Every Property Is Testable on a Natural Class of Scale-Free Multigraphs

In this paper, we introduce a natural class of multigraphs, Hierarchical-Scale-Free (HSF, in short), and consider constant-time testability on the class. We show that a very wide subclass, i.e., the power-low exponent is greater than two, of HSF is hyperfinite. Based on this result, an algorithm of deterministic partitioning oracle can be constructed. And finally we show that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes the result of Newman and Sohler of STOC’11. However, their algorithm is on the bounded-degree model, while it is known that the real scale-free networks usually includes some hubs, which have very large degree. HSF is based on the scale-free property and includes such hubs. This is the first universal result of constant-time testability on the general graph model, and it has a potential to be applicable on very wide real scale-free networks.