A Cluster Algorithm for Graphs

A cluster algorithm for graphs called the Markov Cluster algorithm (MCL algorithm) is introduced. The algorithm provides basically an interface to an algebraic process de ned on stochastic matrices, called the MCL process. The graphs may be both weighted (with nonnegative weight) and directed. Let G be such a graph. The MCL algorithm simulates ow in G by rst identifying G in a canonical way with a Markov graph G1. Flow is then alternatingly expanded and contracted, leading to a row of Markov Graphs G(i). Flow expansion corresponds with taking the k power of a stochastic matrix, where k 2 IN . Flow contraction 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. The heuristic underlying this approach is the expectation that ow between dense regions which are sparsely connected will evaporate. The invariant limits of the process are easily derived and in practice the process converges very fast to such a limit, the structure of which has a generic interpretation as an overlapping clustering of the graph G. Overlap is limited to cases where the input graph has a symmetric structure inducing it. The contraction and expansion parameters of the MCL process in uence the granularity of the output. The algorithm is space and time e cient and lends itself to drastic scaling. This report describes the MCL algorithm and process, convergence towards equilibrium states, interpretation of the states as clusterings, and implementation and scalability. The algorithm is introduced by rst considering several related proposals towards graph clustering, of both combinatorial and probabilistic nature. 2000 Mathematics Subject Classi cation: 05B20, 15A48, 15A51, 62H30, 68R10, 68T10, 90C35.