Kernels on Graphs as Proximity Measures

Abstract

Kernels 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.

Impact of Network Topology on Efficiency of Proximity Measures for Community Detection

“…It is important to note that because of the kernels definitions, the eigenvectors are the same for the Walk and Communicability, and Forest and Heat measures [1]. Consequently, the Spectral clustering will lead to the same partitions for these 2 pairs, and we will use only 3 measures (i.e., Walk, Forest, and PageRank) instead of 5 when discussing results for the Spectral method.…”

“…In [1], a number of proximity measures, including Walk, Communicability, Heat, PageRank, and several logarithmic measures are used to find communities in SBM (stochastic block model) based networks with the Spectral method, and some of the measures lead to better results than others.…”

Measuring Proximity in Attributed Networks for Community Detection

“…In [1], the authors analytically study properties of various proximity measures 6 and kernels on graphs, including Walk, Communicability, Heat, PageR-ank, and several logarithmic measures. Then, these measures are compared in the context of spectral clustering on the stochastic block model.…”

“…Nowadays, we have a surprising variety of measures on the set of graph nodes [9,Chapter 15]. Some of the proximity measures can be defined as kernels on graphs, i.e., symmetric positive semidefinite matrices [1].…”

“…Here, we use the term "proximity measure" in a broaded sense and, unlike[1], do not require a proximity measure to satisfy the triangle inequality for proximities.…”