Do Logarithmic Proximity Measures Outperform Plain Ones in Graph Clustering?

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.

A bag-of-paths framework for network data analysis

“…Table 3 which is not statistically different from the best method. This could be related to the recent comparison in [45] showing that taking the logarithm of some well-known kernels improves the performances in node clustering tasks. Concerning the transformation from distances to inner products of Equations (42) and (43), the Gaussian kernel often provides slightly better results than multidimensional scaling, but not always so.…”

“…Finally, we plan to make a systematic experimental comparison of families of distances and kernels on clustering, semi-supervised classification and dimen-sionality reduction tasks, while trying to analyze the theoretical properties of the proposed distances families by following [39]. In particular, we will investigate the new kernels introduced recently in [45] where it is shown on node clustering tasks that taking the logarithm of well-known kernels improves significantly the performances.…”

Modularity-Driven Kernel k-means for Community Detection

“…Varying the graph node distance between a graph's nodes can have a substantial influence on clustering effectiveness. As shown in prior work [26,43], modern graph distances can improve clustering solutions. However, clustering the nodes using a kernel k-means approach still suffers a major drawback, in that the k-means algorithm always requires that the number of clusters is given.…”

“…The four kernels presented just beforehand equally exist in logarithmic form. We refer to [6][7][8]26] as well as the indicated references for details. Note that ln indicates application of element-wise natural logarithm.…”

“…In this work we therefore (i) propose automatic kernel parameter tuning, in order to improve clustering quality, without requiring user interaction, using the modularity criterion, (ii) identify the natural number of clusters present in a dataset, using also the modularity criterion, without knowledge of the (correct) number of clusters beforehand, (iii) finally, compare this improved kernel k-means approach, using the kernels presented in our previous work [43], as well as additional kernels [26], to the Louvain method [2].…”