度分布宽度对网络鲁棒性的影响：局部攻击与随机攻击的比较
The influence of the broadness of the degree distribution on network's robustness: comparing localized attack and random attack
摘要：
网络的稳定性很大程度上受其度分布的影响，特别是受其宽度的影响。具有更宽度分布的网络通常对随机故障更鲁棒，但对局部攻击的鲁棒性较低。为了更好地理解度分布的宽度的影响，我们在这里研究了两个控制宽度的模型，并比较了它们对局部攻击（LA）和随机攻击（RA）的鲁棒性。我们研究分析和数值模拟的情况下，在网络中的度遵循双泊松分布$P(k)=αe^{-λ_1}\frac{λ_1^k}{k!}+(1-α) e^{-λ_2}\frac{λ_2^k}{k!},α\in[0,1]$，和高斯分布$P(k)=A \cdot exp{(-\frac{(k-μ)^2}{2σ^2})}$与归一化常数$A$其中$k\geq 0$。在双泊松分布中，宽度由$α$、$λ_1$和$λ_2$的值控制，而在高斯分布中，宽度由标准差$σ$控制。我们发现，只有$α=0$或$α=1$，即度服从纯泊松分布，LA和RA是相同的，但对于所有其他情况下，网络更容易受到LA相比RA。对于高斯分布，当平均度$μ$固定时，我们发现当$σ^2$小于$μ$时，网络更容易受到随机攻击。然而，当$σ^2$大于$μ$时，网络变得更容易受到局部攻击。对于相互依赖的网络，也显示了类似的定性结果。
英文摘要：
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 $P(k)=αe^{-λ_1}\frac{λ_1^k}{k!}+(1-α) e^{-λ_2}\frac{λ_2^k}{k!},α\in[0,1]$, and a Gaussian distribution $P(k)=A \cdot exp{(-\frac{(k-μ)^2}{2σ^2})}$ with a normalization constant $A$ where $k\geq 0$. In the Bi-Poisson distribution the broadness 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 for $α=0$ or $α=1$, 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 $μ$ fixed, we find that when $σ^2$ is smaller than $μ$ the network is more vulnerable against random attack. However, when $σ^2$ is larger than $μ$ the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.
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