Network fragility to overload failures: influence of the scale-free property

Recently, it has been found that scale-free networks are fragile against overload failures. More precisely, the critical node-removal fraction $f_{{\rm c}}$ or the critical total load $W_{{\rm c}}$ at the percolation transition by removing overloaded nodes is an increasing function of the scale-free exponent $\gamma $ describing the asymptotic degree distribution function as $P(k)\propto k^{-\gamma }$. In the previous work, however, the decrease of the exponent $\gamma $ reduces the total tolerance for loads. Thus, it is not clear whether scale-free networks are actually fragile against overload failures if the total tolerance dose not change. In this paper, we investigate network robustness against overload failures by calculating analytically the percolation transition point of scale-free networks for various values of $\gamma $ with keeping the total tolerance constant. Our results show that scale-free networks are surely fragile to overload failures even after excluding the effect of the total tolerance.