Visibility graphs

Visibility graphs originate in Computer Science to model the combinatorial structure of intervisible locations. By re-interpreting the set of locations as an ordered sequence of marked events, we proposed to extract visibility graphs from time series, hence providing a combinatorial representation of trajectories and their underlying dynamics.
The concept can also be extended to multivariate time series (via multiplex visibility graphs) and to image processing

Key papers

Introducing natural and horizontal visibility graphs for time series analysis

From time series to complex networks: the visibility graph
Lucas Lacasa, Bartolo Luque, Fernando Ballesteros, Jordi Luque, Juan C. Nuño
PNAS, vol. 105, no. 13 (2008) 4972-4975
Horizontal visibility graphs: exact results for random time series
Bartolo Luque, Lucas Lacasa, Jordi Luque, Fernando J. Ballesteros
Physical Review E 80, 046103 (2009)

Theory for visibility graphs (time series version)

On the degree distribution of horizontal visibility graphs associated to Markov processes and dynamical systems: diagrammatic and variational approaches

Lucas Lacasa
Nonlinearity 27, 2063-2093 (2014)

Sequential visibility-graph motifs
Jacopo Iacovacci, Lucas Lacasa
Physical Review E 93, 042309 (2016)
Horizontal Visibility Graphs from Integer Sequences

Lucas Lacasa

Journal of Physics A: Mathematical and Theoretical 49, 35LT01 (2016)

Canonical horizontal visibility graphs are uniquely determined by their degree sequence

Bartolo Luque, Lucas Lacasa

European Physical Journal Special Topics 226, 383 (2017)

On the spectral properties of Feigenbaum graphs

Ryan Flanagan, Lucas Lacasa, Vincenzo Nicosia

Journal of Physics A: Mathematical and Theoretical 53, 2 (2019)

Visibility graphs and stochastic dynamics

The Visibility Graph: a new method for estimating the Hurst exponent of fractional Brownian motion

Lucas Lacasa, Bartolo Luque, Jordi Luque and Juan Carlos Nuño

EPL 86 (2009) 30001

Time reversibility from visibility graphs of non-stationary processes

Lucas Lacasa and Ryan Flanagan

Physical Review E 92, 022817 (2015)

Visibility graphs and deterministic dynamics

Visibility graphs and symbolic dynamics

Lucas Lacasa, Wolfram Just

Physica D 374, 35-44 (2018)

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

Bartolo Luque, Lucas Lacasa, Fernando J. Ballesteros, Alberto Robledo

Chaos 22, 013109 (2012)

Feigenbaum graphs at the onset of chaos

Bartolo Luque, Lucas Lacasa, Alberto Robledo

Physics Letters A 376 (2012)

Horizontal Visibility graphs generated by type-I intermittency

Angel Nuñez, Bartolo Luque, Lucas Lacasa, Jose Patricio Gómez, Alberto Robledo

Physical Review E 87, 052801 (2013)

Horizontal Visibility graphs generated by type-II intermittency

Angel Nuñez, Jose Patricio Gómez, Lucas Lacasa

Journal of Physics A: Mathematical and Theoretical 47, 035102 (2014)

Quantifying irreversibility

Time series irreversibility: a visibility graph approach

Lucas Lacasa, Angel Nuñez, Edgar Roldán, Juan MR Parrondo, Bartolo Luque

European Physical Journal B 85, 217 (2012)

Time reversibility from visibility graphs of non-stationary processes

Lucas Lacasa and Ryan Flanagan

Physical Review E 92, 022817 (2015)

Arrow of time across five centuries of classical music

Alfredo Gonzalez-Espinosa, Gustavo Martinez-Mekler, Lucas Lacasa

Physical Review Research 2, 033166 (2020)

Featured in Scientific American , Investigacion y Ciencia

Extension to multivariate time series, random fields and images

Network Structure of Multivariate Time Series
Lucas Lacasa, Vincenzo Nicosia and Vito Latora
Scientific Reports 5, 15508 (2015)
Visibility graphs of random scalar fields and spatial data
Lucas Lacasa, Jacopo Iacovacci
Physical Review E 96, 012318 (2017)
Visibility graphs for image processing

Jacopo Iacovacci, Lucas Lacasa

IEEE Transactions in Pattern Analysis and Machine Intelligence 42, 4 (2020)

Software

Horizontal and Directed Horizontal visibility graphs (Fortran 90/95)
This code generates the adjacency matrix and the degree distributions of both HVG and DHVG associated to a series of arbitrary size. The execution time for noisy (stochastic/chaotic) series is O(N).

If you use this code, please cite
[1] Horizontal visibility graphs: exact results for random time series Bartolo Luque, Lucas Lacasa, Jordi Luque, Fernando J. Ballesteros Physical Review E 80, 046103 (2009)
[2] Time series irreversibility: a visibility graph approach Lucas Lacasa, Angel Nuñez, Edgar Roldán, Juan MR Parrondo, Bartolo Luque European Physical Journal B 85, 217 (2012)

Visibility and Directed Visibility graphs (Fortran 90/95)

This code generates the adjacency matrix and the degree distributions of both VG and DVG associated to a series of arbitrary size. The execution time for noisy (stochastic/chaotic) series is O(N^2).

If you use this code, please cite
[1] From time series to complex networks: the visibility graph Lucas Lacasa, Bartolo Luque, Fernando Ballesteros, Jordi Luque, Juan C. Nuño PNAS, vol. 105, no. 13 (2008) 4972-4975
[2] Time series irreversibility: a visibility graph approach Lucas Lacasa, Angel Nuñez, Edgar Roldán, Juan MR Parrondo, Bartolo Luque European Physical Journal B 85, 217 (2012)

Sequential visibility graph motifs (Matlab)

Github repository elaborated by Jacopo Iacovacci This repository contains:
1) 'HVG_motifs.m' a matlab function to extract the horizontal VG motif profile from a time series for motifs of size n=4.
2) 'NVG_motifs.m' a matlab function to extract the natural VG motif profile from a time series for motifs of size n=4.

If you use this code, please cite
[1] Sequential visibility graph motifs Jacopo Iacovacci, Lucas Lacasa Physical Review E 93, 042309 (2016)
[2] Sequential motif profile of natural visibility graphs Jacopo Iacovacci, Lucas Lacasa Physical Review E (in press 2016), arXiv:1605.02645

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