  Recurrence networks are powerful tools used effectively in the nonlinear
analysis of time series data. The analysis in this context is done mostly with
unweighted and undirected complex networks constructed with specific criteria
from the time series. In this work, we propose a novel method to construct
"weighted recurrence network"(WRN) from a time series and show how it can
reveal useful information regarding the structure of a chaotic attractor, which
the usual unweighted recurrence network cannot provide. Especially, we find the
node strength distribution of the WRN, from every chaotic attractor follows a
power law (with exponential tail) with the index characteristic to the fractal
structure of the attractor. This leads to a new class among complex networks,
to which networks from all standard chaotic attractors are found to belong. In
addition, we present generalized definitions for clustering coefficient and
characteristic path length and show that these measures can effectively
discriminate chaotic dynamics from white noise and $1/f$ colored noise. Our
results indicate that the WRN and the associated measures can become
potentially important tools for the analysis of short and noisy time series
from the real world systems as they are clearly demarked from that of noisy or
stochastic systems.
