Before Statistics on network (III): Bootstrapping on directed binary networks via statistical mechanics

Abstract

We explicitly computed and demonstrated that structural information of a seemingly simplistic directed binary network is rather complex with multiple facets. The information is composed of one Parisi adjacency matrix, a power hierarchy and their coupling mechanism. The Parisi adjacency matrix reveals multiscale structural blocks on its undirected binary schedule component, the power hierarchy reveals structural flows on its directed dominance component, while the coupling mechanism couples these two components into a dynamic system. A previously developed Beta Random Field is adopted for such a coupling mechanism. And a two-step bootstrapping algorithm is devised: 1) a schedule matrix is simulated by mimicking a computed Parisi adjacency matrix; 2) the simulated schedule matrix is then turned into a binary ``win-and-loss'' matrix through the Beta Random field functionally governed by the power hierarchy. Such a network bootstrapping can retain all computed global multiscale and flow features. Our computational developments are illustrated through NCAA football network data sets, and manifest the realistic sensitivity. This sensitivity reminds us that intricate multiscale nature and difficult complexity are simultaneously embraced in most of complex systems with directed information flows. Concerns of scale and complexity imply complicate tasks for mimicking, bootstrapping or modeling a directed binary network. ??