The Sum-over-Paths Covariance Kernel: A Novel Covariance Measure between Nodes of a Directed Graph

Abstract

This work introduces a link-based covariance measure between the nodes of a weighted directed graph, where a cost is associated with each arc. To this end, a probability distribution on the (usually infinite) countable set of paths through the graph is defined by minimizing the total expected cost between all pairs of nodes while fixing the total relative entropy spread in the graph. This results in a Boltzmann distribution on the set of paths such that long (high-cost) paths occur with a low probability while short (low-cost) paths occur with a high probability. The sum-over-paths (SoP) covariance measure between nodes is then defined according to this probability distribution: two nodes are considered as highly correlated if they often co-occur together on the same--preferably short--paths. The resulting covariance matrix between nodes (say n nodes in total) is a Gram matrix and therefore defines a valid kernel on the graph. It is obtained by inverting an n\times n matrix depending on the costs assigned to the arcs. In the same spirit, a betweenness score is also defined, measuring the expected number of times a node occurs on a path. The proposed measures could be used for various graph mining tasks such as computing betweenness centrality, semi-supervised classification of nodes, visualization, etc., as shown in Section 7.

Link prediction in complex networks: A survey

“…Otherwise even we can predict the existence of an arc between two nodes, we can not determine its direction. In addition, the path-dependent similarity indices should also be extended to take into account the link direction [149]. The fundamental task of link prediction in weighted networks, namely to predict the existence of links with the help of not only the observed links but also their weights, has already been considered by Murata et al [150] and Lü et al [151].…”

Semi-supervised classification and betweenness computation on large, sparse, directed graphs

“…Despite the growing need for dealing with huge real-world networks, few of the existing methods scale up to large graphs 1 so that semi-supervised classification on large graphs has become one of the current central issues; see the survey [4,Section 6.3]. Indeed, the techniques that scale well [6] are not always competitive when compared to state-of-the-art graph-based metrics [7] such as the regularized Laplacian kernel [8], the sum-over-paths (SoP) covariance [9], the random walk with restart similarity and its normalized version [10,11,7], or the Markov diffusion kernel [12]. A naive application of these graph kernel-based approaches does not scale well since it relies on the computation of a dense similarity matrix, which usually requires a matrix inversion.…”

“…The first approach is based on existing, competitive, kernels on a graph, but it explicitly avoids the computation of the pairwise similarities between the nodes (following an idea suggested by Zhou et al [1,2]). Indeed, as opposed to [11,14,9], Zhou et al suggest to avoid computing each pairwise measure and solving a system of linear equations instead. We design two iterative algorithms along this approach, each based on a different state-of-the-art similarity metric: the SoP covariance kernel [9] and the normalized random walk with restart.…”

“…Indeed, as opposed to [11,14,9], Zhou et al suggest to avoid computing each pairwise measure and solving a system of linear equations instead. We design two iterative algorithms along this approach, each based on a different state-of-the-art similarity metric: the SoP covariance kernel [9] and the normalized random walk with restart. This kernel on a graph was called regularized commute-time kernel in [7] and is closely related to the modified Laplacian matrix [15], as well as the random walk with restart similarity [10,11].…”

Typical process discovery based on affinity propagation

“…For weighted directed graph, the sum-over-paths covariance measure between nodes was defined according to Boltzmann distribution on the set of paths (Mantrach et al 2010). For labeled graph, the similarity of two different graphs depends on both the common features describe for the graph and the set of all their features, and Greedy algorithm is used to calculate the mapping about the correspondence between vertices of the graphs (Champin and Solnon 2003).…”