The stability of networks is greatly influenced by their degree distributions and in particular by their breadth. 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 breadth of the degree distribution we study two models in which the breadth 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, $P\left(k\right)=\alpha e^{-{\lambda }_1}\frac{{\lambda }_1^k}{k!}+(1-\alpha )e^{-{\lambda }_2}\frac{{\lambda }_2^k}{k!},\alpha \in [0,1]$, and a Gaussian distribution, $P\left(k\right)=A\text{exp}(-\frac{{(k-\mu )}^2}{2{\sigma }^2}),$ with a normalization constant $A$ where $k\ge 0$. In the bi-Poisson distribution the breadth 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 when $\alpha =0$ or $\alpha =1$, i.e., degrees obeying a pure Poisson distribution, are LA and RA the same. In all other cases networks are more vulnerable under LA than under RA. For a 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. When ${\sigma }^2$ is larger than $\mu$, however, the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.

• Received 3 June 2015

DOI:https://doi.org/10.1103/PhysRevE.92.032122