Persistent homology in graph power filtrations

The persistence of homological features in simplicial complex representations of big datasets in R resulting from Vietoris-Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r ≥ 1, in such a power filtration is the clique complex of the rth power G of a simple graph G. Because the graph distance in G is the relevant proximity parameter, unlike a Euclidean filtration of a dataset where regional scale differences can be an issue, persistence in power filtrations provides a scale-free insight into the topology of G. It is shown that for a power filtration of G, the girth of G defines an r range over which the homology of the complexes in the filtration are guaranteed to persist in all dimensions. The role of chordal graphs as trivial homology delimiters in power filtrations is also discussed and the related notions of 'persistent triviality', 'transient noise' and 'persistent periodicity' in power filtrations are introduced.