Directed Accelerated Growth: Application in Citation Network

In many real world networks, the number of links increases nonlinearly with the number of nodes. Models of such accelerated growth have been considered earlier with deterministic and stochastic number of links. Here we consider stochastic accelerated growth in a network where links are directed. With the number of out-going links following a power law distribution, the results are similar to the undirected case. As the accelerated growth is enhanced, the degree distribution becomes independent of the “initial attractiveness”, a parameter which plays a key role in directed networks. As an example of a directed model with accelerated growth, the citation network is considered, in which the distribution of the number of outgoing link has an exponential tail. The role of accelerated growth is examined here with two different growth laws. PACS numbers: 05.70.Jk, 64.60.Fr, 74.40.Cx The number of links shows a nonlinear growth in time in many real-world networks which evolve in time [1,2]. Examples of such network are the Internet [3], World Wide Web (WWW) [4], collaboration [5], word web [6], citation [7,8,9], metabolism [10], gene regulatory network [11,12,13] etc. The number of links may increase in two ways: new nodes may tend to get attached to more nodes as the size of the network increases, and there may be additional links generated between the older nodes in a non-linear fashion. These two factors maybe present either singly or simultaneously resulting in the accelerated growth. In some networks like the citation and the gene regulatory network, new links between older nodes are forbidden and therefore only the first scheme is valid, while in collaboration network or internet, the second factor is dominating. In general, in a network with accelerated growth, the total number of links l shows a non-linear increase in time, i.e., l(t) ∼ t with δ > 1. Such an acceleration will be possible when the number of links m(τ) at any time τ Preprint submitted to Elsevier Science 2 February 2008 available to the network is not a constant but increases in such a manner that l(t) = Στ=1m(τ) shows a non-linear increase with time. The case when the new node gets a fixed number of links but older nodes get new links the number of which grows in a non-linear fashion has been considered in both isotropic and directed models of growing networks [5,6,14], showing that it is distinct from networks with a linear growth rule. Networks in which older nodes do not get new links have also been considered recently, with the possibility of a new node to have either deterministic or stochastic number of links [15]. Many real systems like the citation network and WWW are directed networks [9,16]. In the citation network, accelerated growth has been shown to take place [9]. This network also evolves in such a way that new links are forbidden between old nodes. We propose in this paper directed models of accelerated growth with number of outgoing links given by a particular distribution and where older nodes are not allowed to get linked. One of these models mimics the citation network. Directed accelerated growth with power law distribution As in [15], here we assume that in the growing network, the number of outgoing links m is determined by a power-law distribution, i.e., P (m) ∼ m (1) with 1 ≤ m ≤ N(τ) at any time τ when the number of nodes in the network is N(τ). The number of nodes is usually a linear function of time and N(τ) = τ when nodes are added one by one to the network. λ is the parameter controlling the measure of accelerated growth. As a new node comes in, it will get preferentially attached to m existing nodes with the probability Π ∝ A+ kin (2) where kin is the in-degree of the existing node and A is the initial attractiveness of the network. A is a parameter which is necessary in all directed networks to ensure that a node even with kin = 0 has the chance to get linked with the new node. Without acceleration, when m is fixed, one gets a scale-free network with the the degree distribution P (k) ∼ k where [18] γ = 2 + A/m. (3) With m varying (growing) in time, the nature of the degree distribution P (k) is expected to change. This is obvious if one considers the average value of m which shows the following behaviour: