  The spectral graph theory provides an algebraical approach to investigate the
characteristics of weighted networks using the eigenvalues and eigenvectors of
a matrix (e.g., normalized Laplacian matrix) that represents the structure of
the network. However, it is difficult for large-scale and complex networks
(e.g., social network) to represent their structure as a matrix correctly. If
there is a universality that the eigenvalues are independent of the detailed
structure in large-scale and complex network, we can avoid the difficulty. In
this paper, we clarify the Wigner's Semicircle Law for weighted networks as
such a universality. The law indicates that the eigenvalues of the normalized
Laplacian matrix for weighted networks can be calculated from the a few network
statistics (the average degree, the average link weight, and the square average
link weight) when the weighted networks satisfy the sufficient condition of the
node degrees and the link weights.
