Preferential attachment network model with aging and initial attractiveness

In this paper, we generalize the growing network model with preferential attachment for new links to simultaneously include aging and initial attractiveness of nodes. The network evolves with the addition of a new node per unit time, and each new node has m new links that with probability Pi (i) are connected to nodes i already present in the network. In our model, the preferential attachment probability pi (i) is proportional not only to k (i) + A, the sum of the old node i's degree k (i) and its initial attractiveness A, but also to the aging factor tau(-alpha)(i), where tau(i) is the age of the old node i. That is, Pi(i) proportional to(k(i)+A)tau(-alpha)(i). Based on the continuum approximation, we present a mean-field analysis that predicts the degree dynamics of the network structure. We show that depending on the aging parameter alpha two different network topologies can emerge. For alpha < 1, the network exhibits scaling behavior with a power-law degree distribution P(k) proportional to k (-gamma) for large k where the scaling exponent gamma increases with the aging parameter alpha and is linearly correlated with the ratio A/m. Moreover, the average degree k(t(i) , t) at time t for any node i that is added into the network at time t(i) scales as k(t(i),t) proportional to t(i)(-beta) where 1/beta is a linear function of A/m. For alpha > 1, such scaling behavior disappears and the degree distribution is exponential.