On the Emergence of Islands in Complex Networks

Most growth models for complex networks consider networks comprising a single connected block or island, which contains all the nodes in the network. However, it has been demonstrated that some large complex networks have more than one island, with an island size distribution (I-s) obeying a power-law function I-s similar to s(-alpha). This paper introduces a growth model that considers the emergence of islands as the network grows. The proposed model addresses the following two features: (i) the probability that a new island is generated decreases as the network grows and (ii) new islands are created with a constant probability at any stage of the growth. In the first case, the model produces an island size distribution that decays as a power-law I-S similar to s(-alpha). with a fixed exponent alpha = 1 and in-degree distribution that decays as a power-law Q(i) similar to i(-gamma) with gamma = 2. When the second case is considered, the model describes island size and in-degree distributions that decay as a power-law with 2 < alpha < infinity and 2 < gamma < infinity, respectively.