Diapositiva 1

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From Time Series to Complex Networks B. Luque,
L. Lacasa ETSI Aeronáuticos UPM, J. Luque, ETSI
Telecomunicaciones UPC, F. Ballesteros,
Observatorio Astronómico de Valencia UV and J.
C. Nuño, ETSI Montes, UPM, Spain
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Natural visibility How can we extract a graph
from a temporal series?

In the next figure we plot a time series of 20
data. Below it you can see the associated graph.
The number of nodes in the graph is N 20, one
for each data and in the same order.
Two data (nodes) are connected in the graph if
theres a possible visual contact between
themselves (two data heights are in visual
contact if theres a straight line that connects
them, provided that this line doesnt cut any
others data height). For example data (node)
number 15 sees only (is only connected with) data
14 and 16. Note that visual contact or visibility
in the series is possible in the long distance as
happens between data 1 and 16.
Visibility algorithm
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With this method (Natural visibility) two nodes
0 and n will be connected if (see next figure)
for i 1, 2, ... ,n-1 i.e. where xi is
the value (height) of the data i and ?t is the
sample time.
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Graph properties (1) Connected. (2)
Undirected. (3) Sample time
independent. (4) Amplitude size scale
independent. (5) Horizontal and vertical
shift independent.
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Does the graph inherit any structure?

In the pictures below we represent two periodic
series. Note that the associated graph by the
natural visibility method are obviously regular.
Periodic series (period 2)
Periodic series (period 4)
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In this sense, the associated graph inherits the
series structure. We can consider the following
question Is it possible to characterize a series
by its visibility graph? The next example (left)
is part of a random series of 1 million data from
a uniform distribution U(x) with x ? 0, 1. In
the right side we plot the connectivity
distribution of the associated graph this one is
clearly an exponential distribution (plotted in
semi-log).
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the question
If the graph inherits structure from time
series, how is the distribution of connectivity
associated to a self-similar series? Would it
be a scale-free distribution? The following set
of examples may answer to this question
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Brownian motion B(t) is a classical example of
self-similar time series (self-affine properly)
because The next picture (left) is part of a
Brownian motion series of 1 million data. In the
right side we plot the connectivity distribution
of the associated graph (that seems a scale-free
distribution).
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Q series (by Douglas Hofstadter, Gödel, Escher,
Bach, New York Vintage Books, pp. 137-138,
1980. )
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Conway series (by J. Conway, Some Crazy
Sequences, Lecture at ATT Bell Labs, July 15,
1988).
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Stern series (M. A. Stern, Ueber eine
zahlentheoretische Funktion, J. Reine Angew.
Math., 55 (1858), 193-220).
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Thue-Morse series (M. R. Schroeder Fractals,
Chaos, and Power Laws. New York W. H. Freeman,
1991).
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