Horizontal Visibility Graph (HVG)

• Horizontal Visibility Graph is a method that converts a scalar time series into an undirected complex network based on horizontal visibility criteria.
• It employs efficient, stack-based, linear-time algorithms to reveal universal degree distributions for randomness testing, contrasting with the broader NVG approach.
• HVG’s structural constraints yield outerplanar, unigraph networks that facilitate motif profiling, percolation studies, and robust feature extraction in various applications.

A Horizontal Visibility Graph (HVG) is a particular instance of the visibility-graph family, which systematically transforms a real-valued time series or scalar field into a complex network. The transformation is determined by the geometric constraints on "visibility" between points, with HVG using a purely horizontal (min-based) criterion, in contrast to the more general convexity-based linkage of the Natural Visibility Graph (NVG). This mapping allows the fundamental structure and temporal order of the original data to be interrogated using the advanced apparatus of network theory and combinatorics.

1. Formal Definition and Algorithmic Construction

Given a finite, real-valued time series $S = \{s_1, s_2, \ldots, s_N\}$, the HVG is an undirected graph $G = (V, E)$ with $V = \{1, 2, ..., N\}$. Two nodes $i < j$ are connected by an edge if and only if every intermediate datum is smaller than both endpoints: $s_k < \min(s_i, s_j),\quad \forall\, i < k < j$ That is, the horizontal line at elevation $\min(s_i, s_j)$ "sees" no higher or equal intermediate value. The adjacency matrix $A$ is symmetric and binary, and the graph is always connected due to the sequential nature of time series data; adjacent time indices are always linked.

Algorithmic implementation for univariate time series proceeds as follows:

• For all $i = 1, ..., N-1$: For all $j = i+1, ..., N$: If $s_k < \min(s_i, s_j)\ \forall\ i < k < j$, then add undirected edge $G = (V, E)$0 to $G = (V, E)$1.

This minimality condition yields efficient, $G = (V, E)$2 stack-based construction algorithms, in contrast to the $G = (V, E)$3 (or higher) cost of convexity-based NVG, as explicit convexity checks are unnecessary (Lacasa et al., 2015, Lacasa et al., 2017).

2. Geometric Interpretation and Distinction from NVG

While the HVG criterion requires only that all intermediate points be strictly below both $G = (V, E)$4 and $G = (V, E)$5, NVG allows $G = (V, E)$6 to lie below the straight line joining $G = (V, E)$7 and $G = (V, E)$8, thereby relaxing the constraint. Consequently, every HVG is a subgraph of the corresponding NVG. The HVG captures only "short-range" visibilities, as even minor fluctuations block long-range links, while NVG encodes both local and global structure, being sensitive to convexity over time (Adami et al., 2024, Lacasa et al., 2015).

In the spatial (multi-dimensional) case, the HVG extension links nodes along pre-specified lattice directions if, and only if, all intermediate scalar field values along the straight path are less than the minimum at the endpoints: $G = (V, E)$9 with $V = \{1, 2, ..., N\}$0 and $V = \{1, 2, ..., N\}$1 denoting the values at the two grid-points joined by the direction $V = \{1, 2, ..., N\}$2 (Lacasa et al., 2017).

3. Analytical Properties, Degree Distributions, and Randomness Testing

HVGs associated with uncorrelated random series or fields admit closed-form, universal degree distributions that do not depend on the marginal distribution of the input. Specifically, for a $V = \{1, 2, ..., N\}$3-dimensional scalar field with $V = \{1, 2, ..., N\}$4 lattice directions,

$V = \{1, 2, ..., N\}$5

This result arises because each node has exactly $V = \{1, 2, ..., N\}$6 nearest neighbors and each additional visible node represents another consecutive segment with all sub-values below a (random) threshold. This universal property facilitates the use of the degree distribution as a randomness test: empirical deviation from the predicted $V = \{1, 2, ..., N\}$7 indicates the presence of correlation or structure in the original series or field (Lacasa et al., 2017). The functional tail $V = \{1, 2, ..., N\}$8 is thus a diagnostic of nontrivial temporal or spatial ordering.

4. Motif Profiles and Dynamical Information

The motif profile of an HVG refers to the frequencies of all possible small induced subgraphs (motifs). For four-node HVG (and NVG) subgraphs, only certain motifs can occur, reflecting the irreducible combinatorial content of local patterns. The motif frequencies are sensitive to dynamical features such as marginals, correlations, Markov versus deterministic structure, and noise contamination. For HVG, motif extraction admits linear-time algorithms and is robust to sampling noise and modest contamination levels (Iacovacci et al., 2016).

Motif profiles have been used to classify time series as stochastic, deterministic, chaotic, and differentially identify signals with distinct dynamical origins, as well as to analyze noise-to-signal transitions.

5. Computational Complexity and Algorithmic Developments

The stack-based HVG algorithm, due to its minimal intermediate-value check, achieves $V = \{1, 2, ..., N\}$9 offline performance for sequences of length $i < j$0 (Lacasa et al., 2015, Huang et al., 2023). For streaming or online applications, the LOT framework enables both HVG and NVG construction in linear time per window, using adjacency dictionaries and efficient update/elimination operations (Huang et al., 2023). Binary search tree (BST)-based encoders and decoders further increase scalability, offering on-line batch merge and efficient incremental construction (Yela et al., 2019). Practical performance benchmarks confirm orders-of-magnitude acceleration for HVG relative to NVG, particularly for large datasets.

6. Structural and Graph-Theoretic Properties

HVGs are outerplanar by construction, and can be uniquely determined by their degree sequence (unigraph property), which does not extend to NVG. HVGs coincide with certain subclasses of unit-interval graphs, but are not general interval graphs. The absence of induced cycles above certain length, forbidden antihole patterns, and tight constraints on motif orderings further constrain HVG topology, facilitating more efficient motif census and distance computations (Froese et al., 2019).

7. Applications and Limitations

HVG analysis has been effectively applied to problems as diverse as turbulence analysis, medical signal classification, image processing, and the study of complex dynamical systems (Iacobello et al., 2017, Lacasa et al., 2017). In particular, HVG-based metrics are used for time series feature extraction, irreversibility diagnostics, percolation studies (through parametric view-angle extensions), and randomness detection. However, due to the exclusion of convexity information, HVG is less sensitive to long-range correlations or geometric trends than NVG. Its construction is invariant under monotonic transformations, emphasizing the importance of the ordering rather than amplitude.

In summary, the Horizontal Visibility Graph provides a minimal, computationally light, and analytically tractable framework for translating the order structure of a time series into a complex network, making it a standard approach in high-throughput time series and spatial field data mining (Lacasa et al., 2015, Iacovacci et al., 2016, Lacasa et al., 2017, Huang et al., 2023).