Graphlet Laplacians: graphlet-based neighbourhoods highlight topology-function and topology-disease relationships

Motivation Laplacian matrices capture the global structure of networks and are widely used to study biological networks. However, the local structure of the network around a node can also capture biological information. Local wiring patterns are typically quantified by counting how often a node touches different graphlets (small, connected, induced sub-graphs). Currently available graphlet-based methods do not consider whether nodes are in the same network neighbourhood. Contribution To combine graphlet-based topological information and membership of nodes to the same network neighbourhood, we generalize the Laplacian to the Graphlet Laplacian, by considering a pair of nodes to be ‘adjacent’ if they simultaneously touch a given graphlet. Results We utilize Graphlet Laplacians to generalize spectral embedding, spectral clustering and network diffusion. Applying our generalization of spectral clustering to model networks and biological networks shows that Graphlet Laplacians capture different local topology corresponding to the underlying graphlet. In biological networks, clusters obtained by using different Graphlet Laplacians capture complementary sets of biological functions. By diffusing pan-cancer gene mutation scores based on different Graphlet Laplacians, we find complementary sets of cancer driver genes. Hence, we demonstrate that Graphlet Laplacians capture topology-function and topology-disease relationships in biological networks