Limited Improvement of Connectivity in Scale-Free Networks by Increasing the Power-Law Exponent

It has been well-known that many real networks are scale-free (SF) but extremely vulnerable against attacks. We investigate the robustness of connectivity and the lengths of the shortest loops in randomized SF networks with realistic exponents $2.0 < \gamma \leq 4.0$. We show that smaller variance of degree distributions leads to stronger robustness and longer average length of the shortest loops, which means the existing of large holes. These results will provide important insights toward enhancing the robustness by changing degree distributions.