# Covariance and Correlation Kernels on a Graph in the Generalized Bag-of-Paths Formalism

## TL;DR

This paper introduces new covariance and correlation kernels for graphs based on the expectation of node co-occurrences on paths, enabling effective node similarity measures for semi-supervised classification.

## Contribution

It derives closed-form expressions for co-presence and co-occurrence expectations on paths, leading to novel positive semi-definite kernels for graph nodes.

## Key findings

- The new kernels are computationally efficient.
- They outperform some existing similarity measures in classification tasks.
- The methods are applicable to both regular and hitting paths.

## Abstract

This work derives closed-form expressions computing the expectation of co-presence and of number of co-occurrences of nodes on paths sampled from a network according to general path weights (a bag of paths). The underlying idea is that two nodes are considered as similar when they often appear together on (preferably short) paths of the network. The different expressions are obtained for both regular and hitting paths and serve as a basis for computing new covariance and correlation measures between nodes, which are valid positive semi-definite kernels on a graph. Experiments on semi-supervised classification problems show that the introduced similarity measures provide competitive results compared to other state-of-the-art distance and similarity measures between nodes.