Dynamic scaling, data-collapse and self-similarity in Barab\'{a}si-Albert networks

TL;DR

This paper demonstrates that the generalized degree distribution in Barabási-Albert networks exhibits dynamic scaling and self-similarity, revealing universal behavior and universality classes based on node attachment strategies.

Contribution

It introduces the concept of generalized degree and shows its distribution follows a dynamic scaling law, identifying universality classes in BA networks.

Findings

Distribution function exhibits dynamic scaling with time.

Data collapse onto a universal curve for different network sizes.

Universality classes depend on the number of edges new nodes attach with.

Abstract

In this article, we show that if each node of the Barab\'{a}si-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 $F(q,t\rightarrow \infty)\sim t^{-1/2}\phi(q/t^{1/2})$ where $\phi(x)$ is the scaling function. We verified it by showing that a series of distinct $F(q,t)$ vs $q$ curves for different network sizes $N$ collapse onto a single universal curve if we plot $t^{1/2}F(q,t)$ vs $q/t^{1/2}$ 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 ($m>1$).