Theoretical results on the topological properties of the limited penetrable horizontal visibility graph family

The limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks. We extend this visibility graph and create a directed limited penetrable horizontal visibility graph and an image limited penetrable horizontal visibility graph. We define the two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real-value series (or matrices). We perform several numerical simulations to further check the accuracy of our theoretical results. Finally we present an application of the directed limited penetrable horizontal visibility graph for measuring real-value time series irreversibility, and an application of the image limited penetrable horizontal visibility graph that discriminates noise from chaos. The empirical results show the effectiveness of our proposed algorithms.