Random walk with restart (RWR) is widely recognized as one of the most important node proximity measures for graphs, as it captures the holistic graph structure and is robust to noise in the graph. In this paper, we study a novel query based on the RWR measure, called the inbound top-k () query. Given a query node q and a number k, the query aims at retrieving k nodes in the graph that have the largest weighted RWR scores to q. queries can be highly useful for various applications such as traffic scheduling, disease treatment, and targeted advertising. Nevertheless, none of the existing RWR computation techniques can accurately and efficiently process the query in large graphs. We propose two algorithms, namely Squeeze and Ripple, both of which can accurately answer the query in a fast and incremental manner. To identify the top-k nodes, Squeeze iteratively performs matrix-vector multiplication and estimates the lower and upper bounds for all the nodes in the graph. Ripple employs a more aggressive strategy by only estimating the RWR scores for the nodes falling in the vicinity of q, the nodes outside the vicinity do not need to be evaluated because their RWR scores are propagated from the boundary of the vicinity and thus upper bounded. Ripple incrementally expands the vicinity until the top-k result set can be obtained. Our extensive experiments on real-life graph data sets show that queries can retrieve interesting results, and the proposed algorithms are orders of magnitude faster than state-of-the-art method.

Motivation Gene clustering is a widely-used technique that has enabled computational prediction of unknown gene functions within a species. However, it remains a challenge to refine gene function prediction by leveraging evolutionarily conserved genes in another species. This challenge calls for a new computational algorithm to identify gene co-clusters in two species, so that genes in each co-cluster exhibit similar expression levels in each species and strong conservation between the species. Results Here we develop the bipartite tight spectral clustering (BiTSC) algorithm, which identifies gene co-clusters in two species based on gene orthology information and gene expression data. BiTSC novelly implements a formulation that encodes gene orthology as a bipartite network and gene expression data as node covariates. This formulation allows BiTSC to adopt and combine the advantages of multiple unsupervised learning techniques: kernel enhancement, bipartite spectral clustering, consensus clustering, tight clustering, and hierarchical clustering. As a result, BiTSC is a flexible and robust algorithm capable of identifying informative gene co-clusters without forcing all genes into co-clusters. Another advantage of BiTSC is that it does not rely on any distributional assumptions. Beyond cross-species gene co-clustering, BiTSC also has wide applications as a general algorithm for identifying tight node co-clusters in any bipartite network with node covariates. We demonstrate the accuracy and robustness of BiTSC through comprehensive simulation studies. In a real data example, we use BiTSC to identify conserved gene co-clusters of D. melanogaster and C. elegans, and we perform a series of downstream analysis to both validate BiTSC and verify the biological significance of the identified co-clusters. Availability and implementation The Python package BiTSC is open-access and available at https://github.com/edensunyidan/BiTSC.

Gene clustering is a widely-used technique that has enabled computational prediction of unknown gene functions within a species. However, it remains a challenge to refine gene function prediction by leveraging evolutionarily conserved genes in another species. This challenge calls for a new computational algorithm to identify gene co-clusters in two species, so that genes in each co-cluster exhibit similar expression levels in each species and strong conservation between the species. Here we develop the bipartite tight spectral clustering (BiTSC) algorithm, which identifies gene co-clusters in two species based on gene orthology information and gene expression data. BiTSC novelly implements a formulation that encodes gene orthology as a bipartite network and gene expression data as node covariates. This formulation allows BiTSC to adopt and combine the advantages of multiple unsupervised learning techniques: kernel enhancement, bipartite spectral clustering, consensus clustering, tight clustering, and hierarchical clustering. As a result, BiTSC is a flexible and robust algorithm capable of identifying informative gene co-clusters without forcing all genes into co-clusters. Another advantage of BiTSC is that it does not rely on any distributional assumptions. Beyond cross-species gene co-clustering, BiTSC also has wide applications as a general algorithm for identifying tight node co-clusters in any bipartite network with node covariates. We demonstrate the accuracy and robustness of BiTSC through comprehensive simulation studies. In a real data example, we use BiTSC to identify conserved gene co-clusters of D. melanogaster and C. elegans, and we perform a series of downstream analysis to both validate BiTSC and verify the biological significance of the identified co-clusters.