Physica A: Statistical Mechanics and its Applications
Growing scale-free small-world networks with tunable assortative coefficient
Introduction
In the past few years, no issue in the area of network researching attracted more scientists than the one related to the real networks, such as the Internet, the World-Wide Web, the social networks, the scientific collaboration and so on [1], [2], [3], [4], [5], [6]. Recent works on the complex networks have been driven by the empirical properties of real-world networks and the studies on network dynamics [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Many empirical evidences indicate that the networks in various fields have some common topological characteristics. They have a small average distance like random graphs, a large clustering coefficient and power-law degree distribution [1], [2], which are called the small-world and scale-free characteristics. The other characteristic is that the social networks are assortative while almost all biological and technological networks are opposite. The networks with high clustering coefficient and small averaged distance can be generated with an evolution by the small-world model of Watts and Strogatz [1], while the networks with power-law degree distribution can be generated with an evolution by the scale-free model of Barabási and Albert [2]. The BA model is a pioneering work in the studies on networks, which suggests that the growth and preferential attachment are two main self-organization mechanisms. Although BA model can generate the power-law degree distribution, its assortative coefficient r equals to zero in the limit of large size thus fail to reproduce the disassortative property that extensively exists in the real-world networks. Recently, some models that can generate either assortative or disassortative networks have been reported [30], [31], [32], [33], [34], [35]. Wang et al. presented a mutual attraction model for both assortative and disassortative weighted networks. The model found that the initial attraction A of the newly added nodes may contribute to the difference of the assortative and disassortative networks [34]. Liu et al. [36] proposed a self-learning mutual selection model for weighted networks, which demonstrated that the self-learning probability p may be the reason why the social networks are assortative and the technological networks are disassortative. However, one should not expect the existence of a omnipotent model that can completely illuminate the underlying mechanisms for the emergence of disassortative property in various network systems. In this paper, beside the previous studies, we exhibit an alternative model that can generate scale-free small-world networks with tunable assortative coefficient, which may shed some light in finding the possible explanations to the different evolution mechanisms between assortative and disassortative networks.
Dorogovtsev et al. [37] proposed a simple model of scale-free growing networks . In this model, a new node is added to the network at each time step, which connects to both ends of a randomly chosen link undirected. The model can be equally described by the process that the newly added node connects to node i preferentially, then select a neighbor node of the node i randomly. Holme and Kim [38] proposed a model to generate growing scale-free networks with tunable clustering. The model introduced an additional step to get the trial information and demonstrated that the average number of trial information controls the clustering coefficient of the network. It should be noticed that the newly added node connects to the first node i preferentially, while connects to the neighbor node of the first node i randomly. In this paper, we will propose a growing scale-free network model with tunable assortative coefficient. Inspired by the above two models, the new node is added into the network by two steps. In the first step, the newly added node connects to the existing node i preferentially. In the second step, this node selects a neighbor node s of the node i with probability , where is the parameter named preferential exponent and is the neighbor node set of node i. This model will be equal to the Holme–Kim model [37] when , and the MRGN model [35] when . Specifically, the model can generate a nontrivial clustering property and tunable assortativity coefficient. Therefore, one may find explanations to various real-world networks by our microscopic mechanisms.
Section snippets
Construction of the model
Our model is defined as follows:
- (1)
Initial condition: the model starts with connected nodes.
- (2)
Growth: at each time step, one new node with m edges is added at every time step. Time t is identified as the number of time steps.
- (3)
The first step: each edge of is then attached to an existing node with the probability proportional to its degree, i.e., the probability for a node i to be attached to is
- (4)
The second step: if an edge between and was added in the first step, then add one more
Degree distribution
The degree distribution is one of the most important statistical characteristics of networks. Since some real-world networks are scale-free, whether the network is of the power-law degree distribution is a criterion to judge the validity of the model. By adopting the mean-field theory, the degree evolution of individual node can be described aswhere denotes the probability that the node i with degree is selected at the first step, denotes the
Conclusion and discussion
In this paper, we propose a simple rule that generates scale-free small-world networks with tunable assortative coefficient. The inspiration of this model is to introduce the parameter to Holme–Kim model. The simulation results are consistent with the theoretical predictions approximately. Interestingly, we obtain the nontrivial clustering coefficient C and tunable degree assortativity r, depending on the parameters . The model can unify the characterization of both assortative and
Acknowledgment
This work has been supported by the Chinese Natural Science Foundation of China under Grant nos. 70431001, 70271046 and 70471033.
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