  Starting from linear chains of amino acids, the spontaneous folding of
proteins into their elaborate three-dimensional structures is one of the
remarkable examples of biological self-organization. We investigated native
state structures of 30 single-domain, two-state proteins, from complex networks
perspective, to understand the role of topological parameters in proteins'
folding kinetics, at two length scales-- as ``Protein Contact Networks (PCNs)''
and their corresponding ``Long-range Interaction Networks (LINs)'' constructed
by ignoring the short-range interactions. Our results show that, both PCNs and
LINs exhibit the exceptional topological property of ``assortative mixing''
that is absent in all other biological and technological networks studied so
far. We show that the degree distribution of these contact networks is partly
responsible for the observed assortativity. The coefficient of assortativity
also shows a positive correlation with the rate of protein folding at both
short and long contact scale, whereas, the clustering coefficients of only the
LINs exhibit a negative correlation. The results indicate that the general
topological parameters of these naturally-evolved protein networks can
effectively represent the structural and functional properties required for
fast information transfer among the residues facilitating biochemical/kinetic
functions, such as, allostery, stability, and the rate of folding.
