Small-world features of real-world networks

It is well known that there are a large number of complex networks that have vertex-degree distributions in a power-law form of ck <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-γ</sup> , where k is the degree variable and c and γ are scaling and exponent constants. Recently, we found that it is effective to reveal the underlying mechanism of power-law formation in real-world networks by analyzing their vertex-degree sequences. We showed before that, for a scale-free network of size N, if its vertex-degree sequence is k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <; k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <; ⋯ <; k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> , where {k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , ⋯ , k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> } is the set of all non-equal vertex degrees in the network, and if its power exponent satisfies γ > 1, then the length l of the above vertex-degree sequence is of order log N. We underline that this conclusion is important, which proves that the length of the vertex-degree sequence is a fundamental characteristic of a scale-free network. In this paper, we further investigate complex networks with more general distributions and we prove that the same conclusion about the vertex-degree sequences holds even for non-network type of complex systems. We thereby conclude that real-world networks typically possess small-world features. We support this conclusion by verifying a large number of real-world networks and systems. To that end, we discuss some potential applications of the new finding in various fields of science, engineering and society, demonstrating that the conclusion is important with many real applications.