Growing scale-free small-world networks with tunable assortative coefficient

Abstract

In this paper, we propose a simple rule that generates scale-free small-world networks with tunable assortative coefficient. These networks are constructed by two-stage adding process for each new node. The model can reproduce scale-free degree distributions and small-world effect. The simulation results are consistent with the theoretical predictions approximately. Interestingly, we obtain the nontrivial clustering coefficient C and tunable degree assortativity r by adjusting the parameter: the preferential exponent $\beta$. The model can unify the characterization of both assortative and disassortative networks.

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 $k_s^{\beta }/{\sum }_{j\in {\Gamma }_i}{k}_j^{\beta }$, where $\beta$ is the parameter named preferential exponent and ${\Gamma }_i$ is the neighbor node set of node i. This model will be equal to the Holme–Kim model [37] when $\beta =0$, and the MRGN model [35] when $\beta =1$. 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.