DEGREE SEQUENCES BEYOND POWER LAWS IN COMPLEX NETWORKS

Many complex networks possess vertex-degree distributions in a power-law form of <i>ck</i>^-<i>γ</i>, where <i>k</i> is the degree variable and <i>c</i> and <i>γ</i> are constants. To better understand the mechanism of power-law formation in realworld networks, it is effective to analyze their degree variable sequences. We had shown before that, for a scale-free network of size N,if its vertex-degree sequence is <i>k</i>₁ < <i>k</i>₂ <…< <i>k</i>_l, where {<i>k</i>₁, <i>k</i>₂,…, <i>k</i>_l} is the set of all unequal vertex degrees in the network, and if its power exponent satisfies <i>γ</i>>1, then the length <i>l</i> of the vertex-degree sequence is of order <i>logN</i>. In the present paper, we further study complex networks with more general distributions and prove that the same conclusion holds even for non-network type of complex systems. In addition, we support the conclusion by verifying many real-world network and system examples. We finally discuss some potential applications of the new finding in various fields of science, technology and society.