Highly clustered scale-free networks

Abstract

We propose a model for growing networks based on a finite memory of the nodes. The model shows stylized features of real-world networks: power law distribution of degree, linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate. Notably, the degree distribution is conserved even though only the most recently grown part of the network is considered. This feature is relevant because real-world networks truncated in the same way exhibit a power-law distribution in the degree. As the network grows, the clustering reaches an asymptotic value larger than for regular lattices of the same average connectivity. These highclustering scale-free networks indicate that memory effects could be crucial for a correct description of the dynamics of growing networks.Many systems can be represented by networks, i.e. as a set of nodes joined together by links. Social networks, the Internet, food webs, distribution networks, metabolic and protein networks, the networks of airline routes, scientific collaboration networks and citation networks are just some examples of such systems [1][2][3][4][5][6][7][8][9][10][11]. Recently it has been observed that a variety of networks exhibit topological properties that deviate from those predicted by random graphs [1,2]. For instance, real networks display clustering higher than expected for random networks [4]. Also, it has been found that many large networks are scale-free. Their degree distribution decays as a powerlaw that cannot be accounted for by the Poisson distribution of random graphs [12,13]. The type of the degree distribution is of great importance for the functionality of the network [14][15][16]. Beside the degree distribution, other features of the growth dynamics of real-world networks are currently under investigation. For citation networks, the Internet, and collaboration networks of scientists and actors, it has been shown [17,18] that the probability for a node to obtain a new link is an increasing function of the number of links the node already has. This feature of the dynamics is called preferential attachment. Furthermore the aging of nodes is of particular interest [19]. In the network of scientific collaborations, every node stops receiving links a finite time after it has been added to the network, since scientists have a finite time span of being active. Similarly, in citation networks, papers cease to receive links (citations), because their contents are outdated or summarized in review articles, which are then cited instead. Whether a paper is still cited or not, depends on a collective memory containing the popularity of the paper.In the current paper we address the study of growing complex networks from the perspective of the memory of the nodes. First, we present empirical evidence for the age dependence of the growth dynamics of the network of scientific citations. We find that old nodes are less likely to obtain links than nodes added to the network more recently. Second, motivated by this ...

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