IncGraph: Incremental graphlet counting for topology optimisation

<div><p>Motivation</p><p>Graphlets are small network patterns that can be counted in order to characterise the structure of a network (topology). As part of a topology optimisation process, one could use graphlet counts to iteratively modify a network and keep track of the graphlet counts, in order to achieve certain topological properties. Up until now, however, graphlets were not suited as a metric for performing topology optimisation; when millions of minor changes are made to the network structure it becomes computationally intractable to recalculate all the graphlet counts for each of the edge modifications.</p><p>Results</p><p>IncGraph is a method for calculating the differences in graphlet counts with respect to the network in its previous state, which is much more efficient than calculating the graphlet occurrences from scratch at every edge modification made. In comparison to static counting approaches, our findings show IncGraph reduces the execution time by several orders of magnitude. The usefulness of this approach was demonstrated by developing a graphlet-based metric to optimise gene regulatory networks. IncGraph is able to quickly quantify the topological impact of small changes to a network, which opens novel research opportunities to study changes in topologies in evolving or online networks, or develop graphlet-based criteria for topology optimisation.</p><p>Availability</p><p>IncGraph is freely available as an open-source R package on CRAN (incgraph). The development version is also available on GitHub (<a href="https://github.com/rcannood/incgraph" target="_blank">rcannood/incgraph</a>).</p></div

Predicted gene regulatory networks of model organisms are optimised to reduce the false positive rate.

<p>A) The number of redundant edges in each graphlet are counted. B) The network is optimised in order to obtain a lower redundancy over time. Two networks are shown, one before and one after the optimisation procedure. Edges coloured in red have been removed from the network after optimisation, green edges have been added. C) Starting from an empty network, the interactions are modified by iteratively evaluating the increase in redundancy of the next <i>k</i> interactions, and adding the first edge for which its redundancy is less than the 90^th percentile redundancy.</p

The number of mutated genes in relation to a certain number of top-ranked genes.

<p>Mutation plots showing the amount of genes that need to be sequenced (y-axis) in order to find a certain number of mutated genes (depicted on the x-axis), for the six different tumor types. A: colon cancer; B: pancreas cancer; C: breast cancer; D: ovarian cancer: E: glioblastoma; F: medulloblastoma.</p

Overview of the publically available tumor data sets, used in this study.

*<p> <i>one sample was excluded from this study due to a hypermutated profile caused by chemotherapeutic treatment.</i></p><p> <i>CN: copy number; GE: gene expression.</i></p

PPV plot of the fitSNP strategy for the combined tumor entities.

<p>A PPV plot for the fitSNP strategy, performed on the mutation data of all combined tumor entities, in function of different prioritization value cut-offs.</p

IncGraph is significantly faster than non-incremental approaches.

<p>For small networks, the execution time of IncGraph <i>T</i>_<i>IG</i> is already 50 times less than that of non-incremental approaches <i>T</i>_<i>NI</i>. This ratio increases even further for networks with higher numbers of nodes or higher average degrees.</p

A simple graphlet-based scoring method improves predicted regulatory networks.

<p>(A) The F1 score was calculated by calculating the harmonic mean of the AUROC and AUPR scores of the first 1000 interactions. (B) IncGraph is significantly faster than the non-incremental approach. Note that for each interaction added to the network, the graphlet counts of 100 putative interactions were evaluated.</p

Graphlet counting in a network characterises its local topologies.

<p>(A) In total, there are 30 different graphlets containing 2 to 5 nodes, ranging from <i>G</i>₀ to <i>G</i>₂₉. Orbits are an extension of graphlets which also take into account the position of a node within a graphlet. The 73 different orbits are coloured in red. (B) By counting the occurrences of these graphlets in the network, the local topology surrounding a node can be quantified.</p