Methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. The recently developed method of Visibility Graphs transforms a time series into a complex network which inherits several properties of the time series in its structure.

1. Visibility Complex Networks for Chaotic Time
Series
Georgi D. Gospodinov
Rachel L. Maitra
Applied Mathematics Department
Wentworth Institute of Technology
June 11, 2014
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

2. Introduction: Visibility Complex Networks
Temporal sequences of measurements or observations (time
series) are the basic elements for investigating natural
phenomena
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

3. Introduction: Visibility Complex Networks
Temporal sequences of measurements or observations (time
series) are the basic elements for investigating natural
phenomena
Time series analysis aims at understanding the dynamics of
stochastic or chaotic processes
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

4. Introduction: Visibility Complex Networks
Temporal sequences of measurements or observations (time
series) are the basic elements for investigating natural
phenomena
Time series analysis aims at understanding the dynamics of
stochastic or chaotic processes
Methods have been proposed to transform a single time series
to a complex network so that the dynamics of the process can
be understood by investigating the topological properties of
the network
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

5. Introduction: Visibility Complex Networks
Temporal sequences of measurements or observations (time
series) are the basic elements for investigating natural
phenomena
Time series analysis aims at understanding the dynamics of
stochastic or chaotic processes
Methods have been proposed to transform a single time series
to a complex network so that the dynamics of the process can
be understood by investigating the topological properties of
the network
The recently developed method of Visibility Graphs transforms
a time series into a complex network which inherits several
properties of the time series in its structure
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

6. Basic Definitions
Definition
The visibility criterion for mapping a time series into a network is
defined as follows. Two arbitrary data (ta, ya) and (tb, yb) in the
time series are visible if any other data (tc, yc) such that
ta < tb < tc fulfills
yc < ya + (yb − ya)
tc − ta
tb − ta
.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

7. Basic Definitions
Definition
The visibility criterion for mapping a time series into a network is
defined as follows. Two arbitrary data (ta, ya) and (tb, yb) in the
time series are visible if any other data (tc, yc) such that
ta < tb < tc fulfills
yc < ya + (yb − ya)
tc − ta
tb − ta
.
Visibility algorithm for a time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

8. Basic Invariant Properties
a) Original time series
with visibility links
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

9. Basic Invariant Properties
a) Original time series
with visibility links
b) Translation of the
data
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

10. Basic Invariant Properties
a) Original time series
with visibility links
b) Translation of the
data
c) Vertical rescaling
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

11. Basic Invariant Properties
a) Original time series
with visibility links
b) Translation of the
data
c) Vertical rescaling
d) Horizontal rescaling
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

12. Basic Invariant Properties
a) Original time series
with visibility links
b) Translation of the
data
c) Vertical rescaling
d) Horizontal rescaling
e) Addition of a linear
trend to the data
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

13. Basic Invariant Properties
a) Original time series
with visibility links
b) Translation of the
data
c) Vertical rescaling
d) Horizontal rescaling
e) Addition of a linear
trend to the data
Note: The visibility
graph remains
invariant in all of the
above cases
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

14. Basic Properties
the associated visibility graph is connected
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

15. Basic Properties
the associated visibility graph is connected
periodic, random, and fractal time series map into motif-like,
exponential, and scale-free visibility graphs, respectively
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

16. Basic Properties
the associated visibility graph is connected
periodic, random, and fractal time series map into motif-like,
exponential, and scale-free visibility graphs, respectively
the visibility algorithm has been used to estimate the Hurst
exponent in fractional Brownian series via the linear
relationship between the Hurst exponent and the the exponent
of the power law degree distribution in the scale-free
associated visibility graph
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

17. Basic Properties
the associated visibility graph is connected
periodic, random, and fractal time series map into motif-like,
exponential, and scale-free visibility graphs, respectively
the visibility algorithm has been used to estimate the Hurst
exponent in fractional Brownian series via the linear
relationship between the Hurst exponent and the the exponent
of the power law degree distribution in the scale-free
associated visibility graph
the visibility algorithm has been applied to analyze time series
in different contexts, from dynamics, atmospheric sciences, to
finance
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

18. Basic Properties
the associated visibility graph is connected
periodic, random, and fractal time series map into motif-like,
exponential, and scale-free visibility graphs, respectively
the visibility algorithm has been used to estimate the Hurst
exponent in fractional Brownian series via the linear
relationship between the Hurst exponent and the the exponent
of the power law degree distribution in the scale-free
associated visibility graph
the visibility algorithm has been applied to analyze time series
in different contexts, from dynamics, atmospheric sciences, to
finance
the visibility algorithm decomposes time series in a
concatenation of graph motifs, and in this sense acts as a
geometric transform
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

19. Degree Distribution of Scale-Free HVG
5 10 15 20 25 k
2
4
6
8
ln@PHkLD
Linear Fit to ln@PHkLD for x-Henon Time Series
of Length Ranging from 215
to 223
Points
223
Points, l=0.333
222
Points, l=0.323
221
Points, l=0.320
220
Points, l=0.304
219
Points, l=0.305
218
Points, l=0.313
217
Points, l=0.285
216
Points, l=0.280
215
Points, l=0.301
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

20. The Horizontal Visibility Algorithm
the general visibility algorithm was introduced above
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

21. The Horizontal Visibility Algorithm
the general visibility algorithm was introduced above
the horizontal visibility algorithm is a special case of the
general visibility algorithm: ya and yb are visible if ya, yb > yc
for all c such that a < c < b
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

22. The Horizontal Visibility Algorithm
the general visibility algorithm was introduced above
the horizontal visibility algorithm is a special case of the
general visibility algorithm: ya and yb are visible if ya, yb > yc
for all c such that a < c < b
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

23. Random Time Series: HVG Degree Distribution
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

24. Random Time Series: HVG Degree Distribution
(left) First 250 values of R(t), where R is a random series of
107 data values extracted from U[0, 1]
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

25. Random Time Series: HVG Degree Distribution
(left) First 250 values of R(t), where R is a random series of
107 data values extracted from U[0, 1]
(right) Degree distribution P(k) of the visibility graph
associated with R(t) (plotted in semilog)
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

26. Random Time Series: HVG Degree Distribution
(left) First 250 values of R(t), where R is a random series of
107 data values extracted from U[0, 1]
(right) Degree distribution P(k) of the visibility graph
associated with R(t) (plotted in semilog)
The tail is clearly exponential, a behavior due to data with
large values (rare events), which are the hubs.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

27. Random Time Series: HVG Degree Distribution
Theorem
Consider a time series that consists of a periodic orbit of period T.
The mean degree of an horizontal visibility graph associated to an
infinite periodic series of period T (with no repeated values within
a period) is
¯k ≡
#edges
#nodes
= 4 1 −
1
2T
.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

28. Random Time Series: HVG Degree Distribution
Theorem
Consider a time series that consists of a periodic orbit of period T.
The mean degree of an horizontal visibility graph associated to an
infinite periodic series of period T (with no repeated values within
a period) is
¯k ≡
#edges
#nodes
= 4 1 −
1
2T
.
Theorem
Given a sequence {xi } generated by a continuous probability density
f (x), the degree distribution of the associated HVG is
P(k) =
1
3
2
3
k−2
, k ≥ 2
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

29. Random Time Series: HVG Properties
Adjacency matrix of
the HVG of 103
random series data
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

30. Random Time Series: HVG Properties
Adjacency matrix of
the HVG of 103
random series data
The adjacency matrix
is predominantly filled
around the main
diagonal
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

31. Random Time Series: HVG Properties
Adjacency matrix of
the HVG of 103
random series data
The adjacency matrix
is predominantly filled
around the main
diagonal
A sparse structure,
reminiscent of the
Small-World model
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

32. Random Time Series: HVG Properties
Adjacency matrix of
the HVG of 103
random series data
The adjacency matrix
is predominantly filled
around the main
diagonal
A sparse structure,
reminiscent of the
Small-World model
Mean path length
scales logarithmically,
implying the HVG to a
random time series is
Small-World
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

33. HVG Degree Distribution of Chaotic Time Series
(solid line) random series
(squares) time series of 106
points extracted from the Logistic map
xn+1 = µxn(1 − xn) in the chaotic region µ = 4
(black triangles) {xn} time series from the H´enon map
(xn+1, yn+1) = (yn + 1 − ax2
n , bxn) with (a = 1.4, b = 0.3)
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

34. HVGs: Conjectured Distinction of Chaotic vs. Stochastic
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

35. HVGs: Conjectured Distinction of Chaotic vs. Stochastic
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

36. HVGs: Counterexamples (Part I)
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

37. HVGs: Counterexamples (Part II)
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

38. HVGs: Counterexamples (Part III)
Figure : λ values for the HVG degree distribution of chaotic time series
(over 300 chaotic systems plotted, with a degree-11 polynomial fit). The
shaded region shows the range of inflection point values depending on
the different linear fit, and the dotted line shows the λ value for
uncorrelated random time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

39. Introduction: Shannon-Fisher Information Plane
The Shannon-Fisher information plane (SF) is a planar
representation in which the horizontal and vertical axes are
functionals of the PDF: the Shannon Entropy and the Fisher
Information Measure, respectively.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

40. Introduction: Shannon-Fisher Information Plane
The Shannon-Fisher information plane (SF) is a planar
representation in which the horizontal and vertical axes are
functionals of the PDF: the Shannon Entropy and the Fisher
Information Measure, respectively.
A way to represent in the same information plane global and
local aspects of the PDFs associated to the studied system
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

41. Introduction: Shannon-Fisher Information Plane
The Shannon-Fisher information plane (SF) is a planar
representation in which the horizontal and vertical axes are
functionals of the PDF: the Shannon Entropy and the Fisher
Information Measure, respectively.
A way to represent in the same information plane global and
local aspects of the PDFs associated to the studied system
The proposed PDFs here are obtained through the horizontal
visibility graph methodology
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

42. Introduction: Shannon-Fisher Information Plane
The Shannon-Fisher information plane (SF) is a planar
representation in which the horizontal and vertical axes are
functionals of the PDF: the Shannon Entropy and the Fisher
Information Measure, respectively.
A way to represent in the same information plane global and
local aspects of the PDFs associated to the studied system
The proposed PDFs here are obtained through the horizontal
visibility graph methodology
Given a continuous probability distribution function (PDF), its
Shannon entropy is a measure of “global” character that it is
not too sensitive to strong changes in the distribution taking
place on small regions of the PDF’s support
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

43. Introduction: Shannon-Fisher Information Plane
The Shannon-Fisher information plane (SF) is a planar
representation in which the horizontal and vertical axes are
functionals of the PDF: the Shannon Entropy and the Fisher
Information Measure, respectively.
A way to represent in the same information plane global and
local aspects of the PDFs associated to the studied system
The proposed PDFs here are obtained through the horizontal
visibility graph methodology
Given a continuous probability distribution function (PDF), its
Shannon entropy is a measure of “global” character that it is
not too sensitive to strong changes in the distribution taking
place on small regions of the PDF’s support
Fisher’s Information Measure constitutes a measure of the
gradient content of the PDF, thus being quite sensitive even
to tiny localized perturbations
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

44. Definitions: Shannon Entropy, Fisher Measure, Normalized
Shannon Entropy
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

45. Definitions: Shannon Entropy, Fisher Measure, Normalized
Shannon Entropy
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

46. Definitions: Shannon Entropy, Fisher Measure, Normalized
Shannon Entropy
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

47. HVG-PDF Setup
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

48. HVG-PDF Examples
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

49. Shannon-Fisher Plane with HVG-PDFs: Chaotic vs.
Stochastic
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

50. Shannon-Fisher Plane with HVG-PDFs: Zoom
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

51. Shannon-Fisher Plane: Chaotic vs. Stochastic
Figure : Shannon-Fisher values for the HVG degree distribution of
chaotic and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

52. Shannon-Fisher Plane with HVG-PDFs: Zoom
Figure : Shannon-Fisher values for the HVG degree distribution of
chaotic and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

53. Shannon-Lambda Plane: Chaotic vs. Stochastic
Figure : Shannon − λ values for the HVG degree distribution of chaotic
and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

54. Multi-Dimensional Visibility Graphs
Component Visibility Graphs
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

55. Multi-Dimensional Visibility Graphs
Magnitude Visibility Graphs
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

56. Magnitude Visibility Graphs Criterion
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

57. Shannon-Fisher Plane: Chaotic vs. Stochastic
Figure : Shannon-Fisher values for the HVG degree distribution of
chaotic and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

58. Shannon-Lambda Plane: Chaotic vs. Stochastic
Figure : Shannon − λ values for the HVG degree distribution of chaotic
and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

59. Shannon-Lambda Plane: Chaotic vs. Stochastic
Figure : Shannon − λ values for the HVG degree distribution of chaotic
and stochastic time series.
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

60. Current and Future Work
Average degree for multi-dimensional time series
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

61. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

62. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Filtration of VGs of multidimensional time series
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

63. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Filtration of VGs of multidimensional time series
Multi-dimensional dynamical VGs
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

64. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Filtration of VGs of multidimensional time series
Multi-dimensional dynamical VGs
Application of dynamical VGs to Shannon-Fisher analysis
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

65. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Filtration of VGs of multidimensional time series
Multi-dimensional dynamical VGs
Application of dynamical VGs to Shannon-Fisher analysis
Network cluster visibility
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series

66. Current and Future Work
Average degree for multi-dimensional time series
Degree distribution for uncorrelated random multi-dimensional
time series
Filtration of VGs of multidimensional time series
Multi-dimensional dynamical VGs
Application of dynamical VGs to Shannon-Fisher analysis
Network cluster visibility
Data analysis, Manifold learning, Deep learning of hierarchical
data through VGs
Georgi D. Gospodinov Rachel L. Maitra Visibility Complex Networks for Chaotic Time Series