Preferential attachment network model with aging and initial attractiveness

Abstract

Abstract 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 Π i are connected to nodes i already present in the network. In our model, the preferential attachment probability Π 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 τ i − α , where τ i is the age of the old node i. That is, Π i ∝ ( k i + A ) τ i − α . 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 α two different network topologies can emerge. For α < 1, the network exhibits scaling behavior with a power-law degree distribution P(k) ∝ k −γ for large k where the scaling exponent γ increases with the aging parameter α 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 ) ∝ t i − β where 1/β is a linear function of A/m. For α > 1, such scaling behavior disappears and the degree distribution is exponential.