Avalanche homology of digraphs via sandpile dynamics

Abstract

We introduce avalanche homology as a new (di)graph homology theory, based on the dynamics of the sandpile model. Avalanche homology is the simplicial homology of the avalanche complex generated from the sets of unstable vertices at the time steps of the sandpile dynamics. In this work we focus on digraphs, and our main results give the homotopy types of the avalanche complex for directed paths and directed cycles for certain initial configurations of the sandpile dynamics. Even for such simple digraphs a wide range of topologies can arise, and we compare this to the directed flag complex and to the recently introduced burning homology. Furthermore, the dynamics yields very naturally a filtered simplicial complex, and hence persistent avalanche homology.