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) 4972-4975Horizontal 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)
Ryan Flanagan, Lucas Lacasa, Vincenzo Nicosia
Journal of Physics A: Mathematical and Theoretical 53, 2 (2019)
Visibility graphs and stochastic dynamics
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
Lucas Lacasa, Wolfram Just
Physica D 374, 35-44 (2018)
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)
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)
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