Multivariate visibility graphs:from time series to temporal networks

An application to BOLD fMRI data

Lucas LacasaSchool of Mathematical SciencesQueen Mary University of London

@wetuad

Sebastiano StramagliaPhysics Department and INFNUniversity of Bari

@SebinoStram

Speranza SanninoDept of Electrical EngineeringUniversity of Cagliari

Daniele MarinazzoDepartment of Data Analysis

Ghent University@dan_marinazzo

Visibility graphs were defined in computational geometry/computer science as the backbone graph capturing visibility paths (intervisible locations) in landscapes

• Each node represents a location• Two locations are connected by a link if they are visible

Visibility graphs were defined in computational geometry/computer science as the backbone graph capturing visibility paths (intervisible locations) in landscapes

• Each node represents a location• Two locations are connected by a link if they are visible

1D LANDSCAPES CAN BE CONSIDERED AS TIME SERIES

1D LANDSCAPES CAN BE CONSIDERED AS TIME SERIES

1D LANDSCAPES CAN BE CONSIDERED AS TIME SERIES

1D LANDSCAPES CAN BE CONSIDERED AS TIME SERIES

Natural Visibility Algorithm

Lacasa, Luque, Ballesteros, Luque, Nuño, PNAS 105 (2008)

Example of a time series (20 data values) and the associated graph derived from the visibility algorithm.

Lacasa, Luque, Ballesteros, Luque, Nuño, PNAS 105 (2008)

Natural Visibility Horizontal Visibility

Natural version* naïve runtime O(n^2)* optimized runtime O(nlogn)* Analytically cumbersome* Fitted for long-range correlations and non-stationary series

Horizontal version* generates outerplanar graphs* naïve runtime O(nlogn)* analytically tractable* Fitted for short-range correlations

The visibility graph of a time series remains invariant under several transformations

Lacasa, Luque, Ballesteros, Luque, Nuño, PNAS 105 (2008)

EXTENSION TO MULTIVARIATE TIME SERIES:

MULTIPLEX NETWORKS

Lacasa, Nicosia, Latora, Scientific Reports 5, 15508 (2015)

Scalable: runtime complexity O(d) in the number of layers d

Relationship between layers

• Edge overlap

• Mutual information

• …

Spatio-temporal dynamics: diffusively coupled chaotic maps

Captures different dynamical regimes and main features

Empirical study -- Visibility multiplex of multivariate financial series

.com bubble

Great recession

Financially stable

period

Comparison with a standard mutual information multivariate analysis

Multiplex visibility graph Multivariate series

Network structure for correlated data

Why visibility, and why applied to fMRI?

• Useful: both informative and descriptive

• Robust to processing

• Computationally easy and efficient

• Amenable to analytical insight

• Versatile, in both multivariate and univariate settings

• Novel, building a bridge between time series and networks

More motivations specific to fMRIPeaks of BOLD signal encode relevant information on coactivation and correlated activity

Tagliazucchi et al. 2012, Wu et al. 2013, Liu and Duyn 2013

Plus, visibility allows looking both at local and global activity

Examples of visibility in BOLD time series

Visibility graphs for BOLD signals

Modular temporal networks

Different modules: different temporal regimes, mainly adjacent points

TR

Dynamic functional connectivity (Hutchison et al. 2013, Leonardi et al. 2013, Hansen et al. 2015, Hindriks et al. 2016, Kudela et al. 2017)

Dynamical functional connectivity in the visibility framework: partition distance and Sorensen index

Application to a large open fMRI dataset

• Parcellation into 278 ROIs (Shen’s 2013 template)• Natural visibility graph for each ROI• Pairwise mutual information across ROIs• Average MI for each intrinsic connectivity

network, defined according to Yeo

Average MI across RSNs

Differences in the limbic system

• MANCOVAN with age, and framewise displacement as covariates• p-value 0.005, Bonferroni-Holm corrected• Shift function to assess differences across quantiles

In the same regions:• lower Regional Homogeneity (ReHo) Zang et al. (2004)• higher coefficient of variation of the BOLD signal Wu and Marinazzo (2016)• lower value of the fractional amplitude of low-frequency fluctuations (fALFF) Zou et al. (2008)

thanks to neurovault.org

Applications to EEGUnivariate Multivariate

Applications to (neuro) images: segmentation

Thanks!

Daniele MarinazzoDepartment of Data Analysis

Ghent University@dan_marinazzo

http://users.ugent.be/~dmarinaz/Daniele.Marinazzo@UGent.be

Code on

https://github.com/danielemarinazzo/Visibility_LA5C_data

http://www.maths.qmul.ac.uk/~lacasa/Software.html