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Cited by 1,826Complex Systems and Time Series AnalysisComplex Network Analysis TechniquesTime Series Analysis and Forecasting

From time series to complex networks: the visibility graph

Open access
Lacasa, Lucas, Luque, Bartolo, Ballesteros, Fernando, Luque, Jordi, Nuño, Juan Carlos·2008·Proceedings of the National Academy of Sciences
DOI 10.1073/pnas.0709247105PMC PMC2278201
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Abstract

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.

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1073/pnas.0709247105

Cites

22 of 22 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 9 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Resumen · p. 2

    The degree distribution of the visibility graph of a periodic series is formed by a finite number of peaks related to the series period.

    “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.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Resumen · p. 2

    The visibility graph of a time series is invariant under affine transformations of the series data, including rescaling and translation of both axes.

    “( 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).”
    Quote in the paperSubstantial claim
  • Findingown data
    § Resumen · p. 3

    The tail of the degree distribution of the visibility graph of a random series extracted from a uniform distribution fits an exponential distribution.

    “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”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Resumen · p. 1

    The visibility algorithm maps a time series into a network.

    “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.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Resumen · p. 2

    The visibility graph extracted from a time series is always undirected.

    “( ii ) undirected: the way the algorithm is built up, there is no direction defined in the links.”
    Quote in the paperSubstantial claim
  • May combine two claims. 6
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingown data
    § Resumen · p. 3

    The Brownian motion visibility graph shows the Small-World effect, with mean path length L(N) scaling as log(N).

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Resumen · p. 3

    The Conway series visibility graph shows a self-similar relation, with mean path length L(N) scaling as N^β.

    “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.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Resumen · p. 4

    In the deterministic fractal series, nodes A and B have typically the same degree, and their degree is the same as the right degree of C.

    “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”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § Resumen · p. 5

    The summation for the right degree of node C agrees with the number of irreducible polynomials of degree at most n over the Galois field GF(2).

    “Note that this summation agrees with the number of irreducible polynomials of degree at most n over the Galois field GF(2) [20]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [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

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § Resumen · p. 4

    Other fractal series (Q series, Stern series, Thue-Morse series) also yield similar results.

    “some other fractal series have been also studied (Q series [16], Stern series [17], Thue-Morse series [18], etc) with similar results.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [16] D. Hofstadter, G¨ odel, Escher, Bach , New York: Vintage Books, pp. 137-138 (1980).
    • cites [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.

    • cites [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.

  • May combine two claims.

    Interpretationcitation
    § Resumen · p. 6

    The method captures the hub repulsion phenomenon associated to fractal networks and distinguishes scale-free visibility graphs evidencing Small-World effect from those showing scale invariance.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [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.

  • May combine two claims.

    Interpretationnone given
    § Resumen · p. 6

    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.

    “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.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § Resumen · p. 4

    The visibility graph can distinguish different types of fractality.

    “we may argue that the visibility graph can actually distinguish different types of fractality”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § Resumen · p. 3

    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.

    “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].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [4]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

9 of 9

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The resulting visibility graph has characteristics that reflect the equivalent time series temporal structure and can be used to identify trends.

    “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

    The summary (p. 6) says the graph topology conserves time-series structure, supporting the first part of the claim. But the Figure 2 caption (p. 8) says the graph remains unchanged when a linear trend is added, so the paper does not support using the graph to identify such trends. (LLM, conf=0.87)

    Experimental — accuracy not yet measured

  • Recent work on characterizing complex events focuses on using network/graph methodology that can capture non-linear behavior.

    “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

    The Resumen (pp. 1, 6) presents a graph-based method for characterizing time series and shows that graph topology distinguishes periodic, random, and fractal series. It does not establish the broader claim about characterizing complex events or demonstrate that the method captures nonlinear behavior; detecting features of chaotic systems is identified as a subject for future investigation (p. 6). (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • 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.

    “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]

    Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

  • 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.

    “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]

    The visibility-algorithm description (Resumen, p. 1) says that a graph built from scalar time-series values is undirected because its links have no defined direction. It does not discuss VVG, multidimensional vectors, M = 1, or projection-based asymmetry as the reason directionality disappears. Its discussion of a possible directed graph instead relates direction to the temporal axis (Resumen, p. 6). (LLM, conf=0.87)

    Experimental — accuracy not yet measured

  • Time series are transformed into networks that conserve their topology, and allow the identification of varying temporal structures.

    “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

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. J Conway Some Crazy Sequences, 1988).
  2. MA Stern J Reine Angew Math 55, 193–220 (1858).

Cited by

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Related

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)
ORCID
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
DOI
10.1073/pnas.0709247105
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
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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