From Time Series to

Complex Networks

B. Luque, L. LacasaETSI Aeronáuticos UPM,

J. Luque,

ETSI Telecomunicaciones UPC, F. Ballesteros,

Observatorio Astronómico de Valencia UV and

J. C. Nuño, ETSI Montes, UPM, Spain

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”.

Natural visibility:How can we extract a graph from a temporal series?

Visibilityalgorithm

Two data (nodes) are connected in the graph if there’s a possible “visual contact” between themselves (two data heights are in “visual contact” if there’s a straight line that connects them, provided that this line doesn’t cut any other’s 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.

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.

tn

xx

ti

xx ni

00

01 xn

ix

n

ix ni

Graph properties (1) Connected.

(2) Undirected.

(3) Sample time independent.

(4) Amplitude size scale independent.

(5) Horizontal and vertical shift independent.

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)

Does the graph inherit any structure?

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).

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:

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).

a

tBatB 2/1)(

221

110

n;)n-a(n- a)n-a(n- aa(n)

) a() a(

Q series (by Douglas Hofstadter, ‘Gödel, Escher, Bach’, New York: Vintage Books, pp. 137-138, 1980. )

Conway series (by J. Conway, ‘Some Crazy Sequences’, Lecture at AT&T Bell Labs, July 15, 1988).

211

121

n;)n-a(n- a )a(n- aa(n)

) a() a(

Stern series (M. A. Stern, Ueber eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220).

21122

1100

n)a(n a(n)) na( a(n);n) a(

) a(, ) a(

Thue-Morse series: (M. R. Schroeder Fractals, Chaos, and

Power Laws. New York: W. H. Freeman, 1991).