The cluster structure of many real-world graphs is of great interest, as the clusters may correspond e.g. to communities in social networks or to cohesive modules in software systems. Layouts can naturally represent the cluster structure of graphs by grouping densely connected nodes and separating sparsely connected nodes. This article introduces two energy models whose minimum energy layouts represent the cluster structure, one based on repulsion between nodes (like most existing energy models) and one based on repulsion between edges. The latter model is not biased towards grouping nodes with high degrees, and is thus more appropriate for the many real-world graphs with right-skewed degree distributions. The two energy models are shown to be closely related to widely used quality criteria for graph clusterings -namely the density of the cut, Shi and Malik's normalized cut, and Newman's modularity -and to objective functions optimized by eigenvector-based graph drawing methods.

Two natural and widely used representations for the community structure of networks are clusterings, which partition the vertex set into disjoint subsets, and layouts, which assign the vertices to positions in a metric space. This paper unifies prominent characterizations of layout quality and clustering quality, by showing that energy models of pairwise attraction and repulsion subsume Newman and Girvan's modularity measure. Layouts with optimal energy are relaxations of, and are thus consistent with, clusterings with optimal modularity, which is of practical relevance because the two representations are complementary and often used together.

Abstract.We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the best-known force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.

Modularity is a widely used quality measure for graph clusterings. Its exact maximization is NP-hard and prohibitively expensive for large graphs. Popular heuristics first perform a coarsening phase, where local search starting from singleton clusters is used to compute a preliminary clustering, and then optionally a refinement phase, where this clustering is improved by moving vertices between clusters. As a generalization, multilevel heuristics coarsen in several stages, and refine by moving entire clusters from each of these stages, not only individual vertices. This article organizes existing and new single-level and multilevel heuristics into a coherent design space, and compares them experimentally with respect to their effectiveness (achieved modularity) and runtime. For coarsening by iterated cluster joining, it turns out that the most widely used criterion for joining clusters (modularity increase) is outperformed by other simple criteria, that a recent multistep algorithm [Schuetz and Caflisch 2008] is no improvement over simple single-step coarsening for these criteria, and that the recent multilevel coarsening by iterated vertex moving [Blondel et al. 2008] is somewhat faster but slightly less effective (with refinement). The new multilevel refinement is significantly more effective than the conventional single-level refinement or no refinement, in reasonable runtime. A comparison with published benchmark results and algorithm implementations shows that multilevel local search heuristics, despite their relative simplicity, are competitive with the best algorithms in the literature.

Modularity is one of the most widely used quality measures for graph clusterings. Maximizing modularity is NP-hard, and the runtime of exact algorithms is prohibitive for large graphs. A simple and effective class of heuristics coarsens the graph by iteratively merging clusters (starting from singletons), and optionally refines the resulting clustering by iteratively moving individual vertices between clusters. Several heuristics of this type have been proposed in the literature, but little is known about their relative performance.This paper experimentally compares existing and new coarsening-and refinement-based heuristics with respect to their effectiveness (achieved modularity) and efficiency (runtime). Concerning coarsening, it turns out that the most widely used criterion for merging clusters (modularity increase) is outperformed by other simple criteria, and that a recent algorithm by Schuetz and Caflisch is no improvement over simple greedy coarsening for these criteria. Concerning refinement, a new multi-level algorithm is shown to produce significantly better clusterings than conventional single-level algorithms. A comparison with published benchmark results and algorithm implementations shows that combinations of coarsening and multi-level refinement are competitive with the best algorithms in the literature.