Scale-free and small-world properties of a multiple-hub network with fractal structure
Introduction
It has been found that many real-world networks, including such as the World Wide Web and metabolic networks, are not homogeneous networks; instead, their connectivity in heterogeneous. These network have in common two structural characteristics: the small-world effect proposed by Watts–Strogatz [1] (small average path length and high clustering coefficient); and the scale-freeness proposed by Barabasi–Albert [2] (power-law degree distribution).
With the rapid development of fractal geometry in the past few years, the researchers began to pay more attention to the complexity of the network structure and the relationship with its behavior. For example, Yang-Fu-Yu [3] analyzed flow resistance in tree-like branching networks based on the fractal geometry theory. Song–Havlin–Makse [4], [5] revealed that many real networks have self-similarity and fractality, see Gallos–Song–Makse [6], Chen et al. [7], Wang et al. [8], Xi et al. [9], Xue–Zhou [10] for more details. Song et al. [11] and Kim et al. [12] calculated the fractal dimension of a complex network. Komjathy–Simon [13] studied the relation between the degree distribution and the Hausdorff dimension of some networks with fractal structure. The definition of Hausdorff dimension is given in Falconer [14]. Rozenfeld–Song–Makse [15] applied the renormalization group (RG) method to some complex networks.
Zhang et al. [16] studied the average path length of a scale-free and small-world network with single hub and fractal structure. In this paper, we introduce a similar network with multiple hubs, see Section 1.1. Our model is a hierarchical network, and hierarchical networks are quite common in the real-world, such as the metabolic network, see Ravasz et al. [17]. We study the box dimension of the adjacency matrix, degree distribution, average clustering coefficient and average path length. We mainly adopt the definitions of these notions given in Chen–Wang–Li [18]. We will give a list of the main results in Section 1.2.
We start by constructing the networks with a modular structure in an iterative way, and write for the network model of the th () generation.
Our starting point is a small cluster of nodes, which contains hub nodes (coded by ) and external nodes (coded by ). We always assume and in this paper. Denote , then in this case. We connect each hub node with each other node, called . Next we generate replicas of which include replicated central clusters and replicated external clusters. The node in the -th replica is coded by in , where for all . There are four types of nodes in : We connect each hub node with each external node. Then we obtain a large -node module (for instance, see and in Fig. 1). More precisely, when , (the solid squares), (the yellow circles), (the hollow circles), and (the hollow squares). We connect each hub node in with each external node in , then we obtain .
The main results of our work are fourfold.
(1) The box dimension of the adjacency matrix is , see Eq. (2).
(2) When , the degree exponent is ;
when , the degree exponent is , see Eq. (5).
Moreover, when (hub nodes is not more than external nodes), the cumulative degree exponent is ; when , the cumulative degree exponent is .
(3) The average clustering coefficient is , see Eq. (7). In many real-world networks, both and are large enough. In this case, .
(4) The average path length satisfies , see Eq. (12).
Section snippets
Adjacency matrix and its box dimension
For graph with nodes, we order all nodes in lexicographical order. Its adjacency matrix is defined to be the boolean matrix such that if and only if node and node are connected. Obviously for and , .
Several types of nodes
In order to calculate the degree distribution (Section 4) and the average clustering coefficient (Section 5), we divide all nodes in into several types. For each type , we consider four functions: (the number of nodes in this type), (the degree of each node), (the number of complete triangles with each node) and (the clustering coefficient of each node). Notice that, the notation “” means the asymptotic formula for and are large enough. After the list of conclusions, we present
Degree distribution
By the Conclusions (3.1) and (4.3) listed in Section 3.1, for , the number of nodes with degree is equal to In this case, . It is easy to check that the expression of also holds for , see Conclusion (4.2). Moreover and . There exist two positive constants and such that . This means
Average clustering coefficient
Recall that there are nodes in . Then the average clustering coefficient of is By the conclusions listed in Section 3.1, and . Moreover by Eqs. (3), (4), we have the expressions of and . Then
Average path length
If or , we call a pure segment. A maximal pure segment is called a block. For , we use to denote the number of blocks of . For example, when and , .
We denote the geodesic distance between and in . When we write , it should be obvious that . The average path length of the complex network is
In order to calculate , we only need to calculate
CRediT authorship contribution statement
Yuke Huang: Formal analysis, Writing - original draft. Hanxiong Zhang: Formal analysis, Writing - review & editing. Cheng Zeng: Methodology. Yumei Xue: Conceptualization.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (No. 11701024), Projects in the Fundamental Research Funds for the Central Universities, China (No. 2019RC17).
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