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 $ck^{-\gamma}$, where $\emph{k}$ is the degree variable and $\emph{c}$ and $\gamma$
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 \emph{N}, if its vertex-degree sequence is
$ k_{1}1$, then the length \emph{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."