input matrix. The innation operator ?r is shown to map the class of matrices which are diagonally similar to a symmetric matrix onto itself. The term diagonally positive semi{deenite (dpsd) is used for matrices which are diagonally similar to a positive semi{deenite matrix. It is shown that for r 2 IN and for M a stochastic dpsd matrix, the image ?rM is again dpsd. Determinantal inequalities satissed by a dpsd matrix M imply a natural ordering among the diagonal elements of M, generalizing a mapping of nonnegative column allowable idempo-tent matrices onto overlapping clusterings. The spectrum of ?1M, for dpsd M, is of the form f0 n?k ; 1 k g, where k is the number of endclasses of the ordering associated with M, and n is the dimension of M. Reductions of dpsd matrices are given, a connection with Hilbert's distance and the contraction ratio deened for nonnegative matrices is discussed, and several conjectures are made. Note: This report describes mathematical aspects of the MCL process. The process was introduced in 9] as a means for nding cluster structure in graphs. Cluster experiments are described in 11]. The work was carried out under project INS{3.2, Concept Building from Key{Phrases in Scientiic Documents and Bottom Up Classiication Methods in Mathematics. 1. Introduction The subject of this report 1 is an algebraic process deened for stochastic 2 matrices, called the Markov …