On General Laws of Complex Networks

Abstract

By introducing and analyzing a renormalization procedure, Song et al. [1] draw the conclusion that many complex networks exhibit self-repeating patterns on all length scales. First, we aim to demonstrate that the aforementioned conclusion is inadequately justified, mainly because their equation (7) on the invariance of degree distribution under renormalization does not hold in general. Secondly, Barabási and Albert [2] find that many large networks exhibit a scale-free power-law distribution of vertex degrees. They show this common feature to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to those that are already well connected. We show that when vertex degrees of large networks follow a scale-free power-law distribution with the exponent γ ≥ 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order less than log N. Given that many real networks satisfy these two conditions, our results add another necessary characteristic of the scale-free power-law distribution of vertex degrees in such networks. Our method has the benefit of relying on conditions that are static and easily verified. They are verified by many experimental results of diverse real networks.Keywordscomplex networkself-similarityscale-freecomputer networkcomputer communication