Modeling the Evolution of Degree Correlation in Scale-Free Topology Generators

In this paper, we examine the asymptotic behavior of degree correlation (i.e., the joint degree distribution of adjacent nodes) in several scale-free topology generators GED [13], PLRG [1], GLP [10], BA [3], AB [2]. We present a unifying analytical framework that allows tractable analysis of degree correlation in all studied models and derive asymptotic formulas of two degree correlation metrics - assortativity and clustering. Our results indicate that all studied generators become uncorrelated as graph size increases, which is inconsistent with time-invariance of these metrics in real networks such as the Internet [36], [48], [50]. Since the class of degree-based generators is incapable of reproducing evolving characteristics of the Internet, we study three other models that evolve graphs using different rules than preference of degree (e.g., based on random walks [50], optimization [17], and geometry [23]) and show using simulations that these models are much more viable alternatives for replicating the complex structure of Internet-like graphs.