We introduce a modified small-world network adding new links with nonlinearly preferential connection instead of adding randomly, then we apply Bak-Sneppen (BS) evolution model on this network. We study several important structural properties of our network such as the distribution of link-degree, the maximum link-degree, and the length of the shortest path. We further argue several dynamical characteristics of the model such as the important critical value f_c, the f₀ avalanche, and the mutating condition, and find that those characteristics show particular behaviors.