Research on multi-fractals of weighted hypernetworks

Hypernetwork is a kind of network that is more complex than ordinary complex networks, while hypernetwork dimension is a feasible tool for measuring it. For the case that both node weight and hyperedge weight in weighted hypernetworks can value among positive real number, negative real number, pure imaginary number and complex number, etc., fractal dimensions for various weighted hypernetworks is proposed and their multi-fractals are discussed. It is shown that among all types of weighted hypernetworks, except those with both node weight and hyperedge weight being positive or negative real numbers, other types of weighted hypernetworks share multi-fractals and can be divided into seven different categories that are all distributed in the entire complex plane. The analytic expressions of multi-fractal dimensions for all types of weighted hypernetworks are also presented. Finally, some important properties of the multi-fractal dimensions of these weighted hypernetworks are analyzed.