Physica A 391 (2012) 2163–2165

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Physica A

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Comments on ‘‘Scale-free networks without growth’’Weicai Zhong a,b,1, Jing Liu c,∗

a School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australiab College of Information Engineering, Northwest A&F University, Yangling 712100, Chinac Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, China

a r t i c l e i n f o

Article history:Received 2 May 2011Received in revised form 22 October 2011Available online 25 November 2011

Keywords:Scale-freeRewiring modelsSelf-loopMultiple-link

a b s t r a c t

In [Y.-B. Xie, T. Zhou, B.-H. Wang, Scale-free networks without growth, Physica A 387(2008) 1683–1688], a nongrowing scale-free network model has been introduced, whichshows that the degree distribution of the model varies from the power-law form to thePoisson form as the free parameter α increases, and indicates that the growth may notbe necessary for a scale-free network structure to emerge. However, the model implicitlyassumes that self-loops and multiple-links are allowed in the model and counted in thedegree distribution. Inmany real-life networks, such an assumptionmay not be reasonable.We showed here that the degree distribution of the emergent network does not obey apower-law form if self-loops and multiple-links are allowed in the model but not countedin the degree distribution.We also observed the same result when self-loops andmultiple-links are not allowed in the model. Furthermore, we showed that the effect of self-loopsand multiple-links on the degree distribution weakens as α increases and even becomesnegligible when α is sufficiently large.

© 2011 Elsevier B.V. All rights reserved.

Xie et al. (2008) [1] proposed a network rewiring model and showed that a scale-free structure can emerge without thenetwork growth. In their model, the number of nodes N and the number of edges E are both fixed. The model starts witha random network or any other type of network. Then, the network evolves based on the following rewiring process: ateach time step, an edge is randomly selected and removed from the network; at the same time, a node is selected with thepreferential probability

Π(ki) =ki + α∑

j(kj + α)

(1)

where ki is the degree of the ith node and α > 0 is a constant representing the original attraction of each node. Another nodeis selected with the same preferential probability given in Eq. (1). Then, a new edge between these two nodes is created.

Let ps(k) be the degree distribution in a stationary state. The theoretical analysis shows that ps(k) has the followingexpression

ps(k) =

α

α + γ

α γ

α + γ

k 1k!

α(α + 1) · · · (α + k − 1). (2)

where γ = 2E/N represents the density of connectivity. The validity of Eq. (2) is restricted in the interval 0 ≤ k ≤ E.

∗ Corresponding author.E-mail addresses:wakenov@gmail.com, w.zhong@adfa.edu.au (W. Zhong), neouma@163.com (J. Liu).

1 Present address: SPSS Statistics R&D Division, IBM CDL, Xi’an 710065, China.

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.10.025

2164 W. Zhong, J. Liu / Physica A 391 (2012) 2163–2165

Fig. 1. The degree distributions of the Xie et al. model with α = 0.01 and γ = 5. The solid curve represents the analytical prediction. The squares andcircles denote the degree distributions with andwithout counting self-loops andmultiple-links, respectively. The diamonds denote the degree distributionof the model that does not allow self-loops and multiple-links.

When α ≪ 1 and γ ≽ 1 (the symbol ‘‘≽’’ means moderately larger than), a power-law degree distribution emerges

ps(k) =

α

γ

αα

k. (3)

When α ≫ 1, the preferential attachment mechanism is destroyed, and the degree distribution obeys a Poissondistribution

ps(k) =e−γ

k!γ k. (4)

The Xie et al. model above implicitly assumed that self-loops and multiple-links are allowed and counted in the degreedistribution, where a self-loop is defined as a link whose two end nodes are the same and amultiple-link occurs when thereare more than one links between two nodes. Such an assumption may not be reasonable in real-life networks. For instance,in friendship networks [2–4], self-loops have no practical meaning, and two friends have only one link connecting them.To investigate the effect of self-loops and multiple-links on the degree distribution, we run a series of simulations. In ourstudy, 106 time steps are performed in all simulations, but only the final 2× 105 time steps are used to obtain the statisticalaverage, and the network size is fixed at N = 1000.

Fig. 1 shows the simulation results with α = 0.01 and γ = 5. It can be seen that the degree distribution that counts self-loops and multiple-links (squares) obeys a power-law form with the exponent approximately equal to one, which agreeswith the analytical results (solid curve). However, in the emergent network, we found that 84% of the edges are self-loopsor multiple-links, and about 94% of the nodes have zero degrees. Thus, if we remove self-loops and multiple-links from theemergent network, the resulted degree distribution (circles) obviously deviates from a power-law form. Furthermore, if wedisallow self-loops and multiple-links in the Xie et al. model, we found that about 78% of the nodes are isolated, and thedegree distribution (diamonds) does not obey a power-law form either. Therefore, we conclude that the scale-free structuredoes not emerge from the Xie et al. model.

Fig. 2 shows the analytical and simulation results about the degree distributions generated by six groups of parameters,α and γ . It can be seen that as α increases, the departure of the degree distribution (squares) from the power law becomeslarger, and the degree distribution (squares) with α = 100 approximately obeys a Poisson distribution rather than a power-law distribution. In addition, the degree distribution without counting self-loops and multiple-links (circles) deviates lessfrom the analytical prediction as α increases. Particularly, when α = 100 and γ = 10, only approximately 0.01% of all edgesare self-loops or multiple-links and approximately 0.6% of all nodes have zero degrees. This result occurs because a largerα indicates a blunter preferential attachment mechanism; thus, two end nodes of a new edge are unlikely to be the sameor mutual neighbours. A similar result of the degree distribution is obtained when self-loops andmultiple-links (diamonds)are disallowed in the model.

In conclusion, we found that if self-loops and multiple-links are allowed in the model but not counted in the degreedistribution or are disallowed in the rewiring model proposed by Xie et al. [1], the scale-free structure does not emergeunder the parameter setting α ≈ 0 and γ ≽ 1. In addition, as α increases, the degree distribution deviates from the power-law formand approaches the Poisson form, and the effect of self-loops andmultiple-links on the degree distribution becomessmaller. Especially, when α is large, such an effect is negligible. In general, most rewiring models [5,6] have a similar issue.Thus, further theoretical and simulation studies on the rewiring mechanism are necessary.

W. Zhong, J. Liu / Physica A 391 (2012) 2163–2165 2165

a b

c d

e f

Fig. 2. The degree distributions of the Xie et al. model for 6 groups of parameter settings, namely (a) (α = 0.01, γ = 10), (b) (α = 0.1, γ = 5),(c) (α = 1, γ = 10), (d) (α = 10, γ = 5), (e) (α = 100, γ = 5), and (f) (α = 100, γ = 10). The symbols have the same denotations as those depicted inFig. 1.

Acknowledgements

Thisworkwas partially funded by theNational Science Foundation of China (NSFC) under Grant 61103119 and 60872135.

References

[1] Y.-B. Xie, T. Zhou, B.-H. Wang, Scale-free networks without growth, Physica A 387 (2008) 1683–1688.[2] A. Rapoport, W.J. Horvath, A study of a large sociogram, Behav. Sci. 6 (1961) 279–291.[3] B. Hu, X.-Y. Jiang, J.-F. Ding, Y.-B. Xie, B.-H. Wang, A weighted network model for interpersonal relationship evolution, Physica A 353 (2005) 576–594.[4] L. Lü, T. Zhou, Link prediction in complex networks: a survey, Physica A 390 (2011) 1150–1170.[5] J. Lindquist, J. Ma, P. Driessche, F.H. Willeboordse, Network evolution by different rewiring schemes, Physica D 238 (2009) 370–378.[6] X.-J. Xu, X.-M. Hu, L.-J. Zhang, Network evolution by nonlinear preferential rewiring of edges, Physica A 390 (2011) 2429–2434.