A method is proposed to construct the weighted amino acid network. The weight of the link is based on the contact energy between residues. For the 197 proteins with low homology, the "small-world" property was studied based on this method. Additionally, analyses were carried out for the statistic characteristics of the network parameters, the influence of the weight on the network parameters, the network parameter difference of amino acids, and the links between the hydrophobic and hydrophilic residues. Using this method, we studied the network parameter change for the protein chymotrypsin inhibitor 2 (CI2) on its high-temperature unfolding pathway. It is found that the unfolding of the protein is mainly exhibited as the derogation of the hydrophobic core and the shortest path length rise in the unfolding process. This work is helpful for studies of protein folding and the relationship between structure and function using complex network theory.