Do logarithmic proximity measures outperform plain ones in graph clustering?

TL;DR

This paper investigates various graph kernels and proximity measures, demonstrating that logarithmic transformations generally improve clustering performance by aligning multiplicative kernels with additive distance properties.

Contribution

The study systematically compares plain and logarithmic graph proximity measures, revealing the superiority of logarithmic measures in clustering tasks across multiple datasets and random graph models.

Findings

Logarithmic measures outperform plain measures in class distinction.

Logarithmic Communicability is often the best performing measure.

Transforming kernels logarithmically aligns multiplicative and additive properties.

Abstract

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.