Constant-Time Algorithms for Complex Networks

In this paper, we study constant-time algorithms on complex networks. In a recent work presented in ESA~2016, the author introduced a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs and showed that a very wide subclass of HSF is hyperfinite. Based on this result, a surprising result such that every property is constant-time testable for the subclass was obtained. However, this result is theoretical, and the algorithm obtained directly from it may not be practical. In this paper, first we present the result of our paper of ESA~2016, and next we consider what kind of properties are "efficiently" testable in practice. For this purpose, we consider hereditary properties. We show that for hyperfinite graph class, e.g., HSF, any hereditary property can be represented by the forbidden graph subset such that the number of graphs in the subset is bounded by a constant. From this result, it can be considered that to test a hereditary property is relatively easy.