Graphlet-based hyperbolic embeddings capture evolutionary dynamics in genetic networks.

Motivation: Spatial Analysis of Functional Enrichment (SAFE) is a popular tool for biologists to investigate the functional organization of biological networks via highly intuitive 2D functional maps. To create these maps, SAFE uses Spring embedding to project a given network into a 2D space in which nodes connected in the network are near each other in space. However, many biological networks are scale-free, containing highly connected hub nodes. Because Spring embedding fails to separate hub nodes, it provides uninformative embeddings that resemble a 'hairball'. In addition, Spring embedding only captures direct node connectivity in the network and does not consider higher-order node wiring patterns, which are best captured by graphlets, small, connected, nonisomorphic, induced subgraphs. The scale-free structure of biological networks is hypothesized to stem from an underlying low-dimensional hyperbolic geometry, which novel hyperbolic embedding methods try to uncover. These include coalescent embedding, which projects a network onto a 2D disk.

Results: To better capture the functional organization of scale-free biological networks, whilst also going beyond simple direct connectivity patterns, we introduce Graphlet Coalescent (GraCoal) embedding, which embeds nodes nearby on a disk if they frequently co-occur on a given graphlet together. We use GraCoal to extend SAFE-based network analysis. Through SAFE-enabled enrichment analysis, we show that GraCoal outperforms graphlet-based Spring embedding in capturing the functional organization of the genetic interaction networks of fruit fly, budding yeast, fission yeast and Escherichia coli. We show that depending on the underlying graphlet, GraCoal embeddings capture different topology-function relationships. We show that triangle-based GraCoal embedding captures functional redundancies between paralogs.

Availability and implementation: https://gitlab.bsc.es/swindels/gracoal_embedding.