  Whether comparing networks to each other or to random expectation, measuring
dissimilarity is essential to understanding the complex phenomena under study.
However, determining the structural dissimilarity between networks is an
ill-defined problem, as there is no canonical way to compare two networks.
Indeed, many of the existing approaches for network comparison differ in their
heuristics, efficiency, interpretability, and theoretical soundness. Thus,
having a notion of distance that is built on theoretically robust first
principles and that is interpretable with respect to features ubiquitous in
complex networks would allow for a meaningful comparison between different
networks. Here we introduce a theoretically sound and efficient new measure of
graph distance, based on the "length spectrum" function from algebraic
topology, which compares the structure of two undirected, unweighted graphs by
considering their non-backtracking cycles. We show how this distance relates to
structural features such as presence of hubs and triangles through the behavior
of the eigenvalues of the so-called non-backtracking matrix, and we showcase
its ability to discriminate between networks in both real and synthetic data
sets. By taking a topological interpretation of non-backtracking cycles, this
work presents a novel application of Topological Data Analysis to the study of
complex networks.
