Visibility Graph Analysis (VGA)

• Visibility Graph Analysis (VGA) is a method that converts ordered data into networks through geometric visibility criteria to reveal underlying temporal and spatial dynamics.
• VGA employs constructions like HVG, NVG, and LPHVG to extract network invariants that help distinguish randomness, chaos, and memory effects in sequential data.
• Advances in algorithmic frameworks, including linear-time online methods, facilitate scalable implementations of VGA for diverse applications from physiology to urban design.

Visibility Graph Analysis (VGA) is a methodological framework that maps ordered sets—most commonly univariate time series or spatial point sequences—into graphs based on geometric visibility relations among their elements. This approach, grounded in combinatorial and graph-theoretic principles, facilitates the extraction of statistical, structural, and dynamical properties from sequential data by recasting them as network invariants. VGA encompasses a variety of construction schemes, including the Horizontal Visibility Graph (HVG), Natural Visibility Graph (NVG), and their extensions; each variant encodes different aspects of the original sequence’s structure and correlations. Analytical results, computational algorithms, and application-specific diagnostics have established VGA as a tool of choice for distinguishing randomness, chaos, memory effects, correlations, and more, across disciplines from nonlinear dynamics to spatial geometry.

1. Visibility Graph Constructions

Let $X = (x_1, x_2, \dots, x_n)$ denote an ordered set of real numbers, commonly a time series. The principal constructions are:

• Horizontal Visibility Graph (HVG): The undirected simple graph $G = (V, E)$ with $V = \{x_1, x_2, \dots, x_n\}$, where two distinct nodes $x_i$ and $x_j$ ($i < j$) are adjacent if and only if $x_k < \min\{x_i, x_j\}$ for every $i < k < j$. Each consecutive pair $(x_i, x_{i+1})$ is always connected (Gutin et al., 2010).
• Natural Visibility Graph (NVG): Two nodes $x_i, x_j$ ($i < j$) are adjacent if $x_k < x_i + \frac{x_j - x_i}{j - i}(k - i)$ holds for every $i < k < j$. This encodes convexity along the line segment between $(i, x_i)$ and $(j, x_j)$ (Cádiz et al., 12 Sep 2025, Gutin et al., 2010).
• Limited Penetrable Horizontal Visibility Graph (LPHVG$_\rho$): With penetrability parameter $\rho$, $x_i$ and $x_j$ are adjacent if at most $\rho$ intermediates $x_k$ ($i < k < j$) are not less than $\min(x_i, x_j)$ (Wang et al., 2017).
• Spatial and Polygonal VG: For simple polygons or spatial configurations, the visibility graph encodes which pairs of points are mutually “visible” by an unobstructed line segment, e.g., in pseudo-triangle polygons (Mehrpour et al., 2019).

The construction generalizes to weighted graphs that incorporate environmental attenuations or obstacle-induced modifications (Schwartz et al., 2021).

2. Characterization Theorems and Structural Properties

The mathematical structure of several VGA variants is characterized by rigorous theorems:

• HVG Characterization: A finite graph $G$ is an HVG if and only if it is outerplanar and contains a Hamiltonian path. This positions HVGs as a strict subset of outerplanar graphs and enables linear-time recognition via known algorithms (Gutin et al., 2010).
• Combinatorics on Words: Ordered numeric sets can be encoded as words over an alphabet. Edges in HVGs correspond to “visible pairs”; subfamilies such as unimodal HVGs and maximal HVGs are determined by word patterns, e.g., avoidance of specific ascent/descent motifs or “zig-zag” permutations (Gutin et al., 2010).
• Polygonal VGA: Visibility graphs of pseudo-triangle polygons are characterized by a set of geometric–combinatorial properties (e.g., corner detection, visibility intervals on concave chains, strong ordering in subgraphs) that are necessary and sufficient for realizability. These admit $O(|E|)$-time recognition and reconstruction algorithms (Mehrpour et al., 2019).

Analytical results also establish degree distributions, clustering bounds, and shortcut probabilities for various random and deterministic input ensembles, including LPHVG$_\rho$ and image-based generalizations (Wang et al., 2017).

3. Algorithmic Frameworks and Complexity

Algorithmic developments in VGA focus on scalable construction and efficient update in dynamic or streaming contexts:

• Classical Complexity: Naive VGA implementations have $O(n^2)$ time complexity. Divide-and-conquer and BST-based algorithms reduce average-case complexity to $O(n \log n)$ for both NVG and HVG (Yela et al., 2019).
• LOT Framework: The Linear-time Online Transformation (LOT) framework achieves $O(N)$ per-window update for both NVG and HVG via efficient adjacency dictionary management and backward sweeps. This makes real-time visibility graph extraction feasible for streaming data with sliding windows (Huang et al., 2023).
• On-Line/Streaming: Incremental tree-merge and online BST techniques allow batch updates and dynamic recomputation with no added computational cost over offline methods (Yela et al., 2019).
• Polynomial-Time Property Checking: For polygonal VGA, all necessary properties (e.g., chain visibility, blocking structure) can be checked in $O(|E|)$ time (Mehrpour et al., 2019).

These algorithmic advances are critical for deployment in high-throughput scenarios, latency-sensitive analysis, and interactive geometric applications.

4. Statistical and Dynamical Diagnostics

VGA translates sequence properties into network statistics that serve as diagnostics for underlying dynamics:

• Degree Distribution: For HVGs derived from i.i.d. white noise, $$P(k) = \frac{1}{3} \left( \frac{2}{3} \right)^{k-2}$$ for $k \geq 2$, with mean degree 4. For random time series, HVG and NVG degree distributions exhibit either exponential or power-law decay, depending on the process (Gutin et al., 2010, Cádiz et al., 12 Sep 2025).
• Randomness vs. Chaos: Chaotic maps yield non-universal (distinct exponential) decay rates in degree distributions, enabling discrimination from true stochasticity (Gutin et al., 2010).
• Hurst Exponent and Correlation Analysis: The VG degree exponent $\gamma$ provides a diagnostic for persistence and antipersistence. For processes with Hurst exponent $H \leq 0.5$, empirical findings indicate $\gamma \geq 2$. Linear relations $\gamma(H) \simeq 3.1 - 2H$ are observed for fractional Brownian motion in HVG (Cádiz et al., 12 Sep 2025).
• Motif Analysis: Sequential visibility motifs, or isomorphism types of small-window induced subgraphs, form robust, high-discrimination signatures for time series classification, including distinguishing deterministic and stochastic dynamics in both synthetic and physiological data (Iacovacci et al., 2015).

Across applications such as sandpile models and earthquake temporal sequences, VGA-based diagnostics recover scale-free properties, identify long-range correlations, and elucidate intermittency and memory (Adami et al., 2024, Kundu et al., 2020).

5. Geometric, Spatial, and Weighted Variants

Spatial extensions and environmental adaptations of VGA have broadened its relevance:

• Polygon and Pseudo-Triangle Visibility Graphs: For simple polygons (with or without concave chains), visibility relations are mapped considering geometric constraints. Linear-time algorithms allow reconstruction and recognition of such graphs when the geometric structure is partially known (Mehrpour et al., 2019).
• Weighted Visibility Graphs: Integrating attenuation mechanisms for rain, snow, and fog allows the derivation of contrast-weighted visibility graphs, with edge weights computed as $w_{ij} = \exp(-\sigma d_{ij})$, where $\sigma$ is environment-dependent (Schwartz et al., 2021). Weighted centrality and connectivity measures adapt naturally to this framework.
• Limited Penetrable and Image Visibility Graphs: Generalizations to allow a bounded number of occlusions ($\rho$) and to multidimensional image arrays further expand analytical reach (Wang et al., 2017).

Such variants are crucial for analyzing visibility and connectivity in architectural, urban, and spatially distributed settings.

6. Applications and Interpretive Insights

VGA has been applied to problems as diverse as self-organized critical phenomena in sandpiles, discrimination of seismic event subtypes, analysis of economic uncertainty indices, physiological signal classification, spatial design (with weather-aware visibility), and more:

• Sandpile Models: Analysis of avalanche time series via NVG reveals scale-free topology in both degree and betweenness, with higher-order simplicial structure and persistent homology exposing multiscale complexity (Adami et al., 2024).
• Seismology: Both magnitude and inter-event time sequences benefit from VGA diagnostics, revealing memory, long-range clustering, and intermittency patterns distinctive to earthquake classes (Kundu et al., 2020).
• Economic Systems: Mapping economic policy uncertainty indices to NVGs demonstrates persistence (large $H$), heavy-tailed connectivity, high clustering, and assortative mixing, supporting models of bursty volatility and transmission (Dai et al., 2020).
• Dynamic Motif Analysis: Motif profile vectors efficiently discriminate between meditative and non-meditative heart-rate series, showing robustness to noise and strong practical separability (Iacovacci et al., 2015).
• Design and Planning: Weighted VGAs incorporating environmental obstacles and attenuation coefficients guide optimal placement in urban and architectural spaces under adverse weather conditions (Schwartz et al., 2021).

In all cases, VGA acts as a bridge translating geometrical or temporal structure into graph-theoretic invariants, enabling the application of network science methods to previously inaccessible domains.

7. Limitations and Open Problems

While VGA provides rigorous structural insight, certain theoretical and practical limitations remain:

• Non-Universality and Thresholds: Not all dynamical systems yield universal or easily parameterized degree statistics; for highly regular or complex deterministic maps (e.g., logistic map in full chaos), power-law signatures may break down or require interpretation via HVG (Cádiz et al., 12 Sep 2025).
• Recognizability and Generalization: For polygonal VGA, full characterization and reconstruction is known for towers, spirals, and pseudo-triangles, but the case of arbitrary simple polygons remains open, with general recognition still in PSPACE (Mehrpour et al., 2019).
• Finite-Size Effects: Accurate statistical inference using VGA requires sufficiently large datasets (empirical lower bound $N \gtrsim 1000$), as estimation variances and fitting error grow rapidly for small $N$ (Cádiz et al., 12 Sep 2025).
• Resource Constraints: For very large or dense windows, NVG may become memory-bound ($O(N^2)$ edges), and implementation overhead may be prohibitive in embedded or resource-limited environments (Huang et al., 2023).

A plausible implication is that future research will focus both on theoretical characterizations for broader geometric families and on scalable computation for resource-constrained and true real-time scenarios. As the field advances, extension to directed, weighted, and multiplex VGs, and deeper integration with topological data analysis, remain active areas of development.