Energy/Stress Functions for Dimension Reduction and Graph Drawing: Power Laws and Their Clustering Properties∗

We introduce a parametrized family of energy/stress functions useful for proximity analysis, nonlinear dimension reduction, and graph drawing. The functions are inspired by the physics intuitions of attractive and repulsive forces common in graph drawing. Their minimization generates low-dimensional configurations (embeddings, graph drawings) whose interpoint distances match input distances as best as possible. The functions model attractive and repulsive forces with so-called Box-Cox transformations (essentially power transformations with the logarithmic transformation analytically embedded and negative powers made monotone increasing). Attractive and repulsive forces are balanced so as to attempt exact matching of input and output distances. To stabilize embeddings when only local distances are used, we impute infinite non-local distances that exert infinitesimal repulsive forces. The resulting energy/stress functions form a three-parameter family that contains many of the published energy functions in graph drawing and stress functions in multidimensional scaling. The problem of selecting an energy/stress function is therefore translated to a parameter selection problem which can be approached with a meta-criterion previously proposed by the authors (Chen and Buja 2009). Of particular interest is the tuning of a parameter associated with the notion of “clustering strength” proposed by Noack (2003). Such tuning greatly helps identifying “zero-dimensional manifolds”, i.e., clusters.