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From time series to complex networks: the visibility graph
Open accessAbstract
In this work we present a simple and fast computational method, the visibility algorithm , that converts a time series into a graph. The constructed graph inherits several properties of the series in its structure. Thereby, periodic series convert into regular graphs, and random series do so into random graphs. Moreover, fractal series convert into scale-free networks, enhancing the fact that power law degree distributions are related to fractality, something highly discussed recently. Some remarkable examples and analytical tools are outlined to test the method's reliability. Many different measures, recently developed in the complex network theory, could by means of this new approach characterize time series from a new point of view.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
The paper introduces the visibility algorithm, which maps a time series into a graph by connecting data points that can “see” each other without intermediate heights blocking the line of sight. It shows that periodic series yield regular graphs, random series yield random graphs with exponential degree tails, and fractal series yield scale-free visibility graphs.
Key findings
- Periodic time series map to regular graphs whose discrete degree distribution has peaks related to the series period; the example in Fig. 1 is a period-4 series.
- A random series of \(10^6\) data from \(U[0,1]\) produces a visibility graph whose degree distribution has an exponential tail, attributed to rare large values acting as hubs.
- Brownian motion visibility graphs are scale-free with \(P(k) \sim k^{-\alpha}\), \(\alpha = 2.00 \pm 0.01\), and show a Small-World effect with \(L(N) = 1.21 + 0.51\log(N)\).
- Conway series visibility graphs are scale-free with \(\alpha = 1.2 \pm 0.1\) and scale-invariant path length \(L(N) = 0.76 N^{0.38}\); a deterministic fractal example is also shown analytically to yield a power-law degree distribution.
Methods
- The visibility algorithm maps each time-series datum to a node and links two nodes if the straight line between their data values does not intersect any intermediate data height.
- The paper analyzes periodic series, a random series of \(10^6\) points from \(U[0,1]\), Brownian motion of \(10^6\) data, a Conway series of \(4 \cdot 10^6\) data, and a deterministic iterated fractal series.
- Main analyses include degree distributions \(P(k)\), mean path length \(L(N)\), and analytical recurrence relations for the deterministic fractal example.
Limitations
- The paper states that information is inevitably lost because the network structure is completely determined by the binary adjacency matrix; for example, periodic series \(T_1 = \{\ldots,3,1,3,1,\ldots\}\) and \(T_2 = \{\ldots,3,2,3,2,\ldots\}\) have the same visibility graph despite being quantitatively different.
- The paper states that it only tackles undirected graphs, while a directed version related to temporal axis direction is left for further work.
- The paper notes that the simple deterministic fractal example does not provide a general explanation of why fractality translates into power-law distributions, though it may stand as a generic way to deal with deterministic fractal series generated by iteration.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The degree distribution of this graph is formed by a finite number of peaks related to the series period, much in the vein of the Fourier Power Spectrum of a time series.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“( iii ) invariant under affine transformations of the series data: the visibility criterium is invariant under rescaling of both horizontal and vertical axis, as well as under horizontal and vertical translations (see figure 2).”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The tail of this distribution fits quite well an exponential distribution, as expected.”
- Also stated in: § Resumen · p. 3 · plain
“random series convert into exponential random graphs”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Resumen · p. 3 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this letter we introduce a new tool in time series analysis: the visibility graph . This algorithm maps a time series into a network.”
- Also stated in: § Resumen · p. 1 · plain
“the visibility algorithm , that converts a time series into a graph.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Resumen · p. 1 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“( ii ) undirected: the way the algorithm is built up, there is no direction defined in the links.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“( i ) connected: each node sees at least its nearest neighbors (left and right).”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This one is a power law P ( k ) ∼ k -α with α = 1 , 2 ± 0 , 1.”
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“that is, the degree distribution related to the C-nodes is a power law.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This one is a power law P ( k ) ∼ k -α with α = 2 , 00 ± 0 , 01.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is indeed due to these 'unlikely' large values (the hubs) that the tail of the degree distribution deviates from the Poisson distribution.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- It may state two separate claims as one. 6
- Probably not a substantial claim. 3
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“while the Brownian visibility graph seems to evidence the SmallWorld effect (right top figure 4) as L ( N ) ∼ log( N )”
- Also stated in: § Referencias · p. 12 · plain
“The best fitting provides a logarithmic scaling L ( N ) = 1 , 21 + 0 , 51log( N ). This network shows Small-World effect in addition to be scale-free.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § Referencias · p. 12 · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the Conway series shows in turn a self-similar relation (right bottom figure 4) of the shape L ( N ) ∼ N β”
- Also stated in: § Referencias · p. 12 · plain
“The best fitting provides a power law scaling L ( N ) = 0 , 76 N 0 , 38 . Then, this network is scale-invariant.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § Referencias · p. 12 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“nodes A and B have typically the same degree.”
- Also stated in: § Resumen · p. 4 · plain
“The degree of A and B is the same as the right degree of C”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Also stated in: § Resumen · p. 4 · plain
In the paper
“Note that this summation agrees with the number of irreducible polynomials of degree at most n over the Galois field GF(2) [20]”
What the sources say
- Not checked — no full text of the sourcecites [20] K.H. Hicks, G.L. Mullen, and I. Sato, Distribution of irreducible polynomials over F 2 , in Finite Fields with Applications to Coding Theory, Cryptography and Related Areas (Oaxaca, 2001), 177-186, Springer, Berlin, 2002. source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“some other fractal series have been also studied (Q series [16], Stern series [17], Thue-Morse series [18], etc) with similar results.”
What the sources say
- Not checked — no full text of the sourcecites [16] D. Hofstadter, G¨ odel, Escher, Bach , New York: Vintage Books, pp. 137-138 (1980).
- Not matchedcites [17] M.A. Stern, J. Reine Angew. Math. , 55, pp. 193-220 (1858).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [18] M.R. Schroeder, Fractals, Chaos, and Power Laws , New York: W. H. Freeman (1991).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“the method captures the hub repulsion phenomenon associated to fractal networks [8] and thus distinguishes scale free visibility graphs evidencing Small-World effect from those showing scale invariance.”
What the sources say
- Not matchedcites [8] C. Song, S. Havlin, H.A. Makse, Nat. Phys. 2 ,275 (2006).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The structure of the time series is conserved in the graph topology: periodic series convert into regular graphs, random series into random graphs and fractal series into scale-free graphs.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“we may argue that the visibility graph can actually distinguish different types of fractality”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“In every case the algorithm captures the random nature of the series, and the particular shape of the degree distribution of the visibility graph is related to the particular random process [4].”
What the sources say
- Not matchedcites [4]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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Claims other papers make about this one
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The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The resulting visibility graph has characteristics that reflect the equivalent time series temporal structure and can be used to identify trends.”
From PyIOmica: longitudinal omics analysis and trend identification (2020) · cites this paper as Lacasa et al., 2008
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Recent work on characterizing complex events focuses on using network/graph methodology that can capture non-linear behavior (Lacasa et al., 2008).”
From PyIOmica: longitudinal omics analysis and trend identification (2020) · cites this paper as Lacasa et al., 2008
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“The challenges in capturing relationships and dependencies among emotions indicate the necessity for an architecture that is both easy to construct and preserves the structure of the underlying signal in its topology to improve both performance and interpretability [21], [22].”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“In VVG, edge directionality stems from the projection-based asymmetry between multidimensional vectors.When M = 1, vectors reduce to scalars, eliminating projections and making visibility symmetric, resulting in an undirected graph [22], [25].”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Time series are transformed into networks that conserve their topology, and allow the identification of varying temporal structures.”
From PyIOmica: longitudinal omics analysis and trend identification (2020) · cites this paper as Lacasa et al., 2008
The cited source says what the claim attributes to it.
In the citing paper
“To this end, visibility graphs offer an effective means of representing sequential signals by connecting data points according to their value-based visibility, thereby preserving the underlying temporal structure while introducing geometric relationships [22], [23].”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
The cited source says what the claim attributes to it.
In the citing paper
“The vector visibility graph (VVG) algorithm [25] extends the univariate visibility graph [22], [24] concept to multivariate time series data”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
The cited source says what the claim attributes to it.
In the citing paper
“This simple yet versatile transformation supports both scalar [22], [24] and vector representations [25], with applications spanning time series analysis and language processing [26], [27].”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
The cited source says what the claim attributes to it.
In the citing paper
“The natural visibility graph (NVG) algorithm [22] is a foundational method for mapping univariate (scalar) time series into graph networks.”
From Emotion Classification With Visibility Graphs (2025) · cites this paper as [22]
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Self-citation
Metadata
- Authors
- Lacasa, L., Luque, B., Ballesteros, F., Luque, J., & Nuño, J. C.
- Affiliations
- Universidad Politécnica de Madrid (ES)
- Universitat de València (ES)
- Universitat Politècnica de Catalunya (ES)
- Venue
- Proceedings of the National Academy of Sciences
- Publisher
- National Academy of Sciences
- Published
- 2008-04
- Volume
- 105
- Issue
- 13
- Pages
- 4972-4975
- ISSN
- 0027-8424 (ISSN-L)
- 1091-6490
- PMC
- PMC2278201
- OpenAlex
- W2055538060
- Semantic Scholar
- a0c1483e0e34fb9bc1baf929ce892f111e329fa6
- Cited by
- 1,826
- References
- 22
- Citations by year
- 2026: 102 · 2025: 180 · 2024: 149 · 2023: 156 · 2022: 157 · 2021: 151 · 2020: 157 · 2019: 154 · 2018: 137 · 2017: 105 · 2016: 84 · 2015: 60 · 2014: 61 · 2013: 47 · 2012: 51
- FWCI
- 56.61
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Proceedings of the National Academy of Sciences (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- PubMed Central (submittedVersion)
- arXiv (Cornell University)
- Topics
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Physical Sciences › Computer Science › Signal Processing › Time Series Analysis and Forecasting
- Keywords
- visibility graph, time series, complex networks, scale-free networks, power-law degree distribution, fractality, regular graphs, random graphs, network metrics, time series characterization
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