Multilevel Graph Clustering with Density-Based Quality Measures

3 The Multi-Level Refinement Algorithmmove to other clustermove to new clustermerge clustersFigure 3.1: Operations for local cluster modification. The operations move thecluster boundaries, displayed as dashed lines. This changes which edges are in theedge cut. These are highlighted by red lines.input graphcoarseningrefinementrecursion orinitial clusteringFigure 3.2: The Multi-Level Scheme3.1 The Multi-Level SchemeAs already mentioned the refinement heuristic requires a good initial clustering.Unfortunately also the number of clusters is not known in advance. Assigning allvertices to a big cluster cannot be good because the number of clusters is muchtoo small. Similarly assigning each vertex to a separate cluster starts with toomany and there is no good reason why a random clustering should be better. Onthe other hand agglomerative clustering methods perform relatively well at findinggood clusterings.To accelerate and widen the refinement the algorithm initially trades search rangeagainst accuracy by moving whole groups of vertices at once (cf. Fig. reffig:alg:multilevel).Afterwards smaller groups or single vertices are moved for local improvements.The underlying data structures are unified by contracting these vertex groupsinto a coarse graph. This approach is applied recursively to produce a hierarchy ofbigger groups and coarser graphs. Refinement is applied at each of these coarseninglevels beginning on the coarsest. The question of good vertex groups is coupled tothe initial clustering: The coarsest groups already are a clustering and can be used26