Yuan Xin, Shao Shuai, Stanley H Eugene, Havlin Shlomo
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Sep;92(3):032122. doi: 10.1103/PhysRevE.92.032122. Epub 2015 Sep 16.
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(k)=αe^{-λ_{1}}λ_{1}^{k}/k!+(1-α)e^{-λ_{2}}λ_{2}^{k}/k!,α∈[0,1], and a Gaussian distribution, P(k)=Aexp(-(k-μ)^{2}/2σ^{2}), with a normalization constant A where k≥0. In the bi-Poisson distribution the breadth is controlled by the values of α, λ_{1}, and λ_{2}, while in the Gaussian distribution it is controlled by the standard deviation, σ. We find that only when α=0 or α=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 μ fixed, we find that when σ^{2} is smaller than μ the network is more vulnerable against random attack. When σ^{2} is larger than μ, however, the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.
网络的稳定性受其度分布的显著影响，尤其是受其广度的影响。具有更广度分布的网络通常对随机故障更具鲁棒性，但对局部攻击的鲁棒性较差。为了更好地理解度分布广度的影响，我们研究了两个控制广度的模型，并比较它们对局部攻击（LA）和随机攻击（RA）的鲁棒性。我们通过解析和数值模拟研究了网络中度数服从双泊松分布(P(k)=\alpha e^{-\lambda_1}\lambda_1^k/k!+(1-\alpha)e^{-\lambda_2}\lambda_2^k/k!)，(\alpha\in[0,1])以及高斯分布(P(k)=A\exp(-(k-\mu)^2/2\sigma^2))（其中(k\geq0)，(A)为归一化常数）的情况。在双泊松分布中，广度由(\alpha)、(\lambda_1)和(\lambda_2)的值控制，而在高斯分布中，它由标准差(\sigma)控制。我们发现只有当(\alpha = 0)或(\alpha = 1)，即度数服从纯泊松分布时，LA和RA才相同。在所有其他情况下，网络在LA下比在RA下更易受攻击。对于平均度数(\mu)固定的高斯分布，我们发现当(\sigma^2)小于(\mu)时，网络对随机攻击更易受攻击。然而，当(\sigma^2)大于(\mu)时，网络对局部攻击变得更易受攻击。相互依存的网络也显示出类似的定性结果。
