Degree distributions and motif profiles of limited penetrable horizontal visibility graphs

The algorithm of limited penetrable horizontal visibility graphs (LPHVGs) including the limited penetrable horizontal visibility graph [LPHVG(ρ)], the directed limited penetrable horizontal visibility graph [DLPHVG(ρ)] and the image limited penetrable horizontal visibility graph [ILPHVGn(ρ))] are used to map time series (or matrices) on graphs and are powerful tools for analyzing time series. We derive the degree distributions of LPHVGs using an iterative LPHVGs construction process. We propose a more intuitive method of reproducing the construction process of limited penetrable horizontal visibility graphs that is simple to calculate. We find that the results confirm the analytical results from previous methods. We then introduce the concept of sequential LPHVG(ρ) motifs and present a theoretical way of computing the exact motif profiles associated with unrelated random series. We perform several numerical simulations to further check the accuracy of our theoretical results. Finally we use the analytical results of LPHVG(ρ) motif profiles to distinguish among random, periodic, and chaotic signals and find that the frequency of the type-I motif captures sufficient information to easily distinguish among different processes.