Physica A 387 (2008) 1361–1368www.elsevier.com/locate/physa

Detrended fluctuation analysis of particle condensation oncomplex networks

Ming Tang, Zonghua Liu∗

Institute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai, 200062, China

Received 23 August 2007; received in revised form 21 September 2007Available online 22 October 2007

Abstract

It has been found that the structure of complex network has significant influence on the condensation of particles and theequilibrium state of particles is dynamical. We study here the fluctuation of particles on each individual node at equilibrium statusand find that the particle distributions can be normalized to the same framework of Gaussian distribution, which is independent ofthe nodes and network structures, while their differences are reflected by the asymmetric degree of the distribution. By the approachof detrended fluctuation analysis we reveal that the correlation exponents of particle fluctuations may reflect the information ofparticle condensation and be useful in exploring the network structure.c© 2007 Elsevier B.V. All rights reserved.

PACS: 89.75.Fb; 05.40.-a; 05.70.Fh; 05.60.Cd

Keywords: Condensation of particles; Gaussian distribution; Detrended fluctuation analysis; Correlation exponents; Particle fluctuations

1. Introduction

The recent study on complex networks has attracted an increasing interest and revealed that most of the realisticnetworks have small world and scale-free features [1–3]. It has been found that these features have significant influenceon the dynamical behaviors on complex networks, one of which is the particle condensation [4–9]. The particles ona node may jump to its neighbors because of the interaction among them and then condense on hubs under certainconditions. In the equilibrium status, the number of particles on a node fluctuates around an average and has a steadydistribution. As the fluctuated particles come from the neighbors of a node and the latter is closely related to thestructure of network, the network structure will definitely influence the fluctuation. Here we not only focus on howthe network structure influences the particle fluctuation but also wonder whether it is possible to get some usefulinformation by searching the structure of network using the inverse process, i.e., inferring the network structure fromthe fluctuation of particles.

Many physical and biological systems have output signals with erratic fluctuation, heterogeneity, andnonstationarity. A conventional approach to measure these signals is the detrended fluctuation analysis (DFA), which

∗ Corresponding author. Tel.: +86 21 62233216.E-mail address: zhliu@phy.ecnu.edu.cn (Z. Liu).

0378-4371/$ - see front matter c© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2007.10.039

1362 M. Tang, Z. Liu / Physica A 387 (2008) 1361–1368

can reliably quantify scaling features in the fluctuations by filtering out polynomial trends. The DFA method is basedon the idea that a correlated time series can be mapped to a self-similar process by integration [10–16]. The DFAmethod has been successfully applied to detect long-range correlations in highly complex heart beat time series [11],stock index [12], physiological signals [14], and traffic data [15,16]. Recalling the erratic fluctuations on differentnodes in the particle condensation, here we use the DFA method to measure the structure dependence of fluctuationof particles at a node. Our principal results are as follows: (1) We find that all the distributions of particles on differentnodes can be normalized to the same framework of Gaussian distribution with different degree of asymmetry. (2) Thefluctuations of particles on different nodes have different correlations although they belong to the same framework ofnormalized distribution. (3) The correlation exponent depends on both the interaction among particles at a node andthe network structures.

Our results may have potential applications in the searching of network structure which is in fact another importanttopic in complex networks [17–20]. It is well-known that the collective dynamics depend on the network structure [1,2]. In reality, however, many networks have largely unknown aspects of structure, such as networks of neurons,interaction of proteins or genes, and ecological foodwebs. Therefore, it is useful if one can use the dynamical responseto infer the network structure. This inverse method has been successfully used to figure out the information on detailsof connectivity of a network [17–20]. Here we find that the information on unknown network structure may be alsoreflected by the particle distributions and the correlation exponents of particle fluctuations. That is, suppose we donot know the global structure of network but only a part of it, then by doing an experiment of particle jumps in thenetwork, we may get some useful information on the network structure, such as if it is most probably a scale-free,weighted scale-free, or random network etc.

The paper is organized as follows. In Section 2, the particle evolution and its dependence on parameters andnetwork structures are briefly reviewed. More details can be found, e.g., in Ref. [7–9]. Then, we reveal that theparticle distributions on different nodes and different networks can be normalized to the same framework of Gaussiandistribution. In Section 3, the algorithm steps of DFA are briefly reviewed and applied to the time series of particlesat different nodes. We find that the correlation exponents of particle fluctuations may be used to explore the networkstructure. A conclusion is given in Section 4.

2. Particle distributions on different nodes and different networks

The particle condensation on complex networks has been recently studied by both the approach of grand canonicalensemble and the mean-field theory [4–9]. Its evolution process can be briefly reviewed as follows. Suppose Nparticles are randomly put on a network of L nodes and each node i can be occupied by any integral number ofparticles ni = 0, 1, 2, . . . , N . The interactions among different nodes are ignored, i.e., we consider the case of zero-range process (ZRP) where the interaction occurs only when the particles stay at the same node. Because of theinteraction among the particles at the same node, some of the particles will jump out of the node and hop to othernodes, making the particle redistribution among n1, n2, . . . , nL . Hence a microscopic configuration is represented byn = n1, n2, . . . , nL . The particles at node i with ni > 0 may jump out with the jumping rates ρ(ni ) and hop from nodei to one of its neighbors j along the link with the hopping probability T j←i . The jumping rate is taken as ρ(ni ) = nδ

i .δ = 0 means only one of ni particles will jump out each time, indicating that the particles are attracting each other.In the case of δ = 1, all the ni particles will jump out, implying that they are moving independently and the systemreduces to a noninteracting system of N random walkers. δ < 1 means that there exists attractive interaction amongparticles. The next station of a jumping particle is determined by the hopping probability T j←i which is nonzeroif i and j are linked and 0 otherwise, i.e., a particle jumping out of the node i is allowed only to hop to one ofits neighboring nodes. The detailed form of T j←i depends on the structure of network [7–9]. With enough time ofevolution, the system will reach an equilibrium state. It is found that there is a critical δc where the condensation willoccur for δ ≤ δc and does not occur when δ > δc. Here condensation means that a finite fraction of particles arecondensed onto a single site.

Once the condensation occurs, most of the particles will move to those nodes with the largest links, especially thehub. It is pointed out that there is a crossover point kc [7–9]. The average particles on a node will be smaller than onefor k < kc and larger than one for k > kc. Therefore, we may deduce that most of the time, the nodes with k < kc willhave no particles while the nodes with k > kc will have their average particles. That is, the nodes with k < kc will

M. Tang, Z. Liu / Physica A 387 (2008) 1361–1368 1363

Fig. 1. Particle distribution at equilibrium status in BA model for parameters L = 104, N = 5000 (ρ = 0.5) and δ = 0. (a) The average particleson a node 〈nk 〉 versus the degree k; (b) evolution of particles on a given node (k = 10); (c) evolution of particles on the hub.

have fluctuation only on the side of ni >< ni > while the nodes with k > kc will have fluctuation on both the side ofni >< ni > and the side of ni � ni >, see Fig. 1(b) and (c).

Let us first take the famous Barabasi–Albert (BA) network as an example to illustrate the condensation. The BAmodel is a scale-free network with the degree distribution P(k) ∼ k−3 [21]. As BA model is an un-weighted network,the hopping probability T j←i is taken as

T j←i = 1/ki , (1)

i.e., each of its neighboring nodes has the same possibility to capture the jumping particle, and the critical δc = 0.5 [7,8]. For demonstrating the condensation, we put N = 5000 particles randomly on the BA network with size L = 104,i.e., the particle density ρ = 0.5. We find that after the transient time, the average particles on a node are fixed but thetemporal particles are fluctuated around the average. Fig. 1 shows the case of δ = 0 at equilibrium status where nkdenotes the number of particles on a node with degree k, (a) represents how the average particles on a node changewith the degree k, (b) the evolution of particles on a given node, and (c) the evolution of particles on the hub. FromFig. 1(a) it is easy to see that the hub has most of the total particles, indicating the condensation of particles on thehub.

Obviously, there is a most probable number in Fig. 1(b) (zero) and (c) (close to the average). Let us takeit as nop

k , thus we have P(nk) ≤ P(nopk ), where P(nk) represents the possibility of nk . We find that P(nk)

satisfies the framework of Gaussian distribution with the maximum at P(nopk ), i.e., P(nk) ∼ exp(−

(nk−nopk )2

2σ 2 ) where

σ =

√〈(nk − nop

k )2〉. Normalizing P(nopk ) to unity and letting x ≡ (nk − nop

k )/σ , we have P(x) = exp(−x2/2).Interestingly, we find that the expression of P(x) works for every individual node as well as for all the different δ nomatter whether it is larger than δc or smaller than δc. Fig. 2 shows how P(nk) changes with x at four typical degreesfor different δ where the solid curve is the reference framework exp(−x2/2). An obvious difference in the four panelsof Fig. 2 is that panel (a) has only half a branch, panels (b) and (c) are asymmetric with denser points on the rightbranch than the left branch, and panel (d) is approximately symmetric. This can be easily understood as follows: Forthose nodes with small degree k, they have no particles in most of the time and have a few particles, occasionally, seeFig. 1(b). Their optimal nop

k is zero. As the particle number cannot be negative, thus their distribution has only the right

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Fig. 2. Normalized particle distribution on a given node in BA model for different jumping rates where the average degree 〈k〉 = 6, the networksize L = 104, the total particles N = 5000, and the “squares”, “circles”, “up triangles”, and “down triangles” denote δ = 0.0, 0.2, 0.5, and 1.0,respectively, and the solid curve represents the reference framework exp(−x2/2). (a) Denotes the case of k = 10, (b) the case of k = 50, (c) thecase of k = 138, and (d) the case of k = kmax = 301.

branch. While for the hub nodes, they always have a large number of particles because of the condensation. Hencethere is enough space for the particles to fluctuate on both nk > nop

k and nk < nopk sides, resulting in the symmetric

distribution. The middle nodes are just in between these two extremes. Therefore, we conclude that the asymmetricdegree of distribution decreases with the increase of node’s degree k.

Secondly, let us discuss the Barrat, Barthelemy and Vespignani (BBV) model which is a typical weightednetwork [22]. Different from the BA model where each link has the same weight, each link connecting a node iwith a node j in the BBV model is associated with a distinguished weight wi j . The strength of a node i , si , is definedas the sum of all the weights along its links, i.e., si =

∑j∈Bi

wi j where Bi denotes the set of neighbors of the node i .wi j is formed as follows: in the forming process of the BBV network, a new added edge will introduce an additionalstrength to the connected node and the newly introduced strength W will be distributed among the neighbors of thenode according to the weights of the edges, i.e., the weight of edge is related to the forming process of the network.This model will result in a linear relationship between the strength and the degree, i.e., s(k) ∼ k, and a power-lawdegree distribution P(k) ∼ k−γ with γ = (4W + 3)/(2W + 1) [22]. For the weighted network, the hopping rate canbe taken as

T j←i = wi j/si , (2)

i.e., each of its neighboring nodes has different possibilities to capture the jumping particles [23]. It is found that thecritical δc = 5/6 for W = 2 [24].

For demonstrating the condensation, we also put N = 5000 particles randomly on the BBV network with sizeL = 104 and the average degree 〈k〉 = 6, and do the same operations as in BA model. Interestingly, we also findthe same framework of particle distributions, see Fig. 3(a) and (b) for two typical nodes in BBV model. ComparingFig. 3(a) with Fig. 2(b) it is easy to see that their points on the curve exp(−x2/2) are distributed in a different wayalthough their degree are the same (k = 50). That is, for the same δ, the part with x < 0 in the particle distribution isless in the BBV model than that in the BA model, indicating that the BBV model has more significant condensation.Taking the case of δ = 0.5 as an example, it is the critical situation of condensation in the BA model (δ = δc = 0.5)but a normal situation of condensation in the BBV model (δ < δc = 5/6), thus there are “up triangles” on the left-handside of the distribution in Fig. 2(b) but not in Fig. 3(a).

Both the above two examples are heterogeneous networks. Let us now discuss another situation of homogeneousnetwork, i.e., the Erdos and Renyi random (ER) network [25]. The ER network can be constructed as follows: TakeL = 104 nodes and put a link between two randomly chosen nodes until the total links are 4 × 104(〈k〉 = 8)

where the double links between two nodes are prohibited. The resulted network has a Poisson degree distribution

M. Tang, Z. Liu / Physica A 387 (2008) 1361–1368 1365

Fig. 3. Normalized particle distribution on a given node for different jumping rates in BBV model (a) and (b) and in ER model (c) and (d) wherethe network size L = 104, the total particles N = 5000, and the “squares”, “circles”, “up triangles”, and “down triangles” denote δ = 0.0, 0.2, 0.5,and 1.0, respectively, and the solid curve represents the reference framework exp(−x2/2). (a) denotes the case of k = 50 in the BBV model, (b)the case of k = 443 in the BBV model, (c) the case of k = 10 in the ER model, and (d) the case of k = 21 (hub) in the ER model.

P(k) ≈ e−〈k〉 〈k〉k

k! . Here the hopping probability T j←i is taken as the Eq. (1). It is found that the critical δc ≈ 0. Doingthe similar particle evolution and the same operations as in the above two models, we also find the exactly sameexpression of particle distribution, see Fig. 3(c) and (d) for two typical nodes in ER model. Therefore, we concludethat the normalized form exp(−x2/2) is universal for particle condensation in all the different networks. On the otherhand, Fig. 3(d) is the case of hub in ER model, Fig. 3(b) is the case of larger node in BBV model, Fig. 2(c) and (d) arethe cases of larger node and hub in BA model. The influence of network structure may be seriously reflected in thesenodes with larger links. Comparing Fig. 3(d) with Figs. 3(b), 2(c) and (d), respectively, it is easy to see that Fig. 3(d)is an asymmetric distribution while all the Figs. 3(b), 2(c) and (d) are approximate symmetric distributions, indicatingthat the implementation of condensation in ER model is difficult (δc ≈ 0).

3. Detrended fluctuation analysis of particle fluctuations

The DFA method is a modified root-mean-square (rms) analysis of a random walk and its algorithm can be workedout as in the following steps [10–16]:(1) Start with a signal nk(t), where t = 1, . . . , T , and T is the length of the signal, and integrate nk(t) to obtain

y(i) =i∑

t=1

[nk(t)− 〈nk〉], (3)

where 〈nk〉 =1T

∑Tt=1 nk(t).

(2) Divide the integrated profile y(i) into boxes of equal length m. In each box, we fit y(i) to get its local trend yfit byusing a least-square fit.(3) The integrated profile y(i) is detrended by subtracting the local trend yfit in each box:

Ym(i) ≡ y(i)− yfit(i). (4)

(4) For a given box size m, the rms fluctuation for the integrated and detrended signals are calculated:

F(m) =

√√√√ 1T

T∑i=1

[Ym(i)]2. (5)

(5) Repeat this procedure for different box sizes m.

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Fig. 4. F(m) versus m in BA model for δ = 0.2, where (a) represents the F(m) of a given node with k = 10, and (b) the F(m) of the hub.

For scale-invariant signals with power-law correlations, there is a power-law relationship between the rmsfluctuation function F(m) and the box size m:

F(m) ∼ mα. (6)

The correlation exponent α represents the degree of the correlation in the signal: the signal is uncorrelated for α = 0.5and correlated for α > 0.5 [10–16].

We now use the DFA method to quantify the correlation of the fluctuated time series nk(t). From Fig. 1(c) it iseasy to see that there is a local trend in nk(t), thus nk(t) is a good candidate for the DFA method. As the stabilizednk(t) depends on the degree k at the equilibrium status (see Fig. 1(a)), we expect the scaling exponent α in Eq. (6)be different for different k. Our numerical simulations have confirmed this deduction, see Fig. 4 for the case of BAmodel where (a) represents the F(m) of a given node and (b) the F(m) of the hub.

From Fig. 4 it is easy to see that there is a crossover mc, which reflects the correlation length [10–12]. The slopeof F(m) will change at mc and the scaling exponent α is the slope of F(m) with m < mc, i.e., the slope given by thesolid straight line in Fig. 4. Fig. 5(a) shows how the scaling exponent α depends on the degree k for different δ in BAmodel. Obviously, α will gradually increase from 0.5 to 1 with the increase of k for the cases of δ < δc = 0.5 and keepan approximate constant 0.5 for the cases of δ > δc. In detail, the tendency of increasing α is different for differentδ. Both the cases of δ = 0 and 0.2 have a few nodes with α close to unity, and the number of these nodes increaseswith δ, i.e., it is 2 for δ = 0 and 3 for δ = 0.2. While for the critical case of δ = δc = 0.5, there is no nodes with α

close to unity. Considering that the particles are condensed on the hub nodes and the degree of condensation decreaseswith the increase of δ, we conclude that the scaling exponent α reflects the condensation of particles with α = 0.5for no condensation and α = 1 for condensation. This corresponding relationship is easy to be understood. For thecondensed hub, the exchanged particles at each time step are huge and thus are related to most of all the particles,hence it has a global correlation and results in α = 1. While for the un-condensed nodes, the exchanged particles aresmall and thus are only related to a few particles which is purely random, hence resulting in α = 0.5.

The similar situation is observed in the case of BBV network. Comparing with the case of BA network,δ = 0.5 (< δc = 5/6) is not the critical case in the BBV network, and hence the hub nodes will also have α

close to unity. Fig. 5(b) shows the results. While for the case of ER network, we cannot observe the similar situationas in the BA and BBV networks because of its δc ≈ 0. Therefore, except the case of δ = 0, what we can observe is anapproximate constant α for all the different δ in ER model. Fig. 5(c) shows the results.

Comparing the three panels in Fig. 5 one can see that in the middle part, the line with “circles” is higher than theline with “up triangles” in panel (a), lower than it in panel (b), and equal to it in panel (c). A common feature in thethree panels is that all the lines of δ > δc have an approximate constant close to 0.5 and the lines of δ < δc deviatefrom 0.5 and increase with k. In the following we will use this feature and combine it with the normalized distributionof particles to infer the network structure by the inverse analysis method, i.e., use the dynamics to infer the structure.

Suppose the network is very large and we know only a finite part of it, which has the common feature of the wholenetwork. For measuring some useful information on the network structure, let us design the following experiment:

M. Tang, Z. Liu / Physica A 387 (2008) 1361–1368 1367

Fig. 5. α versus k for different δ and different networks, where the “squares”, “circles”, “up triangles”, and “down triangles” denote δ =

0.0, 0.2, 0.5, and 1.0, respectively, and (a) denotes the case of BA model, (b) the case of BBV model, and (c) the case of ER model.

Input N particles to the network and let them evolve according to the ZRP rules with some middle δ, such as δ = 0.2and 0.5. Suppose the weight of the link is reflected by the width of the link. A particle will have a larger possibilityto choose a wide link than a narrow link. In this situation, the particles can choose Eq. (1) for un-weighted networkand Eq. (2) for weighted network. After the transient time, we measure the normalized particle distributions and thescaling exponent α on the known part of nodes. First we check the particle distribution. If all the known nodes havethe asymmetric particle distributions, the network may be the ER model; if there is a small finite percentage of thenodes whose distributions have the left-hand side with x < 0, we may deduce that the network is scale-free, such asthe BA network; and if only a few nodes have the distribution with x < 0, the network may be weighted scaling-free,such as BBV. And then we check the scaling exponent α. If α is an approximate constant 0.5 for all the nodes, thenetwork is confirmed to be the ER model; if α increases with k and the α for δ = 0.5 is smaller than that of δ = 0.2for most of the nodes, the network is confirmed to be the BA model; and if α increases with k but the α for δ = 0.5 isgreater than that of δ = 0.2 for most of the nodes, the network is confirmed to be the BBV model. This approach isbased on the assumption that the particles can detect the weights through the width of the links. If the weights are notreflected by the width of the link and are undetectable, we can only use Eq. (1) as the hopping rate for all the networks.In this situation, we can still distinguish the ER model and the scale-free networks, but fail at distinguishing the BAmodel and BBV model. In sum, we may get the useful information of network structure by measuring the particledistribution and the scaling exponent α.

4. Conclusions

In conclusion, the particle fluctuations on different networks are studied. It is revealed that the fluctuations atdifferent nodes satisfy the same framework of normalized particle distributions although most probable numbersdepend on the individual node and the network structure. The DFA method is used to measure the correlationexponents on different nodes. We find that the scaling exponent α reflects the dependence of particle condensation onthe dynamical parameters and the network structure. An inverse approach is proposed to infer the network structurethrough both the particle distribution and the scaling exponent α. This work may highlight the understanding of therelationship between the structure and its function.

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Acknowledgements

This work was supported by the NNSF of China under Grant No. 10475027 and No. 10635040, by NCET-05-0424and 05SG27, and by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814800.

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