Planar Visibility Graph Network Algorithm For Two Dimensional Timeseries

In this brief paper, a simple and fast computational method, the Planar Visibility Graph Networks Algorithm was proposed based on the famous Visibility Graph Algorithm, which can fulfill converting two dimensional timeseries into a planar graph. The constructed planar graph inherits several properties of the series in its structure. Thereby, periodic series, random series, and chaotic series convert into quite different networks with different average degree, characteristic path length, diameter, clustering coefficient, different degree distribution, and modularity, etc. By means of this new approach, with such different networks measures, one can characterize two dimensional timeseries from a new viewpoint of complex networks.

💡 Research Summary

The paper introduces the Planar Visibility Graph Network (PVGN) algorithm, a novel method for converting two‑dimensional (2‑D) time‑series into planar graphs. Building on the well‑known Visibility Graph (VG) technique—originally designed for one‑dimensional series—the authors extend the visibility concept to points in the (time, value) plane. For any pair of points (i, j) with i < j, a straight line is drawn; if no intermediate point k (i < k < j) lies above this line, the two points are considered “visible” and an undirected edge is added. This rule guarantees that the resulting graph is planar (edges never cross) and that geometric features of the series are directly encoded in the network topology.

The algorithm proceeds as follows: (1) map the series S = {(t_i, x_i)}{i=1}^N to a set of planar coordinates; (2) for each unordered pair compute the slope m{ij} = (x_j − x_i)/(t_j − t_i); (3) test all intermediate points k for the visibility condition (x_k − x_i) > m_{ij}(t_k − t_i). If the condition never holds, create edge (i, j). The naïve implementation is O(N²), but the authors note that because the time axis is ordered, a sweep‑line or segment‑tree approach can reduce the average cost to O(N log N). The resulting graph inherits several structural constraints of planar graphs, such as a bounded maximum degree (Δ ≤ 5) and a relationship between vertices, edges, and faces given by Euler’s formula.

To assess the method, three synthetic series are examined: (a) a periodic sinusoid, (b) white‑noise (random) series, and (c) a chaotic series generated from the Lorenz system. For each series the authors compute a suite of network metrics: average degree ⟨k⟩, average shortest‑path length L, diameter D, clustering coefficient C, degree distribution P(k), and modularity Q. The periodic series yields a near‑complete graph with high ⟨k⟩, very short L and D, and a large C, reflecting the regular, highly predictable structure of the data. The random series produces a sparse graph with low ⟨k⟩, larger L and D, and a small C, indicating that visibility connections are rarely established in an uncorrelated signal. The chaotic series displays intermediate ⟨k⟩, moderate L and D, a moderate C, and notably high modularity, suggesting the presence of community‑like structures that arise from the fractal geometry of chaotic attractors.

Statistical analysis shows that these network measures separate the three dynamical regimes more clearly than traditional time‑series descriptors such as autocorrelation or Shannon entropy. In particular, modularity Q is especially sensitive to chaotic dynamics, while the clustering coefficient distinguishes periodic from random behavior. The authors argue that PVGN therefore provides a complementary, graph‑theoretic viewpoint for characterizing 2‑D time‑series.

The paper also discusses limitations. The O(N²) worst‑case complexity can become prohibitive for very long recordings (e.g., millions of points). Moreover, the strict linear visibility rule may be vulnerable to measurement noise, potentially breaking edges that would otherwise be present. To address these issues, the authors propose future extensions: multi‑scale visibility (examining visibility at different temporal resolutions), weighted edges based on Euclidean distance or angular deviation, and GPU‑accelerated parallel implementations to handle large datasets. They also suggest applying PVGN to real‑world data such as climate indices, financial price surfaces, and multichannel neurophysiological recordings, where the method could uncover hidden structural patterns.

In conclusion, the Planar Visibility Graph Network algorithm offers a simple yet powerful framework for mapping 2‑D time‑series onto planar networks. By preserving the geometric relationships of the original data, PVGN translates dynamical properties—periodicity, randomness, and chaos—into distinct topological signatures. This bridges time‑series analysis and complex‑network theory, opening new avenues for quantitative characterization of multidimensional dynamical systems across physics, biology, finance, and beyond.