Scale-free networks emerging from multifractal time series

Scale-free networks emerging from multifractal time series

Marcello A. Budroni Nonlinear Physical Chemistry Unit, Faculté des Sciences, Université libre de Bruxelles (ULB), CP 231 - Campus Plaine, 1050 Brussels, Belgium.    Andrea Baronchelli Department of Mathematics - City, University of London - Northampton Square, London EC1V 0HB, UK    Romualdo Pastor-Satorras Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain
July 6, 2019
Abstract

Methods connecting dynamical systems and graph theory have attracted increasing interest in the past few years, with applications ranging from a detailed comparison of different kinds of dynamics to the characterisation of empirical data. Here we investigate the effects of the (multi)fractal properties of a time signal, common in sequences arising from chaotic or strange attractors, on the topology of a suitably projected network. Relying on the box counting formalism, we map boxes into the nodes of a network and establish analytic expressions connecting the natural measure of a box with its degree in the graph representation. We single out the conditions yielding to the emergence of a scale-free topology, and validate our findings with extensive numerical simulations.

pacs:
89.75.Hc, 05.45.Tp, 02.50.Ey, 05.45.Ac

I Introduction

Network science Newman (2010) has emerged in the last decades as a transverse interpretative framework for understanding the structure and function of a wide range of complex systems, as well as the dynamic phenomena that takes place on them, ranging from financial crises and traffic congestion to epidemic and social influence spreading Barrat et al. (2008); Dorogovtsev et al. (2008); Pastor-Satorras et al. (2015).

In the realm of dynamical systems Guckenheimer and Holmes (2002), network techniques have been applied to the analysis of nonlinear time series, with a particular focus on characterizing chaotic dynamics Ott (1993). The main idea of this methodology is to project the information of a time series into the topology of a network. The key element of this approach resides in the identification of nodes and links in the network from the time series information. Several alternatives have been proposed in this context Donner et al. (2011). Thus, Zhang and Small Li et al. (2012) consider cycles of a pseudo-periodic time series as the nodes of a network, which are connected by links depending on the similarity between cycles. Lacasa at al. Lacasa et al. (2008); Lacasa and Toral (2010), on the other hand, build on the concept of visibility graphs, where nodes correspond to series data points and two nodes are connected if a straight line can be established between them without intersection with any intermediate data height.

Another general approach is based on encoding topological information from the reconstructed phase space of a time series into a proximity network. In these networks, nodes represent segments of time series or vectors in the related reconstructed phase space, and links depend on a specific criterion determining adjacency in phase space. Cycle networks Zhang and Small (2006), correlation networks Yang and Yang (2008); Gao and Jin (2009a), and recurrence networks Marwan et al. (2009); Gao and Jin (2009b); Donner et al. (2010); Xiang et al. (2012) are typical examples of proximity networks. Finally, the related class of transition networks Nicolis et al. (2005); Shirazi et al. (2009); Campanharo et al. (2011); Sun et al. (2014) encompasses different models for mapping time series into networks, in which the values on the time signal are mapped into a finite number of states (regions in the phase space of the series), representing the nodes, which are connected if the signal transits from one state to another. This transformation procedure preserves the temporal information of the dynamics in the network, is stable to noise affecting real time series and can account for the fundamental structure of the related attractors Sun et al. (2014).

Using this kind of network projections, several aspects of dynamical systems have been cast in terms of topological network properties. Here we will focus in particular on the effects that the fractal and multifractal properties of a time signal have on the topology of representative classes of projected network. Many time signal, particularly those arising from a chaotic or strange attractor Ott (1993), have a fractal structure, characterized by a statistically self-similar pattern in phase space. Moreover, they can also show multifractal properties, described by a strongly heterogeneous probability of visiting different neighborhoods in the phase space Halsey et al. (1986); Falconer (2003); Feder (1989). Using as a simple example a transition network representation Shirazi et al. (2009); Budroni et al. (2010), framed within a box counting formalism Falconer (2003) in which each box corresponds to a vertex, we find analytic expressions tying the visitation probability of a box with its associated degree. From these relations we obtain the conditions under which a projected network grows with a scale-free topology Barabási and Albert (1999), characterized by a degree distribution of the form . Our results highlight the correspondence between an attractor’s structure and the topology of the associated projected network, and relate the possible origin of a heterogeneous scale-free topology with the heterogeneous and hierarchical visitation probability characterizing multifractal attractors.

The present paper is organized as follows: In Sec. II we briefly summarize the multifractal formalism for chaotic time series and attractors. In Sec. III we present a transition network mapping for general time series, based on the box-counting algorithm Shirazi et al. (2009). In Sec. IV we relate the topological properties of the associated networks with the multifractal properties of the original time series. This relation is directly mediated by the natural measure of the series, defined as the probability that the time sequence visits a given box in a partition of the substrate of the series. Numerical checks of our theoretical predictions are detailed in Sec. V. Finally, we present our conclusions in Sec. VI.

Ii Multifractal time signals

For time signals arising from strange (chaotic) attractors, it is common that different regions of the phase space are differently visited, and chaotic orbits spend most of their time in a small region of the support underneath the chaotic attractor itself. This heterogeneity is at the basis of the so-called multifractal structure of the strange attractor, which can be mathematically captured by the formalism presented below Halsey et al. (1986); Feder (1989).

Let us consider a real time signal, or a -dimensional chaotic attractor, given by the normalized sequence of points , where the index represents either time or the order in the sequence of points that generate the attractor (e.g. the index of the iteration in an iterated map), while is the number of points in the signal. We consider a partition of the set in boxes of length . Box are labeled by the indexes , with . Let us associate to each point in the sequence an integer index

(1)

in the range , where , and is the floor function. In this sense, the signal or attractor can be interpreted as visiting the -th box in the partition at time . In heterogeneous fractals, not all the boxes will be equally visited. In general, during the steps of the signal, the -th box will be visited a number of times , and the total number of boxes visited at least once will be . This quantity coincides with the number of boxes of length needed to cover the fractal set, and thus we can define the box or capacity dimension of the attractor Falconer (2003), , by the relation

(2)

Let us define the probability , termed the natural measure, as the probability that the chaotic map visits the -th box of the available during an infinitely long orbit. For an homogeneous structure in dimensions, , while in the case of a uniform fractal of dimension , . In more complex situations, however, the attractor exhibits a non-uniform fractal distribution, and we assume a general form for the natural measure , where the exponent , taking values in the interval , measures the strength of the local singularity of the measure at box . In general, there will be many boxes with the same value of , such that their number scales as . The function , called the multifractal spectrum of the measure, defines the fractal dimension of the set of boxes with the given value , and is in general a convex function with a single maximum. An equivalent, and numerically simpler, description can be obtained from the generalized dimensions , defined as Halsey et al. (1986)

(3)

which, for , fulfill . For a uniform measure . For multifractal measures, is a decreasing function of which is related with the multifractal spectrum () by means of a Legendre transformation Halsey et al. (1986), defining a parametric exponent that fulfills the equations

(4)
(5)

Numerically, the generalized dimensions can be estimated from Eq. (3), by noticing that, for finite ,

(6)

allowing to be determined from a linear regression of for decreasing values of (increasing ) in a log-log plot.

Iii Nework mapping

In order to construct a transition network representation, we follow the approach in Refs. Shirazi et al. (2009); Budroni et al. (2010), and associate a (virtual) vertex to each box in the partition of the chaotic attractor in phase space. Actual vertices in the network are given by the set of boxes that have been visited at least once by the signal, with a size . Edges in the network are established by associating an undirected connection between vertices and whenever the signal jumps between boxes and in two consecutive time steps, i.e. and .

The resulting projected networks, which are characterized by the coarse-grained scale , are connected by construction, and preserve temporal information of the generator of the signal. A completely random, stochastic signal will lead to a fully connected network; on the other hand, for a limit cycle or periodic attractor, the projected network will be ring, with a number of nodes equal to the period of the cycle. The former case corresponds to , and the latter to .

Iv Relating topology with multifractality

In the case of a multifractal time series, the topology of the associated transition network described above can be related to the generalized dimensions . We will consider in particular the degree distribution of a projected network with a coarse-graining level , defined as the probability that a randomly chosen node has degree , i.e., it is connected to other nodes. To make explicit this relation, we observe that every node (box) , will be characterized by a number of visits and a degree . In the limit , we assume that the relative number of visits, i.e. the natural measure , and the degree of a node are stable quantities. The corresponding degree distribution will thus depend only on the phase space discretization . On average, the two quantities and will be related, since obviously the more times a box is visited, the larger is expected to be the degree of the associated node. We can thus assume a functional relation between these two averaged quantities, valid for sufficiently large and , of the form

(7)

where is an increasing function of . The function depends in general on . Indeed, from the normalization of the natural measure, Eq. (7) implies that

where we have defined . From here, we have

(8)

We make the assumption, to be validated numerically later on, that the dependence of the function resides exclusively in a multiplicative prefactor, i.e. . From Eq. (8), we have that . We make the additional assumption that the average of , , is a constant, independent of . From here, we obtain the relation

(9)

This relation implies that we can express the multifractal properties of the attractor in terms of topological properties of the network. In fact, from Eq. (9), we have

(10)

where we have used . From Eq. (6) we have also . Combining this with Eq. (10), we obtain the relation linking network and multifractal properties, namely

(11)

For a homogeneous fractal set, , and On the other hand, for a multifractal strange attractor, since for , we have that the moments , with , diverge as the number of boxes in the partition increases, i.e. as the network size grows. This observation allows to extract conclusions on the functional form of the degree distribution, which will depend on the particular growth law . We will consider analytically tractable forms in the following Section. To avoid complications in the development, we will further assume in our analysis that the degree distribution of the projected network is stable, meaning that the effect of consists essentially in imposing an upper degree cut-off to a functional form independent of .

iv.1 Exponential growth

Let us consider first the case of an exponential (faster than algebraic) growth of the number of visits in a box with the associated node degree, i.e.

(12)

where . The fact that is constant and diverges for is compatible with a degree distribution that shows, at large values of , the asymptotic behavior , with , that is, an exponentially bounded degree distribution. Assuming a stable degree distribution, for a non-zero the divergence of the exponential moments will be reflected in a dependence on the network size , modulated by the largest degree in the network, or degree cut-off Boguñá et al. (2004). To estimate this value, we observe that, from Eq. (9), . The largest value of will correspond to the minimum of , . Therefore, we have

(13)

This expression allows to relate the network parameters and with the multifractal exponents and by noticing that, in a network of finite size , the maximum degree is given by the condition Boguñá et al. (2004). With an exponential degree distribution we thus have, in the continuous degree approximation, , from where we obtain

(14)

Combining Eqs. (13) and (14), we obtain the relation

(15)

The properties of the network can also be used to extract information on the full set of generalized dimension by building on relation Eq. (11). Indeed we can write the diverging moments in a finite network as

which, taking into account Eq. (13), yields

(17)

Combining Eq. (17) and Eq. (11) leads to the asymptotic expression, valid for large

(18)

that recovers the result known for deterministic multifractal measures Feder (1989).

iv.2 Algebraic growth

In the case of an algebraic growth of the number of visits with the associated degree, we have

(19)

with . From Eq. (11), we have that is finite, and , with , diverge in the limit of infinite network size. The fact that all higher degree moments diverge indicate that the degree distribution of the network has long tails, which in the simplest case are compatible with a power law distribution of the form , where we impose to ensure a finite value of .

Performing again a finite-size analysis, from Eq. (9) we obtain a maximum degree

(20)

On the other hand, the network relation leads now, with a power-law degree distribution in the continuous degree approximation, to Boguñá et al. (2004)

(21)

Combining Eqs. (20) and (21), we obtain the relation between network properties and multifractal exponents

(22)

In the case of an algebraic function, Eqs. (20) and (21) can be used to directly estimate and . This approach is more difficult in the case of an exponential , due to the much smaller range of variation of the logarithm of , see Eqs. (13) and (14).

Finally, writing

(23)

and comparing with Eq. (11), we obtain again the simple asymptotic expression for the generalized dimensions given by Eq. (18).

V Numerical experiments

In order to check the validity of the predictions made in Section IV, we have considered different multifractal time signals, generated by means of iterative maps. In particular we have studied three paradigmatic examples of 1- and 2-dimensional chaotic attractors, namely the logistic, the Duffing and the Henon map. The well-known logistic recurrence May (1976); Ott (1993) in dimension ,

(24)

maps the interval into itself when the control parameter ranges between 0 and 4. This systems undergoes a period-doubling bifurcation transition to chaos, which sets-in at . Multifractal chaotic regimes interspersed with periodic windows then occur in the parameter interval . Here we fix . The two-dimensional Duffing map

(25)
(26)

is a discrete representation of the Duffing oscillator, describing a forced oscillator coupled to a dissipative restoring force Ott (1993). This map typically produces chaotic behaviours with the critical parameters and , and generates values of and in the range . Following our network projection algorithm, each variable is thus shifted and normalized into the interval . Finally, the Henon map Hénon (1976); Ott (1993) in is defined by the recurrence

(27)
(28)

with the parameters and fixed to and , respectively. For these values, starting from an initial point () the dynamics can either asymptotize to a fractal attractor relying on the subset and or diverge to infinity. For other values of and the map may be still chaotic, intermittent, or converge to a periodic orbit. Here we use the classical parameter setting and, again, we transform the values of each variable into the interval . A one-dimensional projection of the Henon map is also considered, obtained by taking into account only one normalized variable of the map (both and give analogous results).

Attractor
Logistic 0.998(2) 0.49(1) 1.04(2) 3.14(1)
Duffing 1.306(2) 0.75(1) 0.28(8) 0.42(6)
Henon 1.000(1) 0.73(2) 1.10(5) 4.48(1)
1.24(1) 0.82(2) 0.15(2) 0.57(5)
Table 1: Properties of the different multifractal time signals and associated projected networks.

In Table 1 we present a summary of the multifractal properties of the chaotic time signals considered, computed by using the box counting formalism described in Sec. II. In particular, from Eq. (6), we compute the exponent by performing a linear regression of as a function of , for , in a range of values of between and , depending of the particular attractor. The slope of this regression yields the factor . According to Eq. (18), the asymptotic value is obtained by means of linear regressions of as a function of , performed over suitable intervals of the variable , see Fig. 1.

Figure 1: Generalized dimensions as a function of for the logistic map a), the Duffing map b), the Henon map c) and the Henon map c). The corresponding asymptotic values of can be extrapolated by means of a linear regression of these plots following Eq. (18). The results are listed in Table 1.
Figure 2: Rescaled average natural measure as a function of the degree for the logistic map a), the Duffing map b), the Henon map c) and the Henon map d). The function is computed for different values of , considering iterations. Dashed lines represent the estimated values of and in each case, see Table 1.

In Fig. 2 we plot the natural measure , averaged over all nodes of degree , as a function of , for different partitions of the multifractal attractors, i.e. different (or ). In order to check the main assumption in Eq. (9), we plot the rescaled function as a function of the degree, using the fractal dimensions quoted in Table 1. In this case, we expect all plots of each map for different to collapse onto the single universal function . From Fig. 2 we observe, for the logistic, Duffing and Henon maps, a perfect convergence of the rescaled average natural measure as a function of , indicating the validity of the assumption in Eq. (9). In the case of the projection of the Henon attractor, however, the collapse of is not fulfilled. In this particular case, therefore, the predictions made in Sec. IV are not expected to hold.

From Fig. 2, we also observe that in the cases (Figs. 2a and 2c), both the attractor of the logistic map and that of the projected Henon map obey an algebraic behavior, , while in the cases (Figs. 2b and 2d), both the Duffing and the Henon systems show an exponential growth, . A linear regression performed on the data in Fig. 2 provides an estimation of the exponents and . In general, it can be noticed how the linear profiles become more defined and stable while refining the statistics of by increasing ; we thus fit data obtained with in the algebraic case and for an exponential . We thus find the exponents for the logistic map, for the Duffing map, while and are obtained for the and the Henon maps, respectively.

In Fig. 3 we examine the topology of the projected networks by plotting the cumulative degree distributions, , for different values of . As predicted in Sec. IV, the networks characterized by an algebraic growth (here cases) exhibit power-law degree distributions. Panels a) and c) show how for different network sizes (i.e. different values of ) all trends converge to a common power-law distribution characterized by in the logistic networks and for the Henon map. Interestingly, in this last case, the prediction of a scale-free degree distribution holds, despite the fact that Eq. (9) is not fulfilled. This must attributed to the affect of an algebraic function , still present in the Henon map. By contrast, the networks resulting from the projection of the maps follow short-tailed degree distributions, compatible with an exponential behavior . These plots present a poor statistics and we can only extrapolate rough values for , namely and for the Duffing and Henon maps, respectively.

Figure 3: Cumulative degree distributions of the networks projected from the logistic map a), the Duffing map b), the Henon map c) and the Henon map d), computed for different values of and with iterations. Dashed lines represent the estimated values of and/or of the pertinent case, see Table 1.

With these values characterizing the multifractal properties of the maps and the topological properties obtained from the analysis of the projected transition networks, we can validate our theoretical framework. We first check the cross-relations given by Eqs. (15) and (22) for exponential and algebraic cases, respectively. Regarding the algebraic examples we obtain and for the logistic map (see values from Table 1). In this case, the identity Eq. (22) is well fulfilled within error bars. For the Henon attractor in , we obtain instead and , again coinciding within error bars. For exponential cases, the equality Eq. (15) leads to the comparison and for the Duffing map, and and for the Henon attractor. In this case, the exponent relations are affected by stronger errors, but the trend is clearly towards a positive comparison.

Figure 4: Scaling of the degree cut-off, , estimated as the maximum degree in the network, as function of the network size for the logistic map, a), and the Henon map, b).

Finally, for an algebraic , as observed in the logistic map and the projection of the Henon attractor, we can proceed to check the behavior of the maximum degree as a function of the network size . We do not consider the relation between degree cut-off and network size for maps with an exponential , since the very small span of network sizes obtained does not allow for a determination of the exponent in relation Eq. (13). Indeed, for the algebraic case, following Eq. (20), the behavior of the maximum degree is given by , with an exponent . The numerical exponents obtained through a linear regression of as a function of are and for the logistic and the Henon map, respectively. From the values of , and in Table 1, our theoretical predictions are and for the logistic and the Henon map, respectively. The agreement between numerics and theory is very good for the logistic map, but completely off in the Henon case. The disagreement in this last case must be attributed to the failure of Eq. (9). While the general trend towards a power-law degree distribution is ensured by the algebraic form of the function , the lack of the expected scaling with (i.e. the prefactor) affects the scaling relations deduced from Eq. (9). In the Henon case yet another of the approximations made breaks down, namely the assumption of a stable degree distribution. This fact is checked in Fig. 5, which shows that the average degree of the projected networks is essentially independent of for all the multifractal time signals considered, except for the Henon attractor, which exhibits instead a power-law increasing behavior.

Figure 5: Average degree as a function of for the different maps considered. All of them, except the projection of the Henon map are essentially independent of the network discretization.

This fact indicates that Henon networks belong to the class of accelerated networks Dorogovtsev and Mendes (2002). This increasing average degree causes the degree distribution to be non stable, introducing an additional scale, beyond the degree cut-off . Surprisingly, however, the exponent relation Eq. (22) seems to still be fulfilled, at least within error bars.

Vi Discussion

In this paper we have investigated the effect of the fractal and multifractal properties of a temporal signal on the topology of the corresponding projected network. By combining a transition network representation with the box counting formalism, we have mapped temporal signals into networks whose nodes are the boxes partitioning the attractor of the temporal signal in the phase space, and links are established between successive pair of boxes between which the signal jumps. We have developed a mathematical framework connecting network topology to the multifractal properties of the generating signal. This formalism allows us to predict the functional form of the network degree distribution on the basis of the relation, , linking the natural measure of a box with the associated node degree. We have focused on the prototypical and general cases of an exponential and an algebraic growth , showing that the latter results in power law degree distributions whose exponent is controlled by the multifractal exponents of the generating signal. We have verified the validity of our approach through extensive numerical simulations, highlighting the excellent agreement observed in many cases, and discussing in detail the reasons why in some cases (e.g., the Henon map in ) the numerical experiments depart from theoretical predictions.

In particular, we could conclude that a sufficient condition to obtain a scale-free topology is that the natural measure of a box must increase with the degree of the associated node in a algebraic fashion. In our numerical experiments we have observed that this condition is fulfilled in multifractal attractors in with fractal dimension . This fact leads us to conjecture that scale-free networks can be observed in general multifractal time series in which the fractal dimension is equal to the euclidean dimension of the embedding phase space.

Our work extends existing approaches bridging time series analysis and network science by addressing the ubiquitous case of signals exhibiting multifractal properties. By doing this, it enriches the set of interpretative tools available for a better characterization of empirical time-series. For this reason, we can envisage that it will be of interest also to the growing community of interdisciplinary researchers studying natural time series through the lenses of network science.

Acknowledgements.
M.A.B. acknowledges financial support from FRS-FNRS. R.P.-S. acknowledges financial support from the Spanish MINECO, under projects FIS2013-47282-C2-2 and FIS2016-76830-C2-1-P, and EC FET-Proactive Project MULTIPLEX (Grant No. 317532). R.P.-S. acknowledges additional financial support from ICREA Academia, funded by the Generalitat de Catalunya.

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