Do logarithmic proximity measures outperform plain ones in graph clustering?

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.Comment: 11 pages, 5 tables, 9 figures. Accepted for publication in the Proceedings of 6th International Conference on Network Analysis, May 26-28, 2016, Nizhny Novgorod, Russi

Kernels on Graphs as Proximity Measures

International audienceKernels and, broadly speaking, similarity measures on graphs are extensively used in graph-based unsupervised and semi-supervised learning algorithms as well as in the link prediction problem. We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This can potentially be useful for recommending the adoption of one or another similarity measure in a machine learning method. Also, we numerically compare various similarity measures in the context of spectral clustering and observe that normalized heat-type similarity measures with log modification generally perform the best

Characterizing best–worst voting systems in the scoring context

An increasing body of theoretical and empirical work on discrete choice considers a choice design in which a person is asked to select both the best and the worst alternative in an available set of alternatives, in contrast to more traditional tasks, su