  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.
