Compression-based inference of network motif sets

Physical and functional constraints on biological networks lead to complex topological patterns across multiple scales in their organization. A particular type of higher-order network feature that has received considerable interest is network motifs, defined as statistically regular subgraphs. These may implement fundamental logical and computational circuits and are referred to as “building blocks of complex networks”. Their well-defined structures and small sizes also enable the testing of their functions in synthetic and natural biological experiments. Here, we develop a framework for motif mining based on lossless network compression using subgraph contractions. This provides an alternative definition of motif significance which allows us to compare different motifs and select the collectively most significant set of motifs as well as other prominent network features in terms of their combined compression of the network. Our approach inherently accounts for multiple testing and correlations between subgraphs and does not rely on a priori specification of an appropriate null model. It thus overcomes common problems in hypothesis testing-based motif analysis and guarantees robust statistical inference. We validate our methodology on numerical data and then apply it on synaptic-resolution biological neural networks, as a medium for comparative connectomics, by evaluating their respective compressibility and characterize their inferred circuit motifs.Author summary: Networks provide a useful abstraction to study complex systems by focusing on the interplay of the units composing a system rather than on their individual function. Network theory has proven particularly powerful for unraveling how the structure of connections in biological networks influence the way they may process and relay information in a variety of systems ranging from the microscopic scale of biochemical processes in cells to the macroscopic scales of social and ecological networks. Of particular interest are small stereotyped circuits in such networks, termed motifs, which may correspond to building blocks implementing fundamental operations, e.g., logic gates or filters. We here present a new tool that finds sets of motifs in networks based on an information-theoretic measure of how much they allow to compress the network. This approach allows us to evaluate the collective significance of sets of motifs, as opposed to only individual motifs. We apply our methodology to compare the neural wiring diagrams, termed “connectomes”, of the tadpole larva Ciona intestinalis, the ragworm Platynereis dumerelii, and the nematode Caenorhabditis elegans and the fruitfly Drosophila melanogaster at different developmental stages.