From regular to growing small-world networks

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Physica A 385 (2007) 765–772 www.elsevier.com/locate/physa

From regular to growing small-world networks Zhongzhi Zhanga,b, Shuigeng Zhoua,b,, Zhen Shena,b, Jihong Guanc a

Department of Computer Science and Engineering, Fudan University, Shanghai 200433, China Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China c Department of Computer Science and Technology, Tongji University, 4800 Cao’an Road, Shanghai 201804, China b

Received 3 May 2007; received in revised form 19 June 2007 Available online 25 July 2007

Abstract We propose a growing model which interpolates between one-dimensional regular lattice and small-world networks. The model undergoes an interesting phase transition from large to small worlds. We investigate the structural properties by both theoretical predictions and numerical simulations. Our growing model is a complementarity for the important static Watts–Strogatz network model. r 2007 Elsevier B.V. All rights reserved. PACS: 89.75.Hc; 89.75.k; 89.75.Fb; 05.10.a Keywords: Small-world model; Phase transition; Master equation method

1. Introduction Many real-life systems display both a high degree of local clustering and the small-world effect [1–6]. Local clustering characterizes the tendency of groups of nodes to be all connected to each other, while the smallworld effect describes the property that any two nodes in the systems can be connected by relatively short paths. Networks with these two characteristics are called small-world networks. In the past few years, a number of models have been proposed to describe real-life small-world networks. The first and the most widely studied model is the simple and attractive small-world network model of Watts and Strogatz (WS model) [7], which interpolates between a regular ring lattice and a completely random network. The WS model plays an important role in network science, and has triggered a sharp interest in the studies of small-world networks. Many authors have focused their attention on the analytical computation for different properties of the WS model [8–11], as well as the variations of this seminal model [12–17]. The WS model and its above-mentioned variations are probably reasonable illustrations of how some particular small-world networks are shaped. However, both the WS model and its extensions are static (i.e., the network size is fixed); this does not agree with the growth property of many real-life systems [18]. Corresponding author. Department of Computer Science and Engineering, Fudan University, Shanghai 200433, China.

E-mail addresses: [email protected] (Z. Zhang), [email protected] (S. Zhou), [email protected] (Z. Shen), [email protected] (J. Guan). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.07.024

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Moreover, the small-world effect is much more general than has hitherto been appreciated [1]. Therefore, researchers have begun to explore other mechanisms producing small-world networks. Klemm and Eguı´ luz proposed a model for growing networks based on a finite memory of the nodes [19]. Go´mez-Garden˜es and Moreno introduced a modified Baraba´si–Albert model [20]. Both of the two models can account for high clustering but fail to obtain low average path length (APL). In Refs. [21,22], spatial-growth mechanisms were described to yield to small-world effect. Recently, Ozik, Hunt and Ott have introduced a simple evolution model [23] (OHO model) of growing small-world networks with geographical attachment preference [24]. The OHO model begins with an initial state of m þ 1 (m is even) all-to-all connected nodes on the circumference of a circle. At each increment time the network grows by adding a new node, which is placed in a randomly chosen internode interval along the circle circumference (all intervals have the same probability of being chosen) and makes m links to its m nearest neighbors. Note that nearest here refers to the distance measured in number of intervals along the circumference of the circle. A deterministic version of a special case of the OHO model was presented in Ref. [25], and further expanded in Ref. [26]. It is well known that regular ring lattices are the initial constructions of the WS network. Although lacking the small-world property, regular ring lattices are models of great potential application in designing the topology for computer network, distributed memory multiprocessor systems [27] and so on. It is undeniable that both regular ring lattices and the OHO model provide valuable insights into some existing real-world systems. In this paper, we propose a scenario for constructing growing small-world network model, governed by a tunable parameter q, which can put regular ring lattices and the OHO model into a more general perspective; in other words, it can unify these two specific models to the same framework. By tuning parameter q, the model undergoes a phase transition from large to small worlds as the WS model. We study analytically and numerically the structural characteristics of the model including the degree distribution, the clustering coefficient and the APL, all of which depend on the parameter q.

2. The model In this section, we introduce a growing model which describes networks from regular to small world. The model is constructed in the following way (see Fig. 1): (i) Initial condition: We start from an initial state (t ¼ 2) of three nodes distributed on a ring, all of which form a triangle. (ii) Growth: At each increment of time, a new node is added, which is placed in a randomly chosen internode interval along the ring. Then we perform the following two operations. (iii) Addition of edges: The new node is connected to its two nearest nodes (one on either side) previously existing. Nearest, in this case, refers to the number of intervals along the ring. (iv) Removal of an edge: With probability q, we remove the edge linking the two nearest neighbors of the new node. Note that a similar idea has been used previously in a class of hierarchical lattices [28,29], which have received much recent attention [30–33]. The growing processes are repeated until the network reaches the desired size. There are two limiting cases of the present model. When q ¼ 1, the network is reduced to the onedimensional ring lattice. For q ¼ 0, no edge is deleted, the model coincides with a special case m ¼ 2 of the OHO model for small-world networks [23]. Thus, varying q in the interval (0,1) allows one to study the crossover between the one-dimensional regular lattice and the small-world network. By construction, at every step, the number of nodes increases by one, while the average number of edges added is 2  q. Then we can easily see that, at time t, the network consists of t þ 1 nodes and average ð2  qÞt þ 2q  1 edges. Thus, when t is large, the average node degree at time t is approximately equal to a constant value 4  2q, which shows that our network is sparse like many real-life networks.

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a

767

b 0

0

3

1

2

1

2 t=3

t=2

c

d 4

0

0 4

3

1

2

t=4

3

5

1

2

t=5

Fig. 1. Scheme of the growing network for a particular case, showing the first four time steps of the evolving process. The dashed line represents the edge removed. (a) t ¼ 2, (b) t ¼ 3, (c) t ¼ 4 and (d) t ¼ 5.

3. Topological properties We focus on the behavior of the topological characteristics in terms of the degree distribution, the clustering coefficient and the APL, as a function of the parameter q. 3.1. Degree distribution The degree distribution is one of the most important statistical characteristics of a network. For q ¼ 1, all nodes have the same number of connections 2, the network exhibits a completely homogeneous degree distribution. Next we focus the case 0pqo1. In order to conveniently describe the computation of the network characteristics, we label nodes by their birth times, s ¼ 0; 1; 2; . . . ; t, and use pðk; s; tÞ to denote the probability that at time t a node created at time s has a degree k. At time t, there are t þ 1 internode intervals along the ring and each node has two intervals (one on either side). The master equation [34,35] governing the evolution of the degree distribution of an individual node has the form   2ð1  qÞ 2ð1  qÞ pðk; s; t þ 1Þ ¼ pðk  1; s; tÞ þ 1  pðk; s; tÞ (1) tþ1 tþ1 with the initial condition pðk; s ¼ 0; 1; 2; t ¼ 2Þ ¼ dk;2 and the boundary one pðk; t; tÞ ¼ dk;2 . This accounts for two possibilities for a node: first, with probability 2ð1  qÞ=ðt þ 1Þ, it may get an extra edge from the new node while its existing edges remain undeleted, and thus increase its own degree by one; and, second, with the complementary probability 1  2ð1  qÞ=ðt þ 1Þ, the nodes may remain in the former state with the former degree. It should be noted that Eq. (1) and all the following ones are exact for all tX2.

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The total degree distribution of the entire network can be obtained as t 1 X pðk; s; tÞ. (2) t þ 1 s¼0 P Using this and applying ts¼0 to both sides of Eq. (1), we get the following master equation for the degree distribution:

Pðk; tÞ ¼

ðt þ 2ÞPðk; t þ 1Þ  ðt þ 1ÞPðk; tÞ ¼ 2ð1  qÞPðk  1; tÞ  2ð1  qÞPðk; tÞ þ dk;2 .

ð3Þ

The corresponding stationary equation, i.e. at t!1, takes the form ð3  2qÞPðkÞ  ð2  2qÞPðk  1Þ ¼ dk;2 .

(4)

Eq. (4) implies that PðkÞ is the solution of the recursive equation 8 > < 2  2qPðk  1Þ; k42; PðkÞ ¼ 3  2q > : 1=ð3  2qÞ; k ¼ 2;

(5)

giving PðkÞ ¼

  1 2  2q k2 3  2q 3  2q

(6)

ðkX2Þ

which decays exponentially with k. For q ¼ 0, Eq. (6) recovers the result previously obtained in Ref. [23]. Thus the resulting network is an exponential network. Note that most small-world networks including the WS model belong to this class [11,23,25,26]. In Fig. 2, we report the simulation results of the degree distribution for several values of q. From Fig. 2, we can see that the degree spectrum of the networks is continuous and the degree distribution decays exponentially for large degree values, in agreement with the analytical results and supporting a relatively homogeneous topology similar to most small-world networks [11,23,25,26]. 3.2. Clustering coefficient Most real-life networks show a cluster structure which can be quantified by the clustering coefficient [1–5]. The clustering coefficient of a node gives the relation of connections of the neighborhood nodes connected to it. By definition, clustering coefficient C i of a node i is the ratio of the total number ei of existing edges between 100 q=0.1 q=0.5 q=0.9

10-1

P (k)

10-2 10-3 10-4 10-5 0

5

10

15

20

25

k Fig. 2. Semilogarithmic graph of degree distribution of the networks with order N ¼ 105 . The solid lines are the analytic calculation values given by Eq. (6).

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all its ki nearest neighbors and the number ki ðki  1Þ=2 of all possible edges between them, i.e., C i ¼ 2ei =½ki ðki  1Þ. The clustering coefficient C of the whole network is the average of all individual C 0i s. For the case of q ¼ 1, the network is a one-dimensional chain, the clustering coefficient of an arbitrary node and their average value are both zero. For the case of q ¼ 0, using the connection rules, it is straightforward to calculate exactly the clustering coefficient of an arbitrary node and the average value for the network. When a node i enters the network, ki and ei are 2 and 1, respectively. After that, if the degree ki increases by one, then its new neighbor must connect one of its existing neighbors, i.e. ei increases by one at the same time. Therefore, ei is equal to ki  1 for all nodes at all time steps. So there exists a one-to-one correspondence between the degree of a node and its clustering. For a node v with degree k, the exact expression for its clustering coefficient is 2=k. This expression for the local clustering shows the same inverse proportionality with the degree as those observed in a variety of real-life networks [36]. In this limiting case, the clustering coefficient C of the whole network is given by C¼2

1 X 1 3 PðkÞ ¼ ln 3  1  0:6479. k 2 k¼2

(7)

So in the limit of large t the clustering coefficient is very high. In the range 0oqo1, it is difficult to derive an analytical expression for the clustering coefficient either for an arbitrary node or for the average of them. In order to obtain the result of the clustering coefficient C of the whole network, we have performed extensive numerical simulations for the full range of q between 0 and 1. Simulations were performed for system sizes 105 , averaging over 20 network samples for each value of q. In Fig. 3, we plot the clustering coefficient C as a function of q. It is obvious that C decreases continuously with increasing q. As q increases from 0 to 1, C drops almost linearly from 0.6479 to 0. Note that although the clustering coefficient C changes linearly for all q, we will show below that, in the large limit of q, the APL changes exponentially as q. This is little different from the phenomenon observed in the WS model where C remains practically unchanged in the process of the network transition to a small world. 3.3. Average path length Certainly, the most important property of a small-world network is a logarithmic APL (with the number of nodes). Here, APL means the minimum number of edges connecting a pair of nodes, averaged over all pairs of nodes. It has obvious implications for the dynamics of processes taking place on networks. Therefore, its study has attracted much attention. For the case of q ¼ 1, the network is completely symmetric; the APL of the network is equal to the average distance between one node and all other ones. Thus, we can easily obtain the APL LðNÞ of the network with

0.7 0.6 0.5

C

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

q Fig. 3. The clustering coefficient C of the whole network as a function of q.

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size N as 8 N þ1 > > > < 4 ; LðNÞ ¼ 2 > N > > : 4ðN  1Þ;

N is odd; (8) N is even

which increases linearly with network size N. For the case of q ¼ 0, the network grows stochastically. Generally speaking, for a randomly growing network, the analytical calculation for APL is difficult. Below, we will give an upper bound for the APL of this particular case, which shows that the APL increases at most logarithmically with network size. If dði; jÞ denotes the distance between nodes i and j, we introduce the total distance of the network with size N as sðNÞ: X sðNÞ ¼ dði; jÞ, (9) 0piojpN1

and we denote the APL by LðNÞ, defined as LðNÞ ¼

2sðNÞ . NðN  1Þ

(10)

In the limiting case q ¼ 0, the distances between existing node pairs will not be affected by the addition of new nodes. Then we have the following equation: sðN þ 1Þ ¼ sðNÞ þ

N 1 X

dði; NÞ.

(11)

i¼0

Assume that the node N is added and connected to two nodes w1 ; w2 linked by edge E, then Eq. (11) can be rewritten as sðN þ 1Þ ¼ sðNÞ þ

N1 X

½Dði; wÞ þ 1

i¼0

¼ sðNÞ þ N þ

N 1 X

Dði; wÞ,

ð12Þ

i¼0

where Dði; wÞ ¼ minfdði; w1 Þ; dði; w2 Þg. Constricting the edge E continuously into a single vertex w (here we assume that w  w1 ), we have Dði; wÞ ¼ dði; wÞ. Since dðw1 ; wÞ ¼ dðw2 ; wÞ ¼ 0, Eq. (12) can be rewritten as X sðN þ 1Þ ¼ sðNÞ þ N þ dði; wÞ, (13) i2G

P where G ¼ f0; 1; 2; . . . ; N  1g  fw1 ; w2 g is a node set with cardinality N  2. The sum i2G dði; wÞ can be considered as the total distance from one node w to all the other nodes in the network with size N  1, which can be roughly evaluated by mean-field approximation in terms of LðN  1Þ as [37–39] X dði; wÞ  ðN  2ÞLðN  1Þ. (14) i2G

Since LðNÞ increases monotonously with N, it is clear that ðN  2ÞLðN  1Þ ¼

2sðN  1Þ 2sðNÞ o . N 1 N

(15)

Combining Eqs. (13)–(15), one can obtain the inequation sðN þ 1ÞosðNÞ þ N þ

2sðNÞ . N

(16)

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3000

Average path length

2500 2000 1500 1000 500 0 0

0.2

0.4

0.6

0.8

1

q Fig. 4. Average path length L versus parameter q.

Considering Eq. (16) as an equation and not an inequality, we can provide an upper for the variation of sðNÞ as dsðNÞ 2sðNÞ ¼Nþ , dN N which leads to

(17)

sðNÞ ¼ N 2 ðln N þ aÞ,

(18) 2

where a is a constant. As sðNÞN ln N, we have LðNÞ ln N. Note that as we have deduced Eq. (18) from an inequality, then LðNÞ increases at most as ln N with N. Therefore, we have proved that in the special case of q ¼ 0, there is a slow growth of APL with network size N. For 0oqo1, in order to obtain the variation of the APL with the parameter q, we have performed extensive numerical simulations for different q between 0 and 1. Simulations were performed for system sizes 104 , averaging over 20 network samples for each value of q. In Fig. 4, we plot the APL L as a function of q. We observe that, when lessening q from 1 to 0, APL L drops drastically from a very high value to a small one, which predicts that a phase transition from large world to small world occurs. This behavior is similar to that in the WS model. Why is the APL L low for small q? The explanation is as follows. The older nodes that had once been nearest neighbors along the ring are pushed apart as new nodes are positioned in the interval between them. From Fig. 1 we can see that, when new nodes enter into the networks, the original nodes are not near but, rather, have many newer nodes inserted between them. When q is small, the network growth creates enough ‘‘shortcuts’’ (i.e., long-range edges) attached to old nodes, which join remote nodes along the ring to one another as in the WS model [7]. These shortcuts drastically reduce the APL, leading to a small-world behavior. 4. Conclusions In summary, we have proposed a one-parameter model of growing small-world networks. The presented model interpolates between one-dimensional regular ring and the OHO small-world network, which allow us to explore the crossover between the two limiting cases. We have obtained both analytically and numerically the solution for relevant parameters of the network and observed that our model exhibits the classical phenomenon as that in the WS model. In addition to the property of growth, our model has another desirable character that no nodes ever become disconnected from the rest of the network, whereas this can occur in the WS model. We believe that our model could provide a useful tool to investigate the influence of the clustering coefficient or APL in different dynamics processes taking place on networks. In addition, using the idea presented here, one can also construct models interpolating between homogeneous and heterogeneous networks [40].

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Acknowledgments This research was supported by the National Basic Research Program of China under grant No. 2007CB310806, the National Natural Science Foundation of China under Grant nos. 60496327, 60573183 and 90612007, the Postdoctoral Science Foundation of China under Grant no. 20060400162, the Program for New Century Excellent Talents in University of China (NCET-06-0376) and the Huawei Foundation of Science and Technology (YJCB2007031IN). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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