Integrative Analysis of Many Weighted Co-Expression Networks Using Tensor Computation

Abstract

The rapid accumulation of biological networks poses new challenges and calls for powerful integrative analysis tools. Most existing methods capable of simultaneously analyzing a large number of networks were primarily designed for unweighted networks, and cannot easily be extended to weighted networks. However, it is known that transforming weighted into unweighted networks by dichotomizing the edges of weighted networks with a threshold generally leads to information loss. We have developed a novel, tensor-based computational framework for mining recurrent heavy subgraphs in a large set of massive weighted networks. Specifically, we formulate the recurrent heavy subgraph identification problem as a heavy 3D subtensor discovery problem with sparse constraints. We describe an effective approach to solving this problem by designing a multi-stage, convex relaxation protocol, and a non-uniform edge sampling technique. We applied our method to 130 co-expression networks, and identified 11,394 recurrent heavy subgraphs, grouped into 2,810 families. We demonstrated that the identified subgraphs represent meaningful biological modules by validating against a large set of compiled biological knowledge bases. We also showed that the likelihood for a heavy subgraph to be meaningful increases significantly with its recurrence in multiple networks, highlighting the importance of the integrative approach to biological network analysis. Moreover, our approach based on weighted graphs detects many patterns that would be overlooked using unweighted graphs. In addition, we identified a large number of modules that occur predominately under specific phenotypes. This analysis resulted in a genome-wide mapping of gene network modules onto the phenome. Finally, by comparing module activities across many datasets, we discovered high-order dynamic cooperativeness in protein complex networks and transcriptional regulatory networks.

Population-based 3D genome structure analysis reveals driving forces in spatial genome organization

“…An edge between two nodes is present when there is at least one contact between any of the two corresponding homologous centromere copies. iii) To identify the frequently clustered centromeres, we represent the M projected networks as a third-order tensor and apply our tensor-based recurrent heavy subgraph discovery algorithm (47). We suppose that each heavy subgraph (i) should consist of ≥3 nodes, (ii) occurs in at least ≥1% of the structures in the population, and (iii) has a minimum network density 0.7. iv) Among all projected frequent centromere clusters detected in step iii we only consider those that exist in the original "unprojected" networks.…”

“…We asked whether the 23 chromosomes have different probabilities to participate in centromere clusters. To detect the frequency of clusters with distinct chromosome identities in the population, we translated each genome structure into a centromere interaction graph and applied a frequent dense-subgraph mining algorithm (47). The algorithm revealed 798 specific centromere cluster combinations (i.e., frequent cluster patterns; Materials and Methods) observed in at least 1% of the population (SI Appendix, Fig.…”

Unstable Communities in Network Ensembles

“…Most of the current work has focussed on algorithms based on direct frequency-based subgraph mining, metric-based mining or tensor-based mining [19,77,130,131]. In fact, natural graph ensembles such as timevarying networks have received less attention in general in data mining.…”

Frequent Pattern Discovery in Multiple Biological Networks: Patterns and Algorithms

“…We developed a tensor-based computational method [20] to identify a frequent pattern in multiple weighted networks, a so-called recurrent heavy subgraph (RHS). A heavy subgraph (HS) is a subset of heavily interconnected nodes in a single network.…”

“…However, since biological modules are active across multiple conditions, we can easily filter out spurious edges by looking for patterns that appear frequently in multiple biological networks. In this article, we review algorithms for discovering several types of frequent patterns defined on multiple biological networks: coherent dense subgraphs [12], frequent dense vertex-sets [32], generic frequent subgraphs [13], differential subgraphs [21] and recurrent heavy subgraphs [20]. Although the methods described in this paper are applicable to any type of genome-wide network, we shall demonstrate our algorithms using co-expression networks due to their wide availability.…”