Energy Models for Drawing Clustered Small-World Graphs

We introduce energy models for drawing clustered small-world graphs. Clustered means that there exist (probably unknown) sets of nodes with many edges between nodes of the same set and few edges to nodes outside the set. Small-world means that the graph-theoretic distances between nodes are small relative to the number of nodes. Previous force or energy models do not produce satisfactory results for such graphs. In drawings that have minimum energy with respect to our new energy models, different clusters are clearly separated and have interpretable distances. We formally characterize the minimum energy drawings of our new energy models and of a class of energy models that contains among others the well-known Fruchterman-Reingold model. These characterizations show 1) in what sense the minimum energy drawings of these energy models fulfill requirements like small, uniform edge lengths, well-distributed nodes, or separated clusters, and 2) what properties of the drawn graph can be inferred from the drawings.