Motivation: Graphlets are a useful tool to determine a graph’s small-scale structure. Finding them is exponentially hard with respect to the number of nodes in each graphlet. Therefore, equations can be used to reduce the size of graphlets that need to be enumerated to calculate the number of each graphlet touching each node. Hočevar and Demšar first introduced such equations, which were derived manually, and an algorithm that uses them, but only graphlets with four or five nodes can be counted this way.Results: We present a new algorithm for orbit counting, which is applicable to graphlets of any order. This algorithm uses a tree structure to simplify finding orbits, and stabilizers and symmetry-breaking constraints to ensure correctness. This method gives a significant speedup compared to a brute force counting method and can count orbits beyond the capacity of other available tools.Availability and implementation: An implementation of the algorithm can be found at https://github.com/biointec/jesse.

Graphlets are useful for bioinformatics network analysis. Based on the structure of Hočevar and Demšar’s ORCA algorithm, we have created an orbit counting algorithm, named Jesse. This algorithm, like ORCA, uses equations to count the orbits, but unlike ORCA it can count graphlets of any order. To do so, it generates the required internal structures and equations automatically. Many more redundant equations are generated, however, and Jesse’s running time is highly dependent on which of these equations are used. Therefore, this paper aims to investigate which equations are most efficient, and which factors have an effect on this efficiency. With appropriate equation selection, Jesse’s running time may be reduced by a factor of up to 2 in the best case, compared to using randomly selected equations. Which equations are most efficient depends on the density of the graph, but barely on the graph type. At low graph density, equations with terms in their right-hand side with few arguments are more efficient, whereas at high density, equations with terms with many arguments in the right-hand side are most efficient. At a density between 0.6 and 0.7, both types of equations are about equally efficient. Our Jesse algorithm became up to a factor 2 more efficient, by automatically selecting the best equations based on graph density. It was adapted into a Cytoscape App that is freely available from the Cytoscape App Store to ease application by bioinformaticians.

Background: De Bruijn graphs are key data structures for the analysis of next-generation sequencing data. They efficiently represent the overlap between reads and hence, also the underlying genome sequence. However, sequencing errors and repeated subsequences render the identification of the true underlying sequence difficult. A key step in this process is the inference of the multiplicities of nodes and arcs in the graph. These multiplicities correspond to the number of times eachk-mer (resp.k+1-mer) implied by a node (resp. arc) is present in the genomic sequence. Determining multiplicities thus reveals the repeat structure and presence of sequencing errors. Multiplicities of nodes/arcs in the de Bruijn graph are reflected in their coverage, however, coverage variability and coverage biases render their determination ambiguous. Current methods to determine node/arc multiplicities base their decisions solely on the information in nodes and arcs individually, under-utilising the information present in the sequencing data.Results: To improve the accuracy with which node and arc multiplicities in a de Bruijn graph are inferred, we developed a conditional random field (CRF) model to efficiently combine the coverage information within each node/arc individually with the information of surrounding nodes and arcs. Multiplicities are thus collectively assigned in a more consistent manner.Conclusions: We demonstrate that the CRF model yields significant improvements in accuracy and a more robust expectation-maximisation parameter estimation. Truek-mers can be distinguished from erroneousk-mers with a higher F₁score than existing methods. A C++11 implementation is available athttps://github.com/biointec/detoxunder the GNU AGPL v3.0 license.

Networks and graphs are highly relevant in modeling real-life communities and their interactions. In order to gain insight in their structure, different roles are attributed to vertices, effectively clustering them in equivalence classes. A new formal definition of regular equivalence is presented in this paper, and the relation with other equivalence types is investigated and mathematically proven. An efficient algorithm is designed, able to detect all regularly equivalent roles in large-scale complex networks. We apply it to both Barabási–Albert random networks, as well as real-life social networks, which leads to interesting insights.