The sum-over-paths covariance kernel: a novel covariance measure between nodes of a directed graph.

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.

Pathway discovery in metabolic networks by subgraph extraction.

MOTIVATION: Subgraph extraction is a powerful technique to predict pathways from biological networks and a set of query items (e.g. genes, proteins, compounds, etc.). It can be applied to a variety of different data types, such as gene expression, protein levels, operons or phylogenetic profiles. In this article, we investigate different approaches to extract relevant pathways from metabolic networks. Although these approaches have been adapted to metabolic networks, they are generic enough to be adjusted to other biological networks as well. RESULTS: We comparatively evaluated seven sub-network extraction approaches on 71 known metabolic pathways from Saccharomyces cerevisiae and a metabolic network obtained from MetaCyc. The best performing approach is a novel hybrid strategy, which combines a random walk-based reduction of the graph with a shortest paths-based algorithm, and which recovers the reference pathways with an accuracy of approximately 77%. AVAILABILITY: Most of the presented algorithms are available as part of the network analysis tool set (NeAT). The kWalks method is released under the GPL3 license.