We introduce a graph-orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source–target vertex pairs, the goal is to orient the graph such that a maximum number of pairs admit a directed source-to-target path. We study the complexity and approximability of this problem. We show that the problem is NP-hard even on star graphs and hard to approximate to within some constant factor. On the positive side, we provide an $Ω(log log n/ log n)$ factor approximation algorithm for the problem on n-vertex graphs. We further show that for any instance of the problem there exists an orientation of the input graph that satisfies a logarithmic fraction of all pairs and that this bound is tight up to a constant factor. Our techniques also lead to constant-factor approximation algorithms for some restricted variants of the problem.

Hierarchical organization is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organization. However, in many real-world networks, such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data are collated, the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall "chain of command" makes the determination of such structures extremely challenging.

Here we address the issue of how to reorder a directed network to reveal this type of hierarchy. In doing so, we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random-graph model. This random-graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node-ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically.

Partitioning biological data objects into groups such that the objects within the groups share common traits is a longstanding challenge in computational biology.

Recently, we developed and established transitivity clustering, a partitioning approach based on weighted transitive graph projection that utilizes a single similarity threshold as density parameter. In previous publications, we concentrated on the graphical user interface and on concrete biomedical application protocols. Here, we contribute the following theoretical considerations: (1) We provide proofs that the average similarity between objects from the same cluster is above the user-given threshold and that the average similarity between objects from different clusters is below the threshold. (2) We extend transitivity clustering to an overlapping clustering tool by integrating two new approaches. (3) We demonstrate the power of transitivity clustering for protein-complex detection. We evaluate our approaches against others by utilizing gold-standard data that was previously used by Brohée et al. for reviewing existing bioinformatics clustering tools.

The extended version of this article is available online at http://transclust.mpi-inf. mpg.de.

While sequence homology search has been the main workhorse in protein function prediction, it is not applicable to a significant portion of novel proteins that do not have informative homologues in sequence databases. Similarly, while statistical tests and learning algorithms based purely on gene expression profiles have been popular for analyzing disease samples, critical issues remain in the understanding of diseases based on the differentially expressed genes suggested by these methods. In the past decade, a large number of databases providing information on various types of biological networks have become available. These databases make it possible to tackle these and other biological problems in novel ways. This paper presents a review of biological network databases and approaches to protein function prediction and gene expression profile analysis that are based on biological networks.

Recent advances in systems biology have provided us with massive amounts of pathway data that describe the interplay of genes and their products. The resulting biological networks can be modeled as graphs. By means of "omics" technologies, such as microarrays, the activity of genes and proteins can be measured. Here, data from microarray experiments is integrated with the network data to gain deeper insights into gene expression. We introduce KeyPathwayMiner, a method that enables the extraction and visualization of interesting subpathways given the results of a series of gene expression studies. We aim to detect highly connected subnetworks in which most genes or proteins show similar patterns of expression. Specifically, given network and gene expression data, KeyPathwayMiner identifies those maximal subgraphs where all but k nodes of the subnetwork are expressed similarly in all but l cases in the gene expression data. Since identifying these subgraphs is computationally intensive, we developed a heuristic algorithm based on Ant Colony Optimization. We implemented KeyPathwayMiner as a plug-in for Cytoscape. Our computational model is related to a strategy presented by Ulitsky et al. in 2008. Consequently, we used the same data sets for evaluation. KeyPathwayMiner is available online at http://keypathwayminer.mpi-inf.mpg.de.

Network visualization tools offer features enabling a variety of analyses to satisfy diverse requirements. Considering complexity and diversity of data and tasks, there is no single best layout, no single best file format or visualization tool: one size does not fit all. One way to cope with these dynamics is to support multiple scenarios and workflows. NAViGaTOR (Network Analysis, Visualization & Graphing TORonto) offers a complete system to manage diverse workflows from one application. It allows users to manipulate large graphs interactively using an innovative graphical user interface (GUI) and through fast layout algorithms with a small memory footprint. NAViGaTOR facilitates integrative network analysis by supporting not only visualization but also visual data mining.