Entropy Optimization of Scale-Free Networks Robustness to Random   Failures
  Many networks are characterized by highly heterogeneous distributions of links, which are called scale-free networks and the degree distributions follow $p(k)\sim ck^{-\alpha}$. We study the robustness of scale-free networks to random failures from the character of their heterogeneity. Entropy of the degree distribution can be an average measure of a network's heterogeneity. Optimization of scale-free network robustness to random failures with average connectivity constant is equivalent to maximize the entropy of the degree distribution. By examining the relationship of entropy of the degree distribution, scaling exponent and the minimal connectivity, we get the optimal design of scale-free network to random failures. We conclude that entropy of the degree distribution is an effective measure of network's resilience to random failures. 
Bing Wang, Huanwen Tang, Chonghui Guo, Zhilong Xiu