Complex Network Modeling with Constant Capacity Restriction Based on BA Model Zhang Xiaosong 1, Chen Ting 1, Chen Ruidong 1, Li Hua 2 1
School of Computer Science & Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu, China, 610054
[email protected] 2
Unit 78155 of PLA
Abstract—It is critical to obtain a fine description of the topology of a complex network when modeling its behavior, as the network’s functionality heavily relies on its structure. Constant Capacity Restricted model evolved from BA model (CCRBA) for complex network proposed in this paper is the first attempt to date to take the node’s capacity distribution into modeling. At first, this paper provides the building process of our model then we make a quantitative comparison of the properties of CCRBA with those of BA model, including the degree distribution, clustering coefficient and the average path length. Since majority of networks in practice are capacity–restricted, our model provides a better description of real-life complex networks. Keywords- capacity restriction; complex network; constant distribution; BA model
I.
INTRODUCTION
Complex network topology research has become a hotspot since the end of the 20th century. Many systems in practice or theoretical such as Internet, WWW, ecosystem, social relationship, and food chain etc represent a power-law degree distribution [1-4]. BA model proposed by Barabási and Albert is the first attempt to model the power-law distribution. Although various papers about the improvement of BA model had been proposed these years, they ignored the influence to the network’s topology made by the node’s capacity. In fact it affects node’s degree and the network’s topology obviously. Since majority of networks in practice are capacity-restricted, none of them can simulate real-life complex networks perfectly. To address this drawback, this paper focuses on the influence of the node’s capacity to the complex network topology. We propose a constant capacity restricted model evolved from BA model (CCRBA) for complex network. At first, this paper provides the building process of our model then we make a quantitative comparison of the properties of CCRBA with those of BA model, including the degree distribution, clustering coefficient and the average path length.
3 we present our model CCRBA and quantitative comparison with BA model. The paper finishes with some conclusions and suggestions for future work in Section 4. II.
RELATED WORKS
Since BA model had been proposed in 1990s, various papers about the improvement of BA model sprung up. Based on our research of mass materials, we try to classify all these works into two categories. One category is made up of works which attempt to explain power-law phenomenon in different aspects named Mimetic-BA model. The others are aim to model networks in practice which include power-law and other properties named Compound-BA model. This classification is not strict, thus these two categories may overlap to some extent. A. Mimetic-BA model ZHANG Zhong-zhi and RONG Li-li [5] pointed out that the process of building BA model needed global degree information which was difficult to obtain in many cases. So they introduced a novel method which did not need global degree information to build a BA-like model. For the same reason, CHEN Yu et al. presented a model using only local nodes’ information [6]. Some factors can affect network’s topology obviously, including competition among nodes in the network [7,8], nodes’ attraction [9] and links’ rewire [6]. Some learners took the process of node’s deletion into modeling [6,10,11]. Guo Chonghui and Zhang Liang improved BA model by replacing the preferential attachment of BA model with the pagerank algorithm [12]. S. Bar et al. modified BA model by an incremental super-linear preferential attachment instead of the linear one of BA model [13]. We find that all of mimetic-BA models have one common property: the process of building model is clear but these models may not adapt to explain networks’ phenomena in practice.
The paper is organized as follows. Section 2 provides a brief overview of recent improvements of BA model. In section This work was supported in part by Science and Technology Commission of Shanghai Municipality 09511501600.
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di < min (c j )
B. Compound-BA model Based on BA model, Bar Sagy et al. proposed a directed Internet topology model by introducing the directed property [14]. Many complex networks including Internet, WWW, social relationship network etc have small-world property as well as the power-law distribution. WANG Jian-wei et al. [15] and XU Xin-Ping et al. [16] proposed different methods to build models which satisfied both small-world and power-law properties. Several works focused on many other phenomena which BA model cannot explain, including rich-club, special power-law exponent, triangle coefficient, quadrangle coefficient and so on [17-20]. Guimarães Jr. Paulo R. et al. discovered that the initial network topology intensely affected the final network topology in a small scale network [21]. Compound-BA models also have a common property: they may be competent for explaining some of the real networks’ phenomena but the process of modeling is not easy to comprehend. III.
(1)
1≤ j ≤ m0
Here c j is the capability of node j and d i is the degree of node i . The initial capacity of the network ( tc0 ) is defined as (2), so tc0 > 0 and the initial network can accept new nodes. m0
tc0 = ¦ (ci − di )
In each step, a new node is added with capacity ci and initial degree ei to the existing network. ei is dependent of tci , ci and m . We give its explicit definition in (3).
We believe CCRBA is the first model to date which takes node’s capacity distribution into modeling thus CCRBA obtains a better description of the real-life networks than BA model and the process of modeling is easy to comprehend. So CCRBA can be cast into the intersection of mimetic-BA model and compound-BA model. Moreover, hybrid model of CCRBA with models aforementioned will obtain a better description of the real-life networks and this is our future work. B. Process of Building CCRBA As BA model, CCRBA starts with a small number ( m0 ) of a random network. In every step, a new node is added to the existing network. But the preferential attachment of CCRBA is distinct from that of BA model. The underlying is that the total capacity ( tci ) is limited and network cannot grow anymore if tci = 0 . Therefore we propose an optimal approach to keep network growing permanently:
Start with a small number ( m0 ) of a random network. Degree of each node should satisfy (1).
, ci < tci
min(m, ci ) ° ei = ® min(m, ci − 1) ° min(m, tc ) i ¯
CCRBA MODEL
A. Overview of CCRBA So far, BA model and its’ extended model ignored the influence to the network’s topology made by the node’s capacity. So these models unrestricted the node’s capacity by default. However, in fact the node’s capacity affects node’s degree and the network topology obviously. Majority of real systems are capacity–restricted, for example, the node’s capacity of phone-call network is restricted by the throughput of the telephone switches; the node’s capacity of Internet is restricted by the speed of the hardware; the node’s capacity of WWW is restricted by the design of the software; the node’s capacity of the movie actor network is restricted by their social ability. The node’s capability may represent different probability distribution and it relies on many factors of the specific complex network including the structure, function, design etc.
(2)
i =1
, ci = tci
(3)
, ci > tci
Since the nodes whose di = ci cannot link to new nodes anymore, the probability of a new link connects to node which is existed is present in (4).
(4)
0 °° dj p( j ) = ® ° ¦ dk °¯ k∈V , dk < ck
,d j = cj ,d j < cj
After a new node is added to the network, the total capacity of the network should be modified as (5). tci = tci −1 + ci − 2ei (5) From (1), (2), (3) and (5) we know that tci > 0 and the CCRBA network is able to grow permanently. C. Degree distribution of CCRBA Unfortunately, the degree distribution of CCRBA does not always represent power-law property and in fact the degree distribution mostly relies on the node’s capacity distribution. So networks with different capacity distributions may vary in different degree distributions. Although the degree distributions of CCRBA vary a lot, we find two common properties of those degree distributions. First, the number of low degree nodes is less than that of BA model. Second, the number of nodes whose d j = c j is more than that of BA model. And with the expanding of CCRBA, these two phenomena become more obvious. These two phenomena can be briefly explained as follows:
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And ¦ d ≡ 2 L , So i
i
¦d
So
i∈BA
i
≥
¦
¦d
i
≥
i∈BA
¦
dj .
j∈CCRBA
d j , then p(i ) ≤ p ( j ) where
j∈CCRBA d j < c j
i ∈ BA, j ∈ CCRBA and di = d j < c j .
CCRBAi means CCRBA with a constant distribution i . Fig.1 and Table 1 demonstrate the degree distributions of CCRBA model and the comparison of BA model. The number of nodes whose d = 3 of CCRBA15 (3409) is less than that of the CCRBA50 (3877) and both of them are less than that of BA model (4037). Besides, the number of nodes whose d = 50 of CCRBA50 (52) is larger than that of the CCRBA15 (0) and BA model (2) as well as the number of nodes whose d = 15 of CCRBA15 (934) is larger than that of the CCRBA50 (78) and BA model (53). By the way, this experiment validates the two degree distribution properties of CCRBA. These experiments reflect two conclusions. First, the power-law exponent of the CCRBA model is smaller than that of BA model. The other is that in terms of the CCRBA model, the smaller the capacity restriction is, the smaller the power-law exponent.
D. Clustering coefficient of CCRBA We find that the clustering coefficient of CCRBA is smaller than that of BA model while they have the same size. Generally speaking, the smaller the nodes’ capacity is, the smaller the clustering coefficient. From table 3, we learn that the clustering coefficient of CCRBA is smaller than that of BA model while their sizes are equal to 10000. And the smaller the capacity restriction is, the smaller the clustering coefficient of CCRBA while their sizes are equal to 10000. In fact, from Fig.2, no matter how large of the network size, the conclusion always stands. There are two specific clustering coefficient properties of CCRBA. First, Fig.2 shows that the trend of CCRBA decreases along with the logarithm of the network scale as BA model. So we dare to say that the constant capacity restriction does not affect the decreasing trend of the clustering coefficient along with the evolution of the network. Second, the difference of the clustering coefficient between BA model and CCRBA is not obvious when the network scale is small for reason that the total number of edges is also small, and only a few of nodes whose di = ci , so the topology of CCRBA is close to that of BA model. 0
10
BA CCRBA 50 CCRBA 15 Clustering Coefficient
Proof ∀ a BA network and a ∀ CCRBA network, while N BA = N CCRBA . Here N means the total node number of the network. We ignore the influence made by the initial network topology, so LBA ≥ LCCRBA . Here L means the total edge number of the network.
-1
10
-2
10
BA CCRBA 50 CCRBA 15
3
10
-3
10
2
Node Number
10
3
10 Node Number
4
10
2
10
Figure 2. Clustering coefficient of CCRBA 1
10
E. Average path length of CCRBA
0
10
10
1
2
10 Degree
Figure 1. Degree distribution of CCRBA and BA
TABLE I.
Total Node Number ( N ) Total Edge Number ( L ) Node Number ( d = 3 ) Node Number, ( d = 15 ) Node Number, ( d = 50 ) Average Path Length ( l ) Clustering Coefficient ( C )
NETWORK STATISTICS CCRBA15 10000 30038 3409 934 0 5.1 0.001468
CCRBA 50 10000 30032 3877 78 52 4.6 0.002407
BA 10000 30037 4037 53 2 4.2 0.00637
Average path length ( l ) is inevitably affected by nodes’ capacity distribution of CCRBA. Therefore, the average path length of CCRBA may not present the properties of BA model including the short length and increasing along with the logarithm of the network scale. So, CCRBA may not be a scale-free network and it depends on the nodes’ capacity distribution. There is a common property of the average path length of the networks with different capacity distributions: longer than that of BA model while they have the same size. From table 3, the average path length of BA model is the shortest; the average path length of CCRBA15 is the longest and that of CCRBA50 is in the middle when their sizes are equal to 10000. Fig.3 shows that, this conclusion is always exact with any network scale. Look at Fig.3, the average path length of BA model and CCRBA increase along with the logarithm of the network scale. Therefore, the constant capacity restriction does not influence the increasing trend of the average path length along with the evolution of the network. Then we discover that when
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the network size below a threshold (such as: 2000), the gaps of the average path length of different models with the same size vary a lot along with the changes of the network scale, but once the network size beyond the threshold, the gaps turn to stable. When the network size is smaller than 1000, the diamonds of BA model and that of CCRBA50 are almost overlapped. The reason for this property is that the topology of CCRBA is close to that of BA model when there are only a few nodes whose di = ci .
[2] [3] [4] [5] [6] [7]
BA 10
0.7
C C R B A 50
[8]
Average Path Length
C C R B A 15
gap
10
[9]
gap
0.6
gap
[10] 10
0.5
gap
[11] 10
0.4
10
2
3
10 N ode N um ber
10
4
Figure 3. Average path length of CCRBA [12]
IV.
CONCLUSION AND FUTURE WORK
In conclusion, we propose a capacity restricted model evolved from BA model (CCRBA) for complex network. Our work focuses on the influence of the node’s capacity to the complex network topology. We propose and quantitatively analyze the CCRBA model and compare it with BA model. In a word, CCRBA model is more suitable to simulate real-life network than BA model. Moreover, hybrid model of CCRBA with models mentioned in section 2 will obtain a better description of the networks in practice. This paper is an original attempt to build a model with taking the constant capacity restriction into consideration, so it is too simple to explain all the complex phenomena in real-life network. There are many works to improve our CCRBA model and we conclude them into three steps. The first step of our future work is researching the analytical results and the mathematical proof of CCRBA model. In addition, we would like to consider derived-CCRBA models with different capacity restriction distributions besides the constant distribution since many networks in real represent different capacity restriction distributions. After that, the issue that combines CCRBA and other models aforementioned should be solved, because the other models can refine CCRBA and yield better models.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
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