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Physica A 376 (2007) 692–698

www.elsevier.com/locate/physa

Generation of scale-free networks using a simplepreferential-rewiring dynamics

Suhan Ree�

Department of Industrial Information, Kongju National University, Yesan-Up, Yesan-Gun, Chungnam 340-702, South Korea

Received 29 May 2006; received in revised form 30 September 2006

Available online 27 October 2006

Abstract

We propose a simple dynamical process for non-growing networks, where steady states in the long-time limit exhibit

power-law degree distributions with the exponent 2. At each time step, two nodes, i and j, are randomly selected, and one

incoming link to i is redirected to j with the rewiring probability R, determined only by degrees of two nodes, ki and kj ,

while higher-degree nodes are preferred to get another link. This is an application of the general model introduced earlier

[S. Ree, Phys. Rev. E 73 (2006) 026115]. To take the structure of networks into account, we also consider three possible

distinctions for the model: (i) how we choose a rewiring link out of all incoming links to i (three cases), (ii) whether links are

directed or not (two cases), (iii) types of networks considering the existence of self-loops and multiple links (two cases); as a

result, we specify the total of 12 different cases of the model. We then observe numerically that networks will evolve to

steady states with power-law degree distributions when parameters of the model satisfy certain conditions. This work is

from an effort to find a simple model of the network dynamics generating scale-free networks, and has a potential to

become an underlying mechanism for wide range of scale-free non-growing networks.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Scale-free network; Rewiring dynamics; Stochastic processes

1. Introduction

Complex systems have been studied in various research fields as a general framework to describe systemsthat usually consist of interacting elements, and, in most cases, these systems as a whole exhibit emergingbehavior. We can represent complex systems as networks with nodes (or vertices) and links (or edges), wherenodes represent elements of a given system, and links represent interactions (physical or conceptual) betweentwo elements. Recently, structural and dynamical properties of these networks and their underlyingmechanisms have become the focal point of many researches [1–4]. One well-known property of many naturaland artificial networks alike is the power-law behavior of degree distributions PðkÞ (the degree k of a node isthe number of connected links), and such networks are called scale-free networks; in other words, PðkÞ / k�g,

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.10.011

ing author. Tel.: +82 41 330 1424; fax: +82 41 330 1429.

ess: suhan@kongju.ac.kr.

ARTICLE IN PRESSS. Ree / Physica A 376 (2007) 692–698 693

where g is the exponent, usually in the range of 2pgp3. Examples are citations of scientific papers [5], links inweb pages [6], the Internet [7], metabolic networks [8], and brain functional networks [9], and so on.

Many generating mechanisms have been proposed so far to explain this phenomenon. One popularapproach is using stochastic dynamical processes, in which networks evolve with time and approach scale-freestates, ensembles of network structures sharing specific statistical properties. If we classify these mechanismsinto two classes, one is growing mechanisms, where both the number of nodes, N, and the number of links, E,increase with time, and the other is non-growing mechanisms, where N and E are either constant or varyingbut limited to finite values. One of the most studied growing mechanisms is the preferential-attachment (PA)mechanism [10], where a node and a fixed number of links connected to this node are added at each time step,while other ends of these links are chosen from existing nodes with the probability proportional to theirdegrees. There are many generalizations and extensions of PA mechanism, usually with additional features:for example, rewiring [11], attractiveness of nodes [12], linking between existing nodes [13], age anddeactivation [14], local knowledge [15], link weights [16] and updates on out-degrees of nodes through amultiplicative stochastic process [17]. While PA mechanism is valid to many scale-free growing networks, it isnot suitable for non-growing networks, and there also are numerous non-growing mechanisms that use someadditional features or information: for example, transitive linking and aging [18], fitness of nodes [19,20],adding links using nonlinear PA [21], local optimization using memories of nodes [22], optimization ofHamiltonians [23], and merging and regeneration of nodes [24]. Models mentioned so far share the property ofpower-law degree distributions, but they also attempt to capture different aspects of various types of realnetworks, which we do not describe details here.

In this paper, we propose a non-growing mechanism (both N and E are constant) with a stochasticpreferential-rewiring process, where only degree information of interacting nodes is used for the dynamics. Inother words, nodes and links have no variables that represent memory, attractiveness, or weight, and theglobal structural knowledge of the network is not needed, thereby making the dynamics simple and local. Ournetwork model is an application of the general model introduced in Ref. [25].

2. Model

We first describe the general model [25] briefly, which deals with general N-element systems, whereeach element i is represented by a quantity ki, a non-negative integer. At each time step, two elements, i and j,are randomly chosen out of N elements and one unit of quantity is given from i to j with the exchangeprobability R, determined only by ki and kj. When the exchange occurs, ki and kj become ki � 1 and kj þ 1(the average quantity, a �

PNi¼1ki=N ¼ hki, is constant). The exchange probability is defined using a

parameter b as below

R ¼

1 ð0okipkjÞ;

b ðki4kjÞ;

0 ðki ¼ 0Þ;

8><>:

(1)

where 0pbp1. This is a generalized zero-range process (ZRP) in a fully connected geometry [26], eventhough R as a function of ki and kj in Eq. (1) is rather unique. It was found analytically and numericallythat the distribution of ki’s exhibits the power law when a and b satisfy the condition in the limit oflarge N,

acðbÞ ¼b

1� bln ��1b1=ð1�bÞh i

oa, (2)

where � is a small positive constant, estimated to be about 10�3. This equation defines a power-law regimeinside ða; bÞ-space; i.e., to have a power-law distribution when b is given, a should be greater than the criticalvalue, acðbÞ [or when a is given, b should be less than the critical value, bcðaÞ, which is the inverse of a ¼ acðbÞ].For example, from acð0:5Þ ’ 6, we deduce that b should be less than 0.5 to have a power-law distribution whena ¼ 6. When Eq. (2) is satisfied, the stationary probability distribution, PðkÞ, and the stationary cumulative

ARTICLE IN PRESSS. Ree / Physica A 376 (2007) 692–698694

distribution, PðXkÞ, have been found analytically as below

PðkÞ ¼b

1� b1

½k þ 1=ð1� bÞ�½k þ b=ð1� bÞ�,

PðXkÞ ¼b

1� b1

k þ b=ð1� bÞ, ð3Þ

which is the power-law distribution with g ¼ 2 when kð1� bÞb1.A network with undirected links (a ¼ 2E=N) is an example of such N-element systems, where ki represents

the degree of a node i and R is the rewiring probability (for networks with directed links, ki represents the in-degree, the number of incoming links of a node, and a ¼ E=N). In Fig. 1, the rewiring dynamics is describedusing a schematic diagram. A rewiring process involves three nodes: i and j are randomly chosen, and i0, apivot node, is a neighbor of i, which means a node directly connected to i. When ki and kj are given, rewiringfrom ði0; iÞ to ði0; jÞ can occur with the rewiring probability R in Eq. (1). To be more specific with the modelinvolving networks, we introduce three distinctions: (i) how to choose i0 among neighbors of i (pivot type, threecases), (ii) whether links are directed or not (link type, two cases), (iii) whether self-loops and multiple links areallowed or not (network type, two cases). Then there are 12 different cases of the model. (When two end nodesof a link are the same, we will call it a self-loop, and when there are more than one link between two nodes, wecall them multiple links.)

If we get into the detail, we introduce three pivot types: (i) the node with the smallest degree (‘S’), (ii) therandomly chosen node (‘R’), (iii) the node with the largest degree (‘L’). There are two link types: (i) undirected(‘U ’), (ii) directed (‘D’). Finally, we use two network types: (i) both self-loops and multiple links are allowed(‘1,’ degenerate), (ii) neither a self-loop nor multiple links are allowed (‘2,’ non-degenerate) [27]. Forconvenience, we will use a three-letter notation from here on, where the first letter represents the pivot type,the second letter represents the link type, and the last letter represents the network type. For example, SD2represents the model using non-degenerate networks with directed links, while the pivot node i0 is the neighborof i with the smallest degree.

In addition, three more details should be mentioned for our dynamic rules: (i) while choosing the pivotnode, if there are more than one node with the smallest or largest degree, one node is chosen randomly fromthem, (ii) when the link type is D (directed), out-degrees of neighbors are used when choosing the pivot node,(iii) when the network type is 2 (non-degenerate), a rewiring attempt can be aborted if it results in a self-loopor multiple links.

The dynamical process in the model presented here is a Markov chain, and in most cases the system willreach a steady state, representing a statistical ensemble of network states, regardless of the choice of an initialnetwork. Here we are interested in not only degree distributions but also statistical properties of link structuresin steady states, and they will depend on specific types of the model as given above. For example, the networktype will play an important role (Burda and Krzywicki [28] proposed an algorithm generating an ensemble ofnon-degenerate graphs with an arbitrary degree distribution using rewiring only, and dealt with effects of non-degeneracy). If we consider the link structure of networks, the presence of the pivot type and the network typemakes our network model not just a ZRP but a specialized version of the general model. In the next section,we will observe differences in steady states when different types of our model are used.

i j

i ′

i j

i ′(a) (b)

Fig. 1. A diagram describing the rewiring dynamics. Links of the given network can be either (a) undirected, or (b) directed. For both

cases, two nodes, i and j, are randomly chosen (ki ¼ 2 and kj ¼ 3), and the node i0 is one of the neighbors of i. (For networks with directed

links, ki represents the in-degree of the network, and i0 is one of the neighbors with an outgoing link to i.) Rewiring from ði0; iÞ to ði0; jÞ canoccur with the rewiring probability R.

ARTICLE IN PRESSS. Ree / Physica A 376 (2007) 692–698 695

3. Numerical results

Here we will observe the degree distributions of steady states for all 12 cases of the model, and then focus onnon-degenerate cases while varying the parameters of the model, a, b, and N. Finally, we will look at somestructural properties of non-degenerate undirected networks.

In Fig. 2, we observe stationary degree distributions, PðXkÞ, of 12 different cases when a ¼ 6, b ¼ 0:5, andN ¼ 10 000. If we look at degenerate cases (network type 1), degree distributions are the same as those fromthe general model as expected, because degree distributions do not depend on the pivot type and the link type.These cases can be regarded as ZRPs when we are only interested in degree distributions. On the other hand,for non-degenerate cases (network type 2), the degree distributions are different from those from degeneratecases, because rewiring attempts can be aborted depending on the current link structure of the network; as aresult, cutoffs are usually lower. When the pivot type is S, the distribution is closer to those of degenerate caseseven though cutoffs are still lower, because the probability of aborting a rewiring attempt is lower when thepivot node has the smallest degree [see Figs. 2(a) and (b)], and that is especially true for cases with undirectedlinks.

In Fig. 3, we observe the behavior of the distributions of non-degenerate networks (SU2 and SD2) moreclosely. We vary one parameter while the other two parameters are fixed. In Figs. 3(a) and (b), a is varied whenN and b are fixed, while in Figs. 3(c) and (d), b is varied when N and a are fixed. We observe that the

10-4

10-3

10-2

10-1

100

P(≥

k)

10-4

10-3

10-2

10-1

100

P(≥

k)

100 101 102 103

k

10-4

10-3

10-2

10-1

100

P(≥

k)

100 101 102 103

k

N=10000α=6, β=0.5

N=10000α=6, β=0.5

α=6, β=0.5

α=6, β=0.5

α=6, β=0.5

RD1

N=10000α=6, β=0.5

SD1SU1

RU1RU2

SU2 SD2

RD2

LU1LU2

LD1LD2

(a) (b)

(c) (d)

(e) (f)

N=10000

N=10000

N=10000

Fig. 2. Stationary cumulative degree distributions, PðXkÞ, of 12 different cases, when a ¼ 6, b ¼ 0:5, and N ¼ 10000: (a) SU1 and SU2,

(b) SD1 and SD2, (c) RU1 and RU2, (d) RD1 and RD2, (e) LU1 and LU2, (f) LD1 and LD2. The thin solid line represents the analytic

solution, PðXkÞ ¼ 1=ð1þ kÞ, of the general model when b ¼ 0:5 as N !1.

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P(≥

k)

10-4

10-3

10-2

10-1

100

P(≥

k)

10-4

10-3

10-2

10-1

100

100 101 102 103

k

P(≥

k)

10-4

10-3

10-2

10-1

100

100 101 102 103

k

β=0.1

α=2 α=2α=4α=6α=10α=20

α=4α=6α=10α=20

β=0.3β=0.5β=0.7β=0.9

β=0.1β=0.3β=0.5β =0.7β=0.9

N=1000N=5000N=10000N=20000

N=1000N=5000N=10000N=20000

SU2N=10000

β=0.5

SD2N=10000

β=0.5

SD2N=10000

β=6

SD2α=6β=0.5

SU2α=6β=0.5

SU2N=10000

α=6

(a) (b)

(c) (d)

(e) (f)

Fig. 3. The stationary cumulative degree distributions, PðXkÞ, of networks, SU2 and SD2, when one of three parameters of the model, a,b and N, is varied. (a), (b) a is varied; (c), (d) b is varied; and (e), (f) N is varied. The thin solid line represents the analytic solution,

PðXkÞ ¼ 1=ð1þ kÞ, of the general model when b ¼ 0:5 as N !1. When a ’ acðbÞ, the degree distributions get close to the theoretical

curve, and cutoffs increase as N increases.

S. Ree / Physica A 376 (2007) 692–698696

stationary degree distributions are close to the theoretical curve from the general model when a is close to acðbÞ[or b is close to bcðaÞ] unlike degenerate cases, in which stationary distributions were found to follow the powerlaw when a4acðbÞ [or bobcðaÞ]. In Figs. 3(e) and (f), N is varied when a and b are fixed. We observe thatcutoffs increase as N increases, because the sparser the network gets, the smaller the chance of aborting therewiring attempt becomes. In the general model, Pð0Þ is a good indicator when determining whether the degreedistribution is scale-free or not [when a4acðbÞ, Pð0Þ becomes 1� b and the scale-free degree distribution inEq. (3) can be obtained analytically]. For non-degenerate cases in Fig. 3, Pð0Þ is still a good indicator; Pð0Þ isclose to 1� b when a is close to acðbÞ. For example, when a ¼ 6, b ¼ 0:5, andN ¼ 20 000 [shown in Fig. 3(e)],Pð0Þ ’ 0:495.

In Fig. 4, we observe other structural properties of the SU2 network of the steady state when a ¼ 6, b ¼ 0:5,and N ¼ 20 000. There are several well-known statistical measures [2–4] such as the clustering coefficient [29],C, the average shortest path length, L, and the Pearson correlation coefficient of degrees [30], r. Here an initialnetwork is the Erdos–Renyi (ER) random network, where C ¼ 0:0003, L ¼ 5:73, and r ¼ 0:00008. When thestationary distribution is reached at T=N�105, we find that C ¼ 0:04 ð�0:002Þ, L ¼ 4:53 ð�0:03Þ, andr ¼ 0:2 ð�0:02Þ, showing the small-world behavior [31]. These networks have more than one component: forER networks, the size of the largest component is 19 958 and Pð0Þ ¼ 0:002, and they become 10 199 ð�53Þ and0.467 ð�0:003Þ, respectively, in the steady state. In Fig. 4(a), the average clustering coefficient of nodes with

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0.01

0.1

C(k

)

T/N

C

1 10 100

k

10

100

<k n

n>

T/N

r

10-1

0

0.2

-0.2

10-2

10-3

10-4100 102 104

100 102 104

SU2α= 6, β = 0.5, N = 20000

at T/N = 106

SU2α= 6, β = 0.5, N = 20000

at T/N = 106

(a)

(b)

Fig. 4. Structural properties of stationary SU2 networks when a ¼ 6, b ¼ 0:5, and N ¼ 20 000. Data points (�) are obtained from

averaging values of individual nodes in log-sized bins in both cases. (a) The averaged clustering coefficient, CðkÞ, versus k in a log–log plot

at T=N ¼ 106. Inset: The clustering coefficient, C, versus T=N in a log–log plot. (b) The averaged degree of nearest neighbors of nodes

with degree k, hknni, versus k in a log–log plot at T=N ¼ 106. It shows that the stationary network is assortatively mixed. Inset: The

Pearson correlation coefficient of the degrees, r, versus T=N in a semi-log plot.

S. Ree / Physica A 376 (2007) 692–698 697

degree k, CðkÞ, is found, and we can observe that CðkÞ increases with k. And the inset shows how C increaseswith time until T=N�105. In Fig. 4(b), the averaged degree of nearest neighbors of nodes with degree k, hknni,is found, and we can observe that hknni increases with k also. The inset shows how r changes with time, andboth results imply that this network is assortatively mixed. In both cases, the increasing behavior stops atk�200, and this is probably due to the finite-size effect [the distribution deviates from the theoretical curve atk�200 in Fig. 3(e)].

4. Discussion

We introduced the rewiring model that generates scale-free networks with g ¼ 2, and showed numericalresults for 12 different cases of the model. When the network can have self-loops and multiple links(degenerate cases), we observed that stationary degree distributions follow the same power law when thecondition a4acðbÞ is satisfied, as the general model showed in the previous work [25]. It is more interestingwhen we use networks with neither a self-loop nor multiple links (non-degenerate cases). We observed thatscale-free networks with g ¼ 2 are generated when the condition a ’ acðbÞ is satisfied. The exponent 2 appearsin many real-life scale-free networks. For example, the in-degree distribution of the World Wide Web isknown to exhibit the power law with g ¼ 2 robustly [17,32]. We also looked at some statistical properties ofnon-degenerate undirected networks, and found out that networks from this model are assortatively mixed.

ARTICLE IN PRESSS. Ree / Physica A 376 (2007) 692–698698

This model was not conceived to describe a certain type of real networks, but it was from an effort to find asimple rewiring model generating scale-free networks, with intention of using local degree information onlyand incorporating the rich-get-richer phenomenon. As one of the simplest rewiring models for scale-freenetworks, this model can be extended to meet specific needs of some real non-growing networks.

Acknowledgments

This work was supported by Grant No. R05-2002-000799-0 from the Basic Research Program of the KoreaScience & Engineering Foundation.

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