Performance Criteria for Graph Clustering and Markov Cluster

In 6] a cluster algorithm for graphs was introduced called the Markov cluster algorithm or MCL algorithm. The algorithm is based on simulation of (stochastic) ow in graphs by means of alternation of two operators, expansion and innation. The results in 8] establish an intrinsic relationship between the corresponding algebraic process (MCL process) and cluster structure in the iterands and the limits of the process. Several kinds of experiments conducted with the MCL algorithm are described here. Test cases with varying homogeneity characteristics are used to establish some of the particular strengths and weaknesses of the algorithm. In general the algorithm performs well, except for graphs which are very homogeneous (such as weakly connected grids) and for which the natural cluster diameter (i.e. the diameter of a subgraph induced by a natural cluster) is large. This can be understood in terms of the ow characteristics of the MCL algorithm and the heuristic on which the algorithm is grounded. A generic performance criterion for clusterings of weighted graphs is derived, by a stepwise reenement of a simple and appealing criterion for simple graphs. The most reened criterion uses a particular Schur convex function, several properties of which are established. A metric is deened on the space of partitions, which is useful for comparing diierent clusterings of the same graph. The metric is compared with the metric known as the equivalence mismatch coeecient. The performance criterion and the metric are used for the quantitative measurement of experiments conducted with the MCL algorithm on randomly generated test graphs with 10000 nodes. Scaling the MCL algorithm requires a regime of pruning the stochastic matrices which need to be computed. The eeect of pruning on the quality of the retrieved clusterings is also investigated. Note: Mathematical aspects of the MCL process are described in 8]. 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 Markov cluster algorithm or MCL algorithm was introduced in 6]. Its deenition is repeated here and a summary of results from 6] and 8] is given. The report corresponds with Chapters 9, 10, and 12 in the PhD thesis 7]. The MCL algorithm is basically a shell around an algebraic process deened for stochastic matrices, called the MCL process. The process consists of alternation of two operators, called expansion and innation. The image of a …