A k-bar visibility representation of a digraph G assigns each vertex at most k horizontal segments in the plane so that G has an arc uv if and only if some segment for u “sees” some segment for v above it by a vertical line of sight. The (bar) visibility number b(G) of a digraph G is the least k permitting such a representation. Among other results, we show that b(G) ≤ 4 when G is a planar digr...

Visibility graph has established itself as a powerful tool for analyzing time series. We in this paper develop a novel multiscale limited penetrable horizontal visibility graph (MLPHVG). We use nonlinear time series from two typical complex systems, i.e., EEG signals and two-phase flow signals, to demonstrate the effectiveness of our method. Combining MLPHVG and support vector machine, we detec...

In this paper we consider the problem of drawing a graph in the plane so that the vertices of the graph are drawn as rectangles and the edges are horizontal or vertical line segments. We are in particular interested in drawings where each of the line segments can be thickened to have positive width, and none of these thickened segments (which we call bands of visibility) intersects the interior...

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the per...

A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning them properly may be difficult and, typically, based on a trial-and-error approach. In this work, we adopt a graph-based framework to interpret and characterize internal dynamics of a class of RNNs called echo state networks (ESNs). We design pr...

The widely-used Secchi disk method is re-examined from the modulation transfer aspect. Namely, by assuming a volume scattering function and applying small angle scattering approximation, we show that the Secchi depth and horizontal visibility can be determined using the water modulation transfer function and the corresponding spatial frequencies associated with the disk. A basic equation of Sec...

A set of high-resolution multi-spectral satellite images were collected and processed to estimate the sediment concentration in the sea-breaking zone. The value of Total Suspended Matter (TSM) is empirically related to the sea surface reflectance. However, in order to obtain accurate reflectance measurements from the satellite images, the effect of the atmosphere needs to be accounted for. A se...