Computing Homotopy Types of Directed Flag Complexes

Directed flag complexes are semisimplicial complexes which have recently been used as a tool to explore the global structure of directed graphs, most notably those arising from neuroscience. It is a folklore observation that random flag complexes often have the homotopy type of a wedge of spheres. One might therefore wonder whether this is also the case for graphs arising from nature. To explore this idea, we will take a look at the brain network of the C. Elegans nematode, an important model organism in biology, and show that the homotopy type of its directed flag complex can be computed in its entirety by an iterative approach using elementary techniques of algebraic topology such as homology, simplicial collapses and coning operations. Along the way, we will encounter some other interesting examples and properties of semisimplicial complexes.