Box-covering algorithm for fractal dimension of weighted networks

Dai-Jun Wei, Qi Liu, Hai-Xin Zhang, Yong Hu, Yong Deng, Sankaran Mahadevan
2013 Scientific Reports  
Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected directly and graph-coloring algorithm is based on the node strength. The proposed method is applied to calculate the
more » ... actal dimensions of the "Sierpinski" weighted fractal networks, the E.coli network, the Scientific collaboration network, the C.elegans network and the USAir97 network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal property is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed. C omplex networks have attracted growing interest in various fields of science since they can be used to describe the structure and physical properties of many real complex systems 1-5 . The small-world 6 and the scale-free 7 have been shown as two fundamental properties of complex networks. Recently, fractal and self-similarity properties of complex networks have attracted much attention, since Song et al. found that a variety of real complex networks exist self-similarity property 8 . Especially, the box-covering algorithm is applied to calculate the fractal dimension of many real networks 9 . Though the box-covering algorithm for the complex networks is extensively studied by researchers 10-15 , existing works mainly focus on dealing with the fractal dimension of unweighted networks. In the traditional boxcovering algorithm for complex networks (BCAN), a box size is given in terms of the network distance, which corresponds to the number of edges on the shortest path between two nodes. This means the sizes of these boxed are the integers from 1 to the size of the network. However, the values of edge-weights in weighted networks could be any real numbers excluding zero. For weighted networks, enough numbers of boxes are not been obtained by BCAN. Even the number of box is always one when the size of the weighted network less than one. Thus, using the BCAN to calculate the accurate fractal dimension of weighted networks is unfeasible. Actually, many real-world networks are weighted ones [16] [17] [18] [19] [20] [21] . Some definitions in unweighted networks are extended to weighted networks 22-25 , and some relevant properties and methods of weighted networks are proposed [26] [27] [28] [29] [30] [31] [32] [33] . We believe that the fractal property of weighted networks can be revealed 34,35 . In this paper, an improved box-covering algorithm for weighted networks (BCANw) is proposed. In what follows, we describe the proposed methodology, depict the diversity between BCAN and BCANw, calculate fractal dimension of weighed fractal model such as the "Sierpinski" weighted fractal networks, and apply BCANw to analysis the fractal properties of some real weighted networks. Results BCANw for unweighted networks. For unweighted network, both BCANw and BCAN are the same method. The fractal dimension of unweighted networks by using the BCANw is same as that by using the BCAN. For example, a unweighted network such as the E.coli network with 2859 proteins and 6890 interactions is considered 15 . The correlations between box size (l B ) and number of box (N B ) by using BCAN and BCANw are shown in Figure (1) . By means of the least square, the fractal dimension of the E.coli network is obtained as follows OPEN SUBJECT AREAS: COMPLEX NETWORKS STATISTICAL PHYSICS NONLINEAR PHENOMENA
doi:10.1038/srep03049 pmid:24157896 fatcat:l4uqu3odxjfvjcn6ri3uz3jxca