Community detection in trajectory networks and simplicial complexes for extracting patterns in complex data

Community detection in complex networks has been very successful in understanding several systems where interactions are inherent. First, I introduce a novel trajectory-based method for identifying and predicting subtypes in heterogeneous and longitudinal data, i.e., that are characterized by time-varying interactions between various factors. The conventional Laplacian encodes many dynamical properties of a network, including community structure and flow of information along the network. Through spectral community detection in the graphical domain, I perform community detection on trajectory-based networks, for the application of identifying and predicting subtypes of diseases several years in advance.

While networks are a useful tool to represent data in the graphical domain, most systems naturally evolve to contain simultaneous interactions between more than two entities, represented as simplices - triangles, tetrahedra etc. Here I present the first work on revealing the relationship between a higher order equivalent of the Laplacian (Hodge Laplacian) and higher-order simplicial communities, demonstrating our results on both synthetic networks, as well as social and language networks.

I discuss the implications of Hodge decomposition on simplicial communities, and their relationship with clique communities. Lastly, I present a method to infer higher-order simplicial complexes from the network backbone, a question of some importance as simplicial datasets are rare.