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Multifractality of weighted complex networks
Chinese Journal of Physics 54 (2016) 416e423
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Chinese Journal of Physics journal homepage: http://www.journals.elsevier.com/ chinese-journal-of-physics/
Multifractality of weighted complex networks Daijun Wei a, b, Xiaowu Chen c, Yong Deng a, * a b c
School of Computer and Information Science, Southwest University, Chongqing 400715, China School of Science, Hubei University for Nationalities, Enshi 445000, China School of Computer Science, BeiHang University, Beijing 100191, China
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 25 May 2016
The multifractality property of many complex networks have been investigated. However, existing researches mainly revealed the multifractality of unweighted networks. In this paper, the algorithm for the generalized dimension of weighted networks is proposed. The box-covering algorithm for fractal dimension of weighted networks (BCANw) is modified in the proposed method. The new method is applied to calculate the generalized dimensions of some real networks. The numerical results indicate that the proposed method is efficient for analysis multifractal property of weighted networks. In addition, the proposed method can also involve multifractal property of unweighted networks. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Keywords: Multifractality Box-covering algorithm Weighted networks
1. Introduction In various fields of science, complex networks have attracted growing interest since the structure and physical of many real complex systems can be described by them [1e10]. Some fundamental properties of complex networks, including smallworld, free-scale and fractal were found [11,12]. Inspired by the pioneering work by Song et al. [12], the fractal and self-similar properties of complex networks have been extensively studied and the fractal dimension has been used for fractal analysis of complex networks [13e19]. The box-covering algorithm, which named as the classical box-covering algorithm, was first applied to calculate the fractal dimension of many real networks [20,21]. Subsequently, the classical box-covering algorithm for complex networks was modified by many researchers [22e25]. However, the fractal property of network can not be characterized by unique fractal dimension and the network takes a multifractal structure. In this case, multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns [26,27]. Multifractal analysis has been applied successfully in many different fields such as financial modeling [28], medical science [29], time series analysis [30]. For complex networks, Wang and Yu et al. [31,32] introduced an improved box-covering algorithm for multifractal analysis of complex networks. Moreover, a family of fractal networks was studied by Li et al. [33]. However, the existing works mainly focused on handling the multifractal property of unweighted networks. The main reason is that existing algorithm of complex networks is not suitable for weighted complex networks. In 2013, we given an improved box-covering algorithm, which is named as BCANw, for calculating fractal dimension of weighted networks [34]. By using BCANw, the fractal property of weighted networks can be well described. In this paper, an algorithm for the generalized dimension of weighted networks is proposed. The multifractal property of some real weighted networks is revealed.
* Corresponding author. School of Computer and Information Science, Southwest University, Chongqing 400715, China. E-mail addresses: [email protected], [email protected] (Y. Deng). http://dx.doi.org/10.1016/j.cjph.2016.05.004 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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The paper is organized as follows. BCANw and multifractal method are introduced in Section 2. The proposed model of multifractal analysis for weighted complex networks is described in Section 3. In Section 4, the generalized dimensions of some real weighted networks are calculated by the proposed method. Some conclusions are presented in Section 5. 2. Preliminaries 2.1. The definition of shortest path of complex networks The shortest path of complex networks is the most important factor for calculating fractal dimension of complex networks. For given an unweighted network G ¼ (N,V), N ¼ (1,2,/,n) is a set of nodes and V ¼ (1,2,/,m) is a set of edges. X ¼ ðxij Þnn represent adjacency matrix of G, where xij ¼ 1 represent the connection between node i and node j, and 0 otherwise. The shortest path between node i and node j in unweighted networks is defined as follows, Definition 1. (shortest path in unweighted networks). Denoting dij as the shortest path of between node i and node j, which satisfies
o n dij ¼ min xih1 þ / þ xhk j
(1)
where i,h1,/,hk,j are IDs of nodes. However, definition of the shortest path of weighted network is more complicated. For given a weighted network 0 G ¼ (N,V,W), W ¼ (1,2,/,k) is a set of edge-weight and denoted by wij, where wij is any real number. The network G' is unweighted when wij(i,j ¼ 1,2,/,n) is always equal one. In many weighted networks, value of edge-weight between node i and j is not the distance between them. There are two definitions of the shortest path for weighted networks. One case is that higher the weights, larger the distance of the shortest path. In the kinds of networks, the distance between two connecting nodes is proportional to value of edge-weights. For instance, in the real city network, the edge-weight and distance are both represented by Euclidean distance. The other one is the opposite and the distance between two connecting nodes is inversely proportional to value of edge-weights. For example, in the Scientific collaboration network, the edge-weights correspond to the times of cooperation [35]. The more times of cooperations are, the value of edge-weights more bigger is, but the less distance is. Recently, in Ref. [34], the two cases are uniformly considered by only one parameter. The shortest path of weighted networks was uniformly defined as follows [34], 0
Definition 2. (the shortest path in weighted networks). Denoting dij as the shortest path of between node i and node j, which satisfies any of the following condition,
o n d0ij ¼ min wuij1 þ wuj1 j2 þ / þ wujm1 jm þ wujm j
(2)
where jk(k ¼ 1,2,/,m) are IDs of nodes and u is a real number. In this definition, u > 0 mean that the higher weight is, the further distance is. u < 0 mean that the higher weight is, the less distance is. 2.2. Box-covering algorithm for fractal dimension in weighted networks In this section, a box-covering algorithm for weighted networks (BCANw) is briefly introduced [34]. For the box-covering algorithm of unweighted networks, every covering-box with box-size r is set of nodes where all distances dij between any two nodes i and j in the covering-box are smaller than r. The entire network must be covered by the minimum number of boxes Nb(r). The fractal dimension db is obtained as follows,
db ¼ lim
r/0
ln Nb ðrÞ ln r
(3)
However, weighted networks have different edge weights, which can be non-integers. Thus, values of d0ij may be nonintegers too. Expressly, it is also probable that the value of d0max is less than one. In this case, the minimum number of ij covering boxes is always one. Fractal property of weighted networks cannot be reversed by the classical box-covering algorithm of complex networks [21]. An improved box-covering algorithm for weighted networks (BCANw) is obtained in
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Ref. [34]. The core idea of BCANw is that box size length is given by accumulating the distance between two nodes connected directly. The method is summarized as three steps as follows, Step 1: Values of d0ij ; ði; j ¼ 1; 2; /; nÞ are calculated, where node i is connected directly node j. And then, d0ij is used to sort an array from the lowest value to the highest value. In the array, d0ij only appear one time if the values of two d0ij are equal. Suppose the order of d0ij is denoted by d01 < d02 < / < d0m. P 0 Step 2: Let the value of box size r is obtained from d01 , d01 þ d02 ; /, until di ði ¼ 1; 2; /; mÞ is more than the network i diameter. Step 3: For given a box size r, the minimum number of covering boxes Nb0 ðrÞ is obtained by the classical box-covering algorithm [21]. And then, the fractal dimension of weighted network, which denoted as dB(w) is obtained with a formula similar to the Equation (3) and given as follows,
dB ðwÞ ¼ lim
r/0
ln Nb0 ðrÞ ln r
(4)
2.3. Multifractal method Real word fractals may not be homogeneous. Multifractal property often reveals an inhomogeneous of a physical quantity on a fractal object [36,37]. Multifractal analysis is a useful way to describe fractal property of complex system. The fixed-size box counting algorithm is the most algorithm of multifractal analysis [26,27,31]. For a set E in a metric space, a probability measure 0 m 1 is given. A partition sum is considered as follows,
Zε ðqÞ ¼
X
½mðbÞq
(5)
mðbÞ
where q is a real number and m(b) is m(.) of different non-overlapping boxes b which cover E with a given size ε. The value of Zε(q) follows that Zε(q) 0 and Zε(0) ¼ 1. If Zε(q) has a power-law ε dependence: Zε ðqÞfεtðqÞ . The mass exponent function t(q) is obtained as follows,
tðqÞ ¼ lim
ε/0
ln Zε ðqÞ ln ε
(6)
where t(q) is a linear function of q. When the fluctuation of m(b) over boxes is sufficiently small, Zε(q) is satisfied as follows,
Zε ðqÞ εDq ðq1Þ
(7)
where the Dq is the generalized fractal dimension of the measure m. Therefore, from Equations (1) and (6), we have the linear relation, which is shown as follows,
Dq ¼
tðqÞ ; qs1 q1
(8)
P Z mðbÞln mðbÞ. for q ¼ 1, where D1 ¼ lim lnð1;εÞε and Zð1;εÞ ¼ ε/0 mðbÞs0 A view of life from the other side of multifractal, multifractal spectrum is defined as follows,
f ðaÞ ¼
ln Na ðεÞ ln ε
(9)
where Na(ε) is a number of box in [a,a þ da] and a is called as singularity exponent. It means that f(a) is fractal dimension of all boxes with a. Summating by value of probability, Equation (5) is changed as follows,
Zε ðqÞ ¼
X
NðPÞP q
(10)
where N(P) is number of box, which has probability P. According with P(ε) f εa, Equations (6) and (9), Equation (10) is changed as follows,
Zε ðqÞ ¼
X
εaqf ðaÞ ¼ εtðqÞ
Therefore, there is relationship between f(a) and t(q), which satisfies:
(11)
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f ðaÞ ¼ qaðqÞ tðqÞ
419
(12)
From Equations (8) and (12), the singularity spectrum f(a) contains exactly the same information as Dq. By Equations (6) and (8), the generalized fractal dimensions Dq are estimated through a linear regression of ½ln Zε ðqÞ=ðq 1Þ against lnε for q s 1 [38]. P The value of q is a weighting coefficient for ½mðbÞq. In this equation, subset with high value of m(b) play a major role in mðbÞ
summation processes if the value of q is considerably larger than one. Conversely, subset with low value of m(b) play a major role in summation processes if the value of q is considerably less than one. The generalized fractal Dq determine whether the network is multifractal from the shape of Dq. A monofractal system has the same scaling behavior at any point, Dq should be a constant independent of q [31]. Dq represent property of space of network and has not relationship with probability of uneven P distribution when q equal zero. The value of D0 is named as the Hausdorff dimension. NðPÞP is information entropy of network when q ¼ 1. Therefore, the value of D1 is named as information dimension. 3. Multifractal analysis of weighted networks 3.1. Algorithm analysis of multifractal of networks Most of real fractal networks are inhomogeneous. In scale-free networks, the scale-free property is represented by a power-law degree distribution pðkÞfkg , where k is the number of connections of a node [39]. Thus, the distribution of hubs node density is higher than others node. There is highly inhomogeneous for two subgraphs, which even covered same number of nodes. The inhomogeneous distribution of number of nodes often exhibits the multifractal property [31,33]. For multifractal analysis of unweighted networks, some algorithms are introduced [31]. However, the study is very few about multifractal property of weighted networks. In this section, an algorithm of multifractal of weighted networks is proposed based on BCANw. To our knowledge, the method is the first introduced to analyze the multifractal behavior of weighted network. In the proposed method, firstly, the edge-weights between connecting directly nodes is converted to the distance between them for two types networks. Secondly, the shortest distance between any nodes are calculated and these pairs are saved into matrix ANN, where N is the number of nodes of the network. According to BCANw, set the size of the box with accumulating the distance between two nodes connected directly. Finally, for a given box size r, the minimum number of boxes, which cover the entire network, is obtained by BCANw [34]. For the nonempty boxes b, in Equation (5), mðbÞ is given as follows,
mðbÞ ¼
Nb N
(13)
where Nb is the number of nodes covered by the box b and N is the number of nodes of network. The partition sum as Zr ðqÞ ¼ P mðbÞs0
½mðbÞq is calculated for each value of r. In order to obtained the generalized fractal dimension Dq, lineae regression is an
Fig. 1. The relationship between values of fractal dimension and values of u, jdB ðwÞðuÞ dB ðwÞðuÞj < 0:2 [34].
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Fig. 2. Distribution of edge-weights of the USAir97 network, the C. elegans network and the Scientific collaboration network. The vertical axis y and horizontal w w , respectively. axis x represent value of maxðwÞ and number of edge-weights with maxðwÞ
Table 1 Values of Dq and q for SW network with p ¼ 0.03 and.p ¼ 0.06 q
0
2
3
4
5
(p ¼ 0.03)Dq (p ¼ 0.06)Dq
1.864 2.066
2.304 2.656
2.333 2.707
2.339 2.726
2.338 2.733
essential step for the appropriate range of box size. The linear regression of lnZr ðqÞ=ðq 1Þ against ln(r/d) (q s 1) is considered. In Section 4, we calculate the generalized fractal dimension Dq and determine the multifractality property of complex networks from the shape of Dq. 3.2. Value of u in multifractal analysis of weighted networks For weighted networks, the relationship between distance of nodes and value of edge-weights is core factor. In Ref [40], the Scientific collaboration networks is given by Newman. In the Scientific collaboration networks, two scientists are considered connected if they have coauthored some papers together and edge-weight wij represents the strength of the collaboration between scientists i and j. The distance between author i and j is defined as the inverse of the weight of their
Fig. 3. The Dq curves of the SW network with p ¼ 0.03 and p ¼ 0.06.
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Fig. 4. Multifractal scaling analysis of the USAir97 network, the C. elegans network and the Scientific collaboration network.
collaborative tie. This definition mean that one pair of authors know one another twice as well as another pair, the distance them is half as great, i. e. u ¼ 1 in Equation (2) [40]. For fractal analysis of complex networks, the relationship between distance of nodes and value of edge-weights is very important too. For unweighted networks, we defined the distance of nodes i and j as ki þ kj, where ki is degree of node i [14]. In this case, fractal property of some real unweighted networks can be revealed too. Therefore, the definition of distance between nodes determines the course of our vision for complex networks. For weighted networks, weighted networks changed as unweighted networks when u ¼ 0 in Equation (2). u > 0 mean that the higher weight is, the further distance is. u < 0 mean
Table 2 Values of Dq and q for the USAir97 network [41], the C. elegans network [11] and the Scientific collaboration network [40]. q
0
2
3
4
5
USAir97 C. elegans Scientific
0.7231 1.989 0.5505
1.647 3.916 1.638
1.42 3.095 1.956
1.29 2.76 1.958
1.226 2.588 1.456
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Fig. 5. The Dq curves of the USAir97 network, the C. elegans network and the Scientific collaboration network.
that is opposite. In Ref [34], the relationship between the fractal property of weighted network and u value is considered. We found that the values of fractal dimensions are almost symmetric about values of u, about the vertical. Some real weighted networks such as the USAir97 network, which edge-weights are the number of seats available on the scheduled flights with million per year [41], the Caenorhabditis elegans network, which edge-weights are the number of synapses and gap junctions [11], and the Scientific collaboration network [40] are considered in Ref. [34]. These results is shown in Fig. 1 [34]. In Fig. 1, fractal dimensions of the USAir97 network, the C. elegans network and the Scientific collaboration network are all almost symmetric about values of u, about the vertical. The changing of edge-weights do not affect with fractal property. The symmetric property of fractal dimension is not affected by values of u multiplied to enlarge or decrease [34]. And that, the distributions of edge-weights of this three networks are different each other. All edge-weights of these real networks are divided maximum value of edge-weights and shown in Fig. 2. From Fig. 2, values of edge-weights of the USAir97 network and the Scientific collaboration network have a nice continuous. Value of edge-weights of the C. elegans network has a jump. A lot of edge-weights of the C. elegans network have the same values, which is same as the Scientific collaboration network. The distribution of edge-weights of the USAir97 network is approximate uniform distribution. From Equation (4), fractal dimension dB(w) is obtained by means of the least square fit between number of box and size of box. From Equations (6) and (8), the generalize fractal dimension Dq is obtained by means of the least square fit between number of nodes in the box and size of box. Therefore, the generalized fractal dimension is affected by edge-weights as same as fractal dimension of weighted networks.
4. The numerical simulations Our proposed method is also apply to unweighted networks. For example, a unweighted network such as Small-World network [11] is considered. A small-world network is given as processing: Firstly, we construct a regular ring lattice with n nodes and give a mean degree k, which is the number of neighbors to connect. Finally, shortcuts, which instead of rewiring links, are added with probability p. In our Small-World network, the number of nodes and the mean degree in a given row are 3000 and 4, respectively. The probability p of adding a shortcut is a small value (p ¼ 0.03 and p ¼ 0.06 ). By our proposed method, values of Dq and q are given in Table 1. The relationship between Dq and q is shown in Fig. 3. From Table 1 and Fig. 3, values of Dq have changed little after q ¼ 2, multifractal property of small-world network is not distinct. During the generation, when p increase, more edges are added. Therefore, the value of Dq of network with p ¼ 0.06 is more than network with p ¼ 0.03. These results are consistent with results of Ref [31]. For fractal analysis of weighted networks, the USAir97 network [41], the C. elegans network [11] and the Scientific collaboration network [40] were considered. In the section, multifractal property of these real networks are considered by our proposed method. In last section, we found that the fractal and multifractal of weights network is not affected by value of u. For three weighted networks, the bigger edge-weights is, the less distance is. Therefore, to simplify computation, value of u equal 1 in Equation (2). By our proposed method, lnZr ðqÞ=ðq 1Þ against ln(r/d) (q s 1) of these networks are shown in Fig. 4. From Fig. 4, the generalized fractal dimension Dq and the appropriate range of r2½rmin ; rmax by linear regression. And then, the value range of r is obtained by choosing appropriate points. Value of horizontal coordinates of the USAir97 network is [6, 2.5]. In the C. elegans network and the Scientific collaboration network, we adopt the points from 11th to 30th and from 5th to 20th, respectively. Values of Dq of these weighted networks are given in Table 2 and the relationship between Dq and q is shown in Fig. 5. From Table 2 and Fig. 5, the values of Dq of three real weighted networks have large different when values of q are different. These Dq curves indicate that these weighted networks are multifractal. However, value of Dq of the C. elegans network is
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bigger than Dq of the USAir97 network and the Scientific collaboration network with same q. It indicates the degree of inequality of ratio m(b) of the C. elegans network is bigger than the USAir97 network and the Scientific collaboration network. The value of Dq of the C. elegans network and the USAir97 network reach their peak values around q ¼ 2. However, the maximum value of Dq of the Scientific collaboration network is the interval [3,4]. It shows that the subset with high value of m(b) of the Scientific collaboration has more important role than other networks. 5. Conclusion Fractal property has been shown as one of the fundamental properties of complex networks. The box-covering algorithm is an efficient method to calculate fractal dimension of unweighted complex networks. Recently, the researches on complex networks have shown that some real networks exhibited the property of multifractal scaling. And that, BCANw is presented for analysing fractal property of weighted network [34]. However, the existing studies mainly dealt with the multifractal property of unweighted networks. To address this issue, a multifractal analysis method for weighted networks is proposed based on BCANw. It allows the computation of the generalized fractal dimensions Dq. The numerical examples of unweighted network such as small-world network and some real weighted networks show that the proposed approach not only well reveal mulitfractal property of weighted networks but it can also reveal mulitfractal property of unweighted networks. Acknowledgment The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61174022, 61325011 and 61364030), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20131102130002), R&D Program of China (2012BAH07B01), National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), the funding project of National Bureau of Statistical of China (Grant No. 2014LY128), the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No. BUAA-VR14KF-02). 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] [41]
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Chinese Journal of Physics journal homepage: http://www.journals.elsevier.com/ chinese-journal-of-physics/
Multifractality of weighted complex networks Daijun Wei a, b, Xiaowu Chen c, Yong Deng a, * a b c
School of Computer and Information Science, Southwest University, Chongqing 400715, China School of Science, Hubei University for Nationalities, Enshi 445000, China School of Computer Science, BeiHang University, Beijing 100191, China
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 25 May 2016
The multifractality property of many complex networks have been investigated. However, existing researches mainly revealed the multifractality of unweighted networks. In this paper, the algorithm for the generalized dimension of weighted networks is proposed. The box-covering algorithm for fractal dimension of weighted networks (BCANw) is modified in the proposed method. The new method is applied to calculate the generalized dimensions of some real networks. The numerical results indicate that the proposed method is efficient for analysis multifractal property of weighted networks. In addition, the proposed method can also involve multifractal property of unweighted networks. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Keywords: Multifractality Box-covering algorithm Weighted networks
1. Introduction In various fields of science, complex networks have attracted growing interest since the structure and physical of many real complex systems can be described by them [1e10]. Some fundamental properties of complex networks, including smallworld, free-scale and fractal were found [11,12]. Inspired by the pioneering work by Song et al. [12], the fractal and self-similar properties of complex networks have been extensively studied and the fractal dimension has been used for fractal analysis of complex networks [13e19]. The box-covering algorithm, which named as the classical box-covering algorithm, was first applied to calculate the fractal dimension of many real networks [20,21]. Subsequently, the classical box-covering algorithm for complex networks was modified by many researchers [22e25]. However, the fractal property of network can not be characterized by unique fractal dimension and the network takes a multifractal structure. In this case, multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns [26,27]. Multifractal analysis has been applied successfully in many different fields such as financial modeling [28], medical science [29], time series analysis [30]. For complex networks, Wang and Yu et al. [31,32] introduced an improved box-covering algorithm for multifractal analysis of complex networks. Moreover, a family of fractal networks was studied by Li et al. [33]. However, the existing works mainly focused on handling the multifractal property of unweighted networks. The main reason is that existing algorithm of complex networks is not suitable for weighted complex networks. In 2013, we given an improved box-covering algorithm, which is named as BCANw, for calculating fractal dimension of weighted networks [34]. By using BCANw, the fractal property of weighted networks can be well described. In this paper, an algorithm for the generalized dimension of weighted networks is proposed. The multifractal property of some real weighted networks is revealed.
* Corresponding author. School of Computer and Information Science, Southwest University, Chongqing 400715, China. E-mail addresses: [email protected], [email protected] (Y. Deng). http://dx.doi.org/10.1016/j.cjph.2016.05.004 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
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The paper is organized as follows. BCANw and multifractal method are introduced in Section 2. The proposed model of multifractal analysis for weighted complex networks is described in Section 3. In Section 4, the generalized dimensions of some real weighted networks are calculated by the proposed method. Some conclusions are presented in Section 5. 2. Preliminaries 2.1. The definition of shortest path of complex networks The shortest path of complex networks is the most important factor for calculating fractal dimension of complex networks. For given an unweighted network G ¼ (N,V), N ¼ (1,2,/,n) is a set of nodes and V ¼ (1,2,/,m) is a set of edges. X ¼ ðxij Þnn represent adjacency matrix of G, where xij ¼ 1 represent the connection between node i and node j, and 0 otherwise. The shortest path between node i and node j in unweighted networks is defined as follows, Definition 1. (shortest path in unweighted networks). Denoting dij as the shortest path of between node i and node j, which satisfies
o n dij ¼ min xih1 þ / þ xhk j
(1)
where i,h1,/,hk,j are IDs of nodes. However, definition of the shortest path of weighted network is more complicated. For given a weighted network 0 G ¼ (N,V,W), W ¼ (1,2,/,k) is a set of edge-weight and denoted by wij, where wij is any real number. The network G' is unweighted when wij(i,j ¼ 1,2,/,n) is always equal one. In many weighted networks, value of edge-weight between node i and j is not the distance between them. There are two definitions of the shortest path for weighted networks. One case is that higher the weights, larger the distance of the shortest path. In the kinds of networks, the distance between two connecting nodes is proportional to value of edge-weights. For instance, in the real city network, the edge-weight and distance are both represented by Euclidean distance. The other one is the opposite and the distance between two connecting nodes is inversely proportional to value of edge-weights. For example, in the Scientific collaboration network, the edge-weights correspond to the times of cooperation [35]. The more times of cooperations are, the value of edge-weights more bigger is, but the less distance is. Recently, in Ref. [34], the two cases are uniformly considered by only one parameter. The shortest path of weighted networks was uniformly defined as follows [34], 0
Definition 2. (the shortest path in weighted networks). Denoting dij as the shortest path of between node i and node j, which satisfies any of the following condition,
o n d0ij ¼ min wuij1 þ wuj1 j2 þ / þ wujm1 jm þ wujm j
(2)
where jk(k ¼ 1,2,/,m) are IDs of nodes and u is a real number. In this definition, u > 0 mean that the higher weight is, the further distance is. u < 0 mean that the higher weight is, the less distance is. 2.2. Box-covering algorithm for fractal dimension in weighted networks In this section, a box-covering algorithm for weighted networks (BCANw) is briefly introduced [34]. For the box-covering algorithm of unweighted networks, every covering-box with box-size r is set of nodes where all distances dij between any two nodes i and j in the covering-box are smaller than r. The entire network must be covered by the minimum number of boxes Nb(r). The fractal dimension db is obtained as follows,
db ¼ lim
r/0
ln Nb ðrÞ ln r
(3)
However, weighted networks have different edge weights, which can be non-integers. Thus, values of d0ij may be nonintegers too. Expressly, it is also probable that the value of d0max is less than one. In this case, the minimum number of ij covering boxes is always one. Fractal property of weighted networks cannot be reversed by the classical box-covering algorithm of complex networks [21]. An improved box-covering algorithm for weighted networks (BCANw) is obtained in
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Ref. [34]. The core idea of BCANw is that box size length is given by accumulating the distance between two nodes connected directly. The method is summarized as three steps as follows, Step 1: Values of d0ij ; ði; j ¼ 1; 2; /; nÞ are calculated, where node i is connected directly node j. And then, d0ij is used to sort an array from the lowest value to the highest value. In the array, d0ij only appear one time if the values of two d0ij are equal. Suppose the order of d0ij is denoted by d01 < d02 < / < d0m. P 0 Step 2: Let the value of box size r is obtained from d01 , d01 þ d02 ; /, until di ði ¼ 1; 2; /; mÞ is more than the network i diameter. Step 3: For given a box size r, the minimum number of covering boxes Nb0 ðrÞ is obtained by the classical box-covering algorithm [21]. And then, the fractal dimension of weighted network, which denoted as dB(w) is obtained with a formula similar to the Equation (3) and given as follows,
dB ðwÞ ¼ lim
r/0
ln Nb0 ðrÞ ln r
(4)
2.3. Multifractal method Real word fractals may not be homogeneous. Multifractal property often reveals an inhomogeneous of a physical quantity on a fractal object [36,37]. Multifractal analysis is a useful way to describe fractal property of complex system. The fixed-size box counting algorithm is the most algorithm of multifractal analysis [26,27,31]. For a set E in a metric space, a probability measure 0 m 1 is given. A partition sum is considered as follows,
Zε ðqÞ ¼
X
½mðbÞq
(5)
mðbÞ
where q is a real number and m(b) is m(.) of different non-overlapping boxes b which cover E with a given size ε. The value of Zε(q) follows that Zε(q) 0 and Zε(0) ¼ 1. If Zε(q) has a power-law ε dependence: Zε ðqÞfεtðqÞ . The mass exponent function t(q) is obtained as follows,
tðqÞ ¼ lim
ε/0
ln Zε ðqÞ ln ε
(6)
where t(q) is a linear function of q. When the fluctuation of m(b) over boxes is sufficiently small, Zε(q) is satisfied as follows,
Zε ðqÞ εDq ðq1Þ
(7)
where the Dq is the generalized fractal dimension of the measure m. Therefore, from Equations (1) and (6), we have the linear relation, which is shown as follows,
Dq ¼
tðqÞ ; qs1 q1
(8)
P Z mðbÞln mðbÞ. for q ¼ 1, where D1 ¼ lim lnð1;εÞε and Zð1;εÞ ¼ ε/0 mðbÞs0 A view of life from the other side of multifractal, multifractal spectrum is defined as follows,
f ðaÞ ¼
ln Na ðεÞ ln ε
(9)
where Na(ε) is a number of box in [a,a þ da] and a is called as singularity exponent. It means that f(a) is fractal dimension of all boxes with a. Summating by value of probability, Equation (5) is changed as follows,
Zε ðqÞ ¼
X
NðPÞP q
(10)
where N(P) is number of box, which has probability P. According with P(ε) f εa, Equations (6) and (9), Equation (10) is changed as follows,
Zε ðqÞ ¼
X
εaqf ðaÞ ¼ εtðqÞ
Therefore, there is relationship between f(a) and t(q), which satisfies:
(11)
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f ðaÞ ¼ qaðqÞ tðqÞ
419
(12)
From Equations (8) and (12), the singularity spectrum f(a) contains exactly the same information as Dq. By Equations (6) and (8), the generalized fractal dimensions Dq are estimated through a linear regression of ½ln Zε ðqÞ=ðq 1Þ against lnε for q s 1 [38]. P The value of q is a weighting coefficient for ½mðbÞq. In this equation, subset with high value of m(b) play a major role in mðbÞ
summation processes if the value of q is considerably larger than one. Conversely, subset with low value of m(b) play a major role in summation processes if the value of q is considerably less than one. The generalized fractal Dq determine whether the network is multifractal from the shape of Dq. A monofractal system has the same scaling behavior at any point, Dq should be a constant independent of q [31]. Dq represent property of space of network and has not relationship with probability of uneven P distribution when q equal zero. The value of D0 is named as the Hausdorff dimension. NðPÞP is information entropy of network when q ¼ 1. Therefore, the value of D1 is named as information dimension. 3. Multifractal analysis of weighted networks 3.1. Algorithm analysis of multifractal of networks Most of real fractal networks are inhomogeneous. In scale-free networks, the scale-free property is represented by a power-law degree distribution pðkÞfkg , where k is the number of connections of a node [39]. Thus, the distribution of hubs node density is higher than others node. There is highly inhomogeneous for two subgraphs, which even covered same number of nodes. The inhomogeneous distribution of number of nodes often exhibits the multifractal property [31,33]. For multifractal analysis of unweighted networks, some algorithms are introduced [31]. However, the study is very few about multifractal property of weighted networks. In this section, an algorithm of multifractal of weighted networks is proposed based on BCANw. To our knowledge, the method is the first introduced to analyze the multifractal behavior of weighted network. In the proposed method, firstly, the edge-weights between connecting directly nodes is converted to the distance between them for two types networks. Secondly, the shortest distance between any nodes are calculated and these pairs are saved into matrix ANN, where N is the number of nodes of the network. According to BCANw, set the size of the box with accumulating the distance between two nodes connected directly. Finally, for a given box size r, the minimum number of boxes, which cover the entire network, is obtained by BCANw [34]. For the nonempty boxes b, in Equation (5), mðbÞ is given as follows,
mðbÞ ¼
Nb N
(13)
where Nb is the number of nodes covered by the box b and N is the number of nodes of network. The partition sum as Zr ðqÞ ¼ P mðbÞs0
½mðbÞq is calculated for each value of r. In order to obtained the generalized fractal dimension Dq, lineae regression is an
Fig. 1. The relationship between values of fractal dimension and values of u, jdB ðwÞðuÞ dB ðwÞðuÞj < 0:2 [34].
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Fig. 2. Distribution of edge-weights of the USAir97 network, the C. elegans network and the Scientific collaboration network. The vertical axis y and horizontal w w , respectively. axis x represent value of maxðwÞ and number of edge-weights with maxðwÞ
Table 1 Values of Dq and q for SW network with p ¼ 0.03 and.p ¼ 0.06 q
0
2
3
4
5
(p ¼ 0.03)Dq (p ¼ 0.06)Dq
1.864 2.066
2.304 2.656
2.333 2.707
2.339 2.726
2.338 2.733
essential step for the appropriate range of box size. The linear regression of lnZr ðqÞ=ðq 1Þ against ln(r/d) (q s 1) is considered. In Section 4, we calculate the generalized fractal dimension Dq and determine the multifractality property of complex networks from the shape of Dq. 3.2. Value of u in multifractal analysis of weighted networks For weighted networks, the relationship between distance of nodes and value of edge-weights is core factor. In Ref [40], the Scientific collaboration networks is given by Newman. In the Scientific collaboration networks, two scientists are considered connected if they have coauthored some papers together and edge-weight wij represents the strength of the collaboration between scientists i and j. The distance between author i and j is defined as the inverse of the weight of their
Fig. 3. The Dq curves of the SW network with p ¼ 0.03 and p ¼ 0.06.
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Fig. 4. Multifractal scaling analysis of the USAir97 network, the C. elegans network and the Scientific collaboration network.
collaborative tie. This definition mean that one pair of authors know one another twice as well as another pair, the distance them is half as great, i. e. u ¼ 1 in Equation (2) [40]. For fractal analysis of complex networks, the relationship between distance of nodes and value of edge-weights is very important too. For unweighted networks, we defined the distance of nodes i and j as ki þ kj, where ki is degree of node i [14]. In this case, fractal property of some real unweighted networks can be revealed too. Therefore, the definition of distance between nodes determines the course of our vision for complex networks. For weighted networks, weighted networks changed as unweighted networks when u ¼ 0 in Equation (2). u > 0 mean that the higher weight is, the further distance is. u < 0 mean
Table 2 Values of Dq and q for the USAir97 network [41], the C. elegans network [11] and the Scientific collaboration network [40]. q
0
2
3
4
5
USAir97 C. elegans Scientific
0.7231 1.989 0.5505
1.647 3.916 1.638
1.42 3.095 1.956
1.29 2.76 1.958
1.226 2.588 1.456
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Fig. 5. The Dq curves of the USAir97 network, the C. elegans network and the Scientific collaboration network.
that is opposite. In Ref [34], the relationship between the fractal property of weighted network and u value is considered. We found that the values of fractal dimensions are almost symmetric about values of u, about the vertical. Some real weighted networks such as the USAir97 network, which edge-weights are the number of seats available on the scheduled flights with million per year [41], the Caenorhabditis elegans network, which edge-weights are the number of synapses and gap junctions [11], and the Scientific collaboration network [40] are considered in Ref. [34]. These results is shown in Fig. 1 [34]. In Fig. 1, fractal dimensions of the USAir97 network, the C. elegans network and the Scientific collaboration network are all almost symmetric about values of u, about the vertical. The changing of edge-weights do not affect with fractal property. The symmetric property of fractal dimension is not affected by values of u multiplied to enlarge or decrease [34]. And that, the distributions of edge-weights of this three networks are different each other. All edge-weights of these real networks are divided maximum value of edge-weights and shown in Fig. 2. From Fig. 2, values of edge-weights of the USAir97 network and the Scientific collaboration network have a nice continuous. Value of edge-weights of the C. elegans network has a jump. A lot of edge-weights of the C. elegans network have the same values, which is same as the Scientific collaboration network. The distribution of edge-weights of the USAir97 network is approximate uniform distribution. From Equation (4), fractal dimension dB(w) is obtained by means of the least square fit between number of box and size of box. From Equations (6) and (8), the generalize fractal dimension Dq is obtained by means of the least square fit between number of nodes in the box and size of box. Therefore, the generalized fractal dimension is affected by edge-weights as same as fractal dimension of weighted networks.
4. The numerical simulations Our proposed method is also apply to unweighted networks. For example, a unweighted network such as Small-World network [11] is considered. A small-world network is given as processing: Firstly, we construct a regular ring lattice with n nodes and give a mean degree k, which is the number of neighbors to connect. Finally, shortcuts, which instead of rewiring links, are added with probability p. In our Small-World network, the number of nodes and the mean degree in a given row are 3000 and 4, respectively. The probability p of adding a shortcut is a small value (p ¼ 0.03 and p ¼ 0.06 ). By our proposed method, values of Dq and q are given in Table 1. The relationship between Dq and q is shown in Fig. 3. From Table 1 and Fig. 3, values of Dq have changed little after q ¼ 2, multifractal property of small-world network is not distinct. During the generation, when p increase, more edges are added. Therefore, the value of Dq of network with p ¼ 0.06 is more than network with p ¼ 0.03. These results are consistent with results of Ref [31]. For fractal analysis of weighted networks, the USAir97 network [41], the C. elegans network [11] and the Scientific collaboration network [40] were considered. In the section, multifractal property of these real networks are considered by our proposed method. In last section, we found that the fractal and multifractal of weights network is not affected by value of u. For three weighted networks, the bigger edge-weights is, the less distance is. Therefore, to simplify computation, value of u equal 1 in Equation (2). By our proposed method, lnZr ðqÞ=ðq 1Þ against ln(r/d) (q s 1) of these networks are shown in Fig. 4. From Fig. 4, the generalized fractal dimension Dq and the appropriate range of r2½rmin ; rmax by linear regression. And then, the value range of r is obtained by choosing appropriate points. Value of horizontal coordinates of the USAir97 network is [6, 2.5]. In the C. elegans network and the Scientific collaboration network, we adopt the points from 11th to 30th and from 5th to 20th, respectively. Values of Dq of these weighted networks are given in Table 2 and the relationship between Dq and q is shown in Fig. 5. From Table 2 and Fig. 5, the values of Dq of three real weighted networks have large different when values of q are different. These Dq curves indicate that these weighted networks are multifractal. However, value of Dq of the C. elegans network is
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bigger than Dq of the USAir97 network and the Scientific collaboration network with same q. It indicates the degree of inequality of ratio m(b) of the C. elegans network is bigger than the USAir97 network and the Scientific collaboration network. The value of Dq of the C. elegans network and the USAir97 network reach their peak values around q ¼ 2. However, the maximum value of Dq of the Scientific collaboration network is the interval [3,4]. It shows that the subset with high value of m(b) of the Scientific collaboration has more important role than other networks. 5. Conclusion Fractal property has been shown as one of the fundamental properties of complex networks. The box-covering algorithm is an efficient method to calculate fractal dimension of unweighted complex networks. Recently, the researches on complex networks have shown that some real networks exhibited the property of multifractal scaling. And that, BCANw is presented for analysing fractal property of weighted network [34]. However, the existing studies mainly dealt with the multifractal property of unweighted networks. To address this issue, a multifractal analysis method for weighted networks is proposed based on BCANw. It allows the computation of the generalized fractal dimensions Dq. The numerical examples of unweighted network such as small-world network and some real weighted networks show that the proposed approach not only well reveal mulitfractal property of weighted networks but it can also reveal mulitfractal property of unweighted networks. Acknowledgment The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61174022, 61325011 and 61364030), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20131102130002), R&D Program of China (2012BAH07B01), National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), the funding project of National Bureau of Statistical of China (Grant No. 2014LY128), the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No. BUAA-VR14KF-02). 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] [41]
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