  Ranking node importance is crucial in understanding network structure and
function on complex networks. Degree, h-index and coreness are widely used, but
which one is more proper to a network associated with a dynamical process, e.g.
SIR spreading process, is still unclear. To fill this gap, a, which is
extracted from the fitting function (f(x)=1-1/(e^(2a(x-b))+1)) of the average
number of nodes in each radius of the neighborhood of a node, is proposed.
Experiment results which are carried out on twenty real-world networks show
that a can classify which of the three measures (degree, h-index and coreness)
is more proper to a network in ranking node importance. We also find that [b/3]
is a good indicator for forecasting the optimal radius of the neighborhood of a
node in ranking node importance for a given network. To the best of our
knowledge, it is the first solution of this interesting and open issue.
Furthermore, by extending the range of neighborhood where we construct an
operator H on of a node, we propose a new method to quantify the importance of
a node. The ranking accuracies of most networks can be improved when the radius
is increased from 0 to its forecasting optimal radius and the improvement, for
the best case, reaches up to 111%. The performances will reduce on half of the
networks studied in this paper if we roughly extend the radius of the
neighborhood. Our work deepens the understanding of how to find out the proper
node ranking method for complex networks. The proposed methods bridge the gaps
among network structure and node importance, and may have potential
applications in controlling the outbreak of disease, designing of optimal
information spreading strategies.
