Horizontal visibility graphs from integer sequences

The horizontal visibility graph (HVG) is a graph-theoretical representation of a time series and builds a bridge between dynamical systems and graph theory. In recent years this representation has been used to describe and theoretically compare different types of dynamics and has been applied to characterize empirical signals, by extracting topological features from the associated HVGs which have shown to be informative on the class of dynamics. Among some other measures, it has been shown that the degree distribution of these graphs is a very informative feature that encapsulates nontrivial information of the series's generative dynamics. In particular, the HVG associated to a bi-infinite real-valued series of independent and identically distributed random variables is a universal exponential law , independent of the series marginal distribution. Most of the current applications have however only addressed real-valued time series, as no exact results are known for the topological properties of HVGs associated to integer-valued series. In this paper we explore this latter situation and address univariate time series where each variable can only take a finite number n of consecutive integer values. We are able to construct an explicit formula for the parametric degree distribution , which we prove to converge to the continuous case for large n and deviates otherwise. A few applications are then considered.