The influence of the broadness of the degree distribution on network's robustness: comparing localized attack and random attack

The stability of networks is greatly influenced by their degree distributions and in particular by their broadness. Networks with broader degree distributions are usually more robust to random failures but less robust to localized attacks. To better understand the effect of the broadness of the degree distribution we study here two models where the broadness is controlled and compare their robustness against localized attacks (LA) and random attacks (RA). We study analytically and by numerical simulations the cases where the degrees in the networks follow a Bi-Poisson distribution $P(k)=\alpha e^{-\lambda_1}\frac{\lambda_1^k}{k!}+(1-\alpha) e^{-\lambda_2}\frac{\lambda_2^k}{k!},\alpha\in[0,1]$, and a Gaussian distribution $P(k)=A \cdot exp{(-\frac{(k-\mu)^2}{2\sigma^2})}$ with a normalization constant $A$ where $k\geq 0$. In the Bi-Poisson distribution the broadness is controlled by the values of $\alpha$, $\lambda_1$ and $\lambda_2$, while in the Gaussian distribution it is controlled by the standard deviation, $\sigma$. We find that only for $\alpha=0$ or $\alpha=1$, namely degrees obeying a pure Poisson distribution, LA and RA are the same but for all other cases networks are more vulnerable under LA compared to RA. For Gaussian distribution, with an average degree $\mu$ fixed, we find that when $\sigma^2$ is smaller than $\mu$ the network is more vulnerable against random attack. However, when $\sigma^2$ is larger than $\mu$ the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.