Table Of ContentAmbiguities in recurrence-based complex network representations of time series
Reik V. Donner,1,2,3,∗ Yong Zou,2,† Jonathan F. Donges,2,4 Norbert Marwan,2 and Ju¨rgen Kurths2,4
1Max Planck Institute for Physics of Complex Systems, No¨thnitzer Str. 38, 01187 Dresden, Germany
2Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
3Institute for Transport and Economics, Dresden University of Technology, Wu¨rzburger Str. 35, 01187 Dresden, Germany
4Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
(Dated: January 26, 2010)
Recently, different approaches have been proposed for studying basic properties of time series
from a complex network perspective. In this work, the corresponding potentials and limitations of
networksbased on recurrences in phasespace are investigated in some detail. We discuss themain
requirements that permit a feasible system-theoretic interpretation of network topology in terms
0 of dynamically invariant phase space properties. Possible artifacts induced by disregarding these
1 requirements are pointed out and systematically studied. Finally, a rigorous interpretation of the
0 clustering coefficient and thebetweenness centrality in terms of invariant objects is proposed.
2
PACSnumbers: 05.45.Tp,89.75.Hc,05.45.Ac
n
a
J
Duringthelastdecade,increasinginteresthasarisenin depend – among others – on the length N of the con-
6
structuralanddynamicalpropertiesofcomplexnetworks sidered time series (i.e., the network size), the probabil-
2
[1]. Particular efforts have been spent on reconstruct- ity distribution of the data, embedding, sampling, etc.
] ing network topologies from experimental data, e.g., in In the following, we present a critical discussion of the
D ecology [2], social systems [3], neuroscience [4], or at- basic requirements for the application of recurrence net-
C mospheric dynamics [5]. The latter example yields a worksandshowthattheirinsufficientapplicationleadsto
. complex network representation of a continuous system, pitfalls in the system-theoreticinterpretationof complex
n
whichsuggeststhatapplyinga similarspatialdiscretiza- network measures.
i
l tion to the phase space of dynamical systems and using Threshold selection. The crucial algorithmic param-
n
complex network methods as a novel tool for time series eter of recurrence-based time series analysis is ε. Sev-
[
analysis could be feasible, too [6, 7]. For this purpose, eralinvariantsofadynamicalsystem(e.g.,the2nd-order
1 differentmethodshavebeenproposed(foracomparative R´enyientropy K ) can be estimated by taking its recur-
2
v
review, see [8]) and successfully applied to real-world as rence propertiesfor ε→0 [10], whichsuggeststhat for a
0
well as model systems. feasibleanalysisofrecurrencenetworks,alowεisprefer-
2
Many existing methods for transforming time series ableaswell. Thisissupportedbytheanalogytocomplex
6
4 into complex network representations have in common networksbasedonspatially extended systems, where at-
. that they define the connectivity of a complex network tentionisusuallyrestrictedtothestrongestlinksbetween
1
– similar to the spatio-temporal case – by the mutual individual vertices (i.e., observations from different spa-
0
0 proximity of different parts (e.g., individual states, state tial coordinates) for retrieving meaningful information
1 vectors, or cycles) of a single trajectory. In this work, about relevant aspects of the systems’ dynamics [4, 5].
: we particularly consider recurrence networks, which are In contrast, a high edge density
v
based on the concept of recurrences in phase space [9,
i 2E(ε)
X 10] and provide a generic way for analyzing phase space ρ(ε)= (2)
N(N −1)
r properties in terms of network topology [7, 8, 11, 12].
a
Here, the basic idea is to interpretthe recurrencematrix
(with E(ε) being the total number of edges for a chosen
R (ε)=Θ(ε−kx −x k) (1) ε) does not yield feasible information about the actually
i,j i j relevant structures, because these are hidden in a large
set of mainly less important edges.
associated with a dynamical system’s trajectory, i.e., a
As a consequence, only those states should be con-
binary matrix that encodes whether or not the phase
space distance between two observed “states” x and x nected in a recurrence network that are closely neigh-
i j
bored in phase space, leading to rather sparse networks.
issmallerthanacertainrecurrencethresholdε,asthead-
Following a corresponding rule of thumb recently con-
jacency matrix of an undirected complex network. Since
firmedforrecurrencequantificationanalysis[13],wesug-
a singlefinite-time trajectorymayhowevernotnecessar-
gest choosing ε as corresponding to an edge density
ily represent the typical long-term behavior of the un-
ρ . 0.05 [7, 8], which yields neighborhoods covering
derlying system, the resulting network properties may
appropriately small regions of phase space. Note that
since many topological features of recurrence networks
are closely related to the local phase space properties of
∗Electronicaddress: [email protected] the underlying attractor [8], the corresponding informa-
†Electronicaddress: [email protected] tion is best preserved for such low ε unless the presence
2
Lorenz (x-y-z) Lorenz (x) Rössler(x-y-z) Rössler(x)
A0.03 B C0.08 D0.05
0.04 τ1 0.04 τ1 Ma
τ 0.06 τ n
d/dρε00..0012 d/dρε0.02 τ23 d/dρε00..0024 d/dρε000...000123 τ23 hattan distan
c
e
0 0 0 0
0 20 40 60 80 100 0 20 40 60 80 100 0 10 20 30 40 50 0 10 20 30 40 50
ε ε ε ε
E F0.08 G 0.1 H0.08
0.04 τ1 0.08 τ1 Eu
0.03 0.06 τ2 0.06 τ2 clid
d/dρε0.02 d/dρε0.04 τ3 d/dρε00..0046 d/dρε0.04 τ3 ean d
is
0.01 0.02 0.02 0.02 tan
c
e
0 0 0 0
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 0 10 20 30
ε ε ε ε
I 0.06 J0.08 K 0.1 L 0.1
τ τ
0.06 τ1 0.08 0.08 τ1 Max
0.04 2 2 im
d/dρε d/dρε0.04 τ3 d/dρε00..0046 d/dρε00..0046 τ3 um d
0.02 is
0.02 0.02 0.02 tan
c
e
0 0 0 0
0 10 20 30 40 50 0 10 20 30 40 50 0 5 10 15 20 25 0 5 10 15 20 25
ε ε ε ε
FIG.1: (Coloronline)Effectsofdifferentmetricsandembeddingsontheρ(ε)relationship,expressedintermsofthecorrespond-
ing first derivative. (A,B,C,D): Manhattan distance; (E,F,G,H): Euclidean distance; (I,J,K,L): maximum distance. (A,E,I):
Lorenzsystem(cid:0)x˙ =10(y−x), y˙ =x(28−z), z˙ =xy− 8z(cid:1)withoriginalcomponentsatthreedifferentrandomlychoseninitial
3
conditions. (B,F,J):SameLorenzsystemembeddedfromthexcomponentwithembeddingdelaysτ1 =5,τ2=15,andτ3 =20.
(C,G,K):sameas(A,E,I)for theRo¨ssler system(x˙ =−y−z, y˙ =x+0.2y, z˙ =z(x−5.7)). (D,H,L):sameas(B,F,J) forthe
Ro¨sslersystemwithτ1 =10,τ2 =15andτ3=20. Circlesindicatetherespectivemaxima. Inallcases,timeseriesofN =1,000
points with a sampling time of ∆t = 0.05 have been used, obtained with a 4th-order Runge-Kutta integrator with fixed step
width h=0.01. The values of τ2 are guided by thefirst zeros of the corresponding auto-correlation functions.
of noise requires higher ε [13]. ric used for defining distances in phase space (Fig. 1).
Recently, a heuristic criterion has been proposed by Moreover, experimental time series often contain only a
Gao and Jin, which selects ε as the (supposedly unique) singlescalarvariable,sothatembeddingmightbeneces-
turning point ε in the ρ(ε) relationship of certain dy- sary. Since the detailed shape of the attractor in phase
crit
namical systems [12], formally reading space is affected by the embedding parameters, chang-
ingthe embedding delayhasasubstantialeffectonε ,
dρ(cid:12)(cid:12) d2ρ(cid:12)(cid:12) which is particularly visible in the Ro¨ssler system (crsiete
(cid:12) =max!, (cid:12) =0. (3)
dε(cid:12) dε2(cid:12) Fig. 1 (D,H,L)). An improper choice of embedding pa-
ε=εcrit ε=εcrit
rameterswould further increase the variance of ε and
crit
Incontrasttoouraboveconsiderations,fordifferentreal- will generally not yield meaningful results. In a similar
izationsoftheLorenzsystem,thisturningpointcriterion way, depending on the choice of the other parameters
yields link densities of ρcrit = ρ(ε = εcrit) ∼ 0.15...0.3 the sampling time of the time series may also influence
[12], implying that considerably large regions of the at- the recurrence properties [14] (and, hence, ε ), since
crit
tractorare coveredby the correspondingneighborhoods. temporal coarse-graining can cause a loss of detections
In such cases, it is however not possible to attribute of recurrences.
certain network features to specific small-scale attractor
properties in phase space. More generally, ε should be (ii) The ε-selection should be as independent as pos-
chosen in such a way that small variations in ε do not sible of the particular realization of the studied system,
induce large variations in the results of the analysis. In especiallyfromtheinitialconditionsandthelengthN of
contrast,the turning pointcriterion(3) explicitly selects the time series. The turning point ε after conditions
crit
ε such that small perturbations in its value will result in (3)ishowevernot independentofthespecificinitialcon-
a maximum variation of the results. Moreover, besides ditions (Fig. 1 (A,E,I) and (C,G,K)): while its average
our general considerations supporting low ε, application value does not change much with changing N, there is
of the turning point criterion leads to serious pitfalls: a large variance among the individual trajectories that
(i) ε and, hence, ρ depend on the specific met- convergesonly slowlywith increasingN (Fig.2). Hence,
crit crit
3
A 45 B 30 In particular, L ∼ 1/ε (for given N), since spatial dis-
0.7 1
tances are approximately conserved in recurrence net-
35 ρcrit0.5 25 ρcrit0.6 works, whereas the specific ε-dependence of C varies be-
εcrit25 0.3 N εcrit1250 0.2 N tween different systems.
Inadditiontotheaforementionedglobalnetworkchar-
15
10
acteristics, specific vertex properties characterize the lo-
5 500 1000 1500 2000 2500 5 500 1000 1500 2000 2500 cal attractor geometry in phase space in some more de-
N N
tail, where the spatial resolution is determined by ε. In
particular,thelocal clusteringcoefficientC ,whichquan-
FIG. 2: Mean values (squares) and range (shaded areas) of v
tifies the relative amountoftriangles centeredata given
turning points εcrit in 200 independent realizations of the
vertexv, givesimportantinformationaboutthe geomet-
Lorenz(A)andtheRo¨ssler system(B) independenceon the
ric structure of the attractor within the ε-neighborhood
network size N (Euclidean distance, ∆t = 0.05). The insets
showthecorrespondinglinkdensitiesρcrit forthesamerange ofv inphasespace. Specifically,iftheneighboringstates
of N. form a lower-dimensionalsubset than the attractor,it is
morelikelythatclosedtrianglesemergethanforaneigh-
borhood being more uniformly filled with states [16].
A 0.2 B 4
Hence,highvaluesofC indicatelower-dimensionalstruc-
v
0.15 3 tures that may correspond to laminar regimes [7] or dy-
ρε/d 0.1 ρε/d2 namically invariant objects like unstable periodic orbits
d d (UPOs) [8]. The relationship with UPOs follows from
0.05 1 the fact that trajectories tend to stay in the vicinity of
suchorbits fora finite time [17], which leads to a certain
0 0
0 2 4 6 8 10 0 0.2 0.4 0.6 0.8
ε ε amountofstatesbeingaccumulatedalongtheUPOwith
adistinctspatialgeometrythatdiffersfromthatinother
FIG. 3: Examples for multiple turning points of dρ/dε: (A) parts of a chaotic attractor. However,since there are in-
quasiperiodic trajectory of a continuous system (torus) and finitelymanyUPOsembeddedinchaoticattractors,such
(B)aweaklychaoticorbitofthestandardmap(xn+1=xn+ objects(evenofaloworder)canhardlybedetectedusing
5sin(yn) mod 1, yn+1 =yn+xn+1 mod 1, see [15]). large ε (where the resulting neighborhoods cover differ-
ent UPOs) and short time series as recently suggested
[12]. In contrast,they may be well identified using low ε
for the same system and the same network size, already and long time series [8].
slightly different initial conditions may yield strong dif- Another intensively studied vertex property is be-
ferences in εcrit and ρcrit (Fig. 2, inset) and, hence, the tweenness centrality bv, which quantifies the relative
topological features of the resulting networks. number of shortest paths in a network that include a
(iii) One has to emphasize that the turning point cri- given vertex v [3]. In a recurrence network, vertices
terionis not generallyapplicable, since there are various with high b correspondto regions with low phase space
v
typical examples for both discrete and continuous dy- density that are located between higher density regions.
namical systems that are characterized by several max- Hence, b yields information about the local fragmen-
v
ima of dρ(ε)/dε (Fig. 3). tation of an attractor. In particular, since phase space
The above considerations are mainly of concern when regions close to the outer boundaries of the correspond-
studying properties of (known) dynamical systems. In ing attractors do not contribute to many shortest paths,
applications to real-world time series with typically a the vertices located in these regions are characterizedby
small number of data or even non-stationarities, it is low bv, which is (at least for the Lorenz oscillator) even
stillpossibletoderivemeaningfulqualitative resultsfrom enhanced by a lower state density. For the sharp inner
small time series networks. However, for a detailed boundary of the Ro¨ssler oscillator, one may observe the
system-theoretic interpretation the use of smaller recur- opposite behavior. For phase space regions close to low-
rence thresholds is recommended [7]. periodUPOs,onealsofinds lowervaluesofbv due tothe
Topology ofrecurrencenetworks. Thetopologicalfea- accumulation of states along these structures (many al-
tures of recurrence networks are closely related to in- ternativepaths). As thedistributionofbv (Fig.4(A,B))
variant properties of the observed dynamical system suggests, these features are robust for low ε, but may
[7, 8, 12]. However, a system-theoretic interpretation significantly change if ε gets too large (i.e., ρ=0.2).
of the resulting network characteristics is feasible only Weconcludethatinarecurrencenetwork,bothC and
v
based on a careful choice of ε, avoiding the pitfalls out- b aresensitivetothepresenceofUPOs,butresolvecom-
v
lined above. For example, many paradigmatic network plementary aspects (see Fig. 4). For the Ro¨ssler system,
models as well as real-worldsystems have been reported we find two distinct maxima in the betweenness distri-
to possess small-world properties (i.e., a high clustering bution, which are related to the inner and outer parts
coefficient C and low average path length L). However, of the attractor, respectively. In particular, the abun-
it can be shown that C and L are both functions of ε. dance of low values is promoted by a high state den-
4
A B namically relevant aspects of phase space properties of
10−4 10−4 the attractor, given that (i) the considered time series
is long enough to be representative for the system’s dy-
+1)10−6 ρρ1 +1)10−6 ρρ1 namics and (ii) the threshold distance ε in phase space
pb(v10−8 ρρ23 pb(v10−8 ρρ23 ftohredsecfianleinsgoaf irnetceurrersetn.ceInispcahrotsiecnulsamr,aullseinngoutghhetnoertewsoorlvke-
4 4
10−1100 0 102 104 106 10−1100 2 103 104 105 106 theoretic measures discussed here, the turning point cri-
bv+1 bv+1 terionforthresholdselection[12]oftendoesnot allowfea-
C 1 D 1 sible conclusions about dynamically relevant structures
in phase space. In contrast, for sufficiently low recur-
0.8 0.8
rencethresholds(wesuggestρ.0.05asaruleofthumb),
0.6 0.6
v v small-scale structure may be resolved appropriately by
C0.4 C0.4
complex network measures, which allow identification
0.2 0.2 of invariant objects such as UPOs by purely geometric
0 0 means. We emphasize that although our presented
100 102 104 106 102 103 104 105 106
bv+1 bv+1 considerations have been restricted to paradigmatic ex-
ample systems, recurrence networks and related meth-
FIG. 4: (Color online) Probability distribution function of ods have already been successfully applied to real-world
betweenness centrality bv (in logarithmic scale) for different data, e.g., a paleoclimate record [7] or seismic activ-
edge densities (ρ1 = 0.005, ρ2 = 0.01, ρ3 = 0.015, ρ4 = 0.2) ity [19]. Since these examples aretypically characterized
for the Lorenz (A, N = 20,000) and Ro¨ssler system (B,
by non-stationaritiesandnon-deterministic components,
N = 10,000), and corresponding relationships between lo-
we conclude that recurrence networks are promising for
calclusteringcoefficientCv andbetweennesscentralitybv (C:
futureappliedresearchonvariousinterdisciplinaryprob-
Lorenz,D:Ro¨ssler,ρ=0.01) obtainedfromtheoriginaldata
lems (e.g., [21]).
using theEuclidean distance (∆t=0.05).
Asafinalremark,wenotethattheproblemofparame-
terselectionarisesformostothernetwork-basedmethods
sity at the outer boundary of the attractor near the x- oftimeseriesanalysis(see[8]foradetailedcomparison).
y plane, which coincides with a period-3 UPO [18]. In Importantexamplesincludecyclenetworks[6]withacor-
contrast, for the Lorenz system there is no second max- relation threshold, and k-nearest neighbor networks [11]
imum of p(b ), since the outer parts of the attractor are with a fixed number k of neighbors as free parameters,
v
more diffuse and characterized by a considerably lower respectively. A general framework for parameter selec-
phase space density than in the Ro¨ssler attractor. In tion in the context considered here would consequently
both cases, vertices with a high clustering coefficient C bedesirable. Othermethodsareparameter-free,butmay
v
are characterized by a broad continuum of betweenness sufferfromconceptuallimitationsandstrongintrinsicas-
values, which suggests that b is no universal indicator sumptions. Forexample,thecurrentlyavailablevisibility
v
for the presence of UPOs, whereas C allows an approx- graph concepts [20] are restricted to univariate time se-
v
imate detection of at least low-periodic UPOs in phase ries.
space. Acknowledgments. This work has been financially
In summary, transforming time series into complex supported by the German Research Foundation (SFB
networks yields complementary measures for character- 555, project C4 and DFG project no. He 2789/8-
izing phase space properties of dynamical systems. This 2), the Max Planck Society, the Leibniz Society
workhasprovidedempiricalargumentsthattherecently (project ECONS), and the Potsdam Research Cluster
suggested approach based on the recurrence properties PROGRESS (BMBF). All complex networks measures
in phase space allows a detailed characterization of dy- have been calculated using the igraph package [22].
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