Computing singular simplicial homologies of digraphs and quivers

Abstract

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as networks within numerous scientific disciplines. This homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large networks. By extending the ideas of singular simplicial homology to quivers in categories of different morphism types, several new singular simplicial homology theories have recently been constructed. Computationally efficient homologies for quivers have in general not previously been seriously considered. In this paper we develop further the homotopy theory of quivers necessary to derive functors that realise isomorphisms between the singular simplicial quiver homologies and the homologies of certain spaces. The simplicial chains of these spaces arise in a conveniently compact form that is at least as convenient as the directed flag complex for computations. Moreover, our constructions are natural with respect to the isomorphisms on homology making them suitable for applications in conjunction with persistent homology for practical use. In particular, for each of the singular simplicial homologies considered, we provide efficient algorithms for the computation of their persistent homology.