Dynamic scaling, data-collapse and self-similarity in Barabási-Albert networks

In this paper, we show that if each node of the Barabási-Albert (BA) network is characterized by the generalized degree q, i.e. the product of their degree k and the square root of their respective birth time, then the distribution function F(q, t) exhibits dynamic scaling F(q, t → ∞) ~ t^-1/2phi(q/t^1/2) where phi(x) is the scaling function. We verified it by showing that a series of distinct F(q, t) versus q curves for different network sizes N collapse onto a single universal curve if we plot t^1/2F(q, t) versus q/t^1/2 instead. Finally, we show that the BA network falls into two universality classes depending on whether new nodes arrive with single edge (m = 1) or with multiple edges (m > 1).