网络风险的厚尾分布
Heavy-Tailed Distribution of Cyber-Risks
摘要：
随着互联网的发展，出现了新型的大规模流行病、分布式攻击、虚拟冲突和犯罪。我们提出了一个网络风险的一些显着的统计特性的研究，量化的分布和时间演变的信息风险在互联网上，了解他们的机制，并创造机会，以减轻，控制，预测和确保他们在全球范围内。首先，我们报告了一个异常稳定的幂律分布的个人身份损失，每个事件，${\rm Pr}({\rm ID loss} \geq V) \sim 1/V^b$，与$b =0.7 \pm 0.1$。这一结果是强大的，对一个令人惊讶的强大的非平稳增长的ID损失在2006年7月达到高潮，随后是一个更稳定的阶段。此外，对于不同类型和规模的目标组织，这种分布是相同的。自$b<1$以来，根据$\mathbf{\simeq t^{1/b}}$，在$t$之前的所有事件上的所有丢失的累积数量随着时间以比线性更快的速度增加，这表明以个人身份为特征的隐私必然变得越来越不安全。我们还表明存在的规模效应，这样的最大可能的ID损失，每个事件的增长速度比线性$\sim S^{1.3}$与组织规模$S$。ID损耗的幂律分布的小值$b \simeq 0.7$由齐普夫定律和尺寸效应之间的相互作用来解释。我们还推断，受损的实体表现出基本相同的概率，招致或大或小的损失。
英文摘要：
With the development of the Internet, new kinds of massive epidemics, distributed attacks, virtual conflicts and criminality have emerged. We present a study of some striking statistical properties of cyber-risks that quantify the distribution and time evolution of information risks on the Internet, to understand their mechanisms, and create opportunities to mitigate, control, predict and insure them at a global scale. First, we report an exceptionnaly stable power-law tail distribution of personal identity losses per event, ${\rm Pr}({\rm ID loss} \geq V) \sim 1/V^b$, with $b =0.7 \pm 0.1$. This result is robust against a surprising strong non-stationary growth of ID losses culminating in July 2006 followed by a more stationary phase. Moreover, this distribution is identical for different types and sizes of targeted organizations. Since $b<1$, the cumulative number of all losses over all events up to time $t$ increases faster-than-linear with time according to $\mathbf{\simeq t^{1/b}}$, suggesting that privacy, characterized by personal identities, is necessarily becoming more and more insecure. We also show the existence of a size effect, such that the largest possible ID losses per event grow faster-than-linearly as $\sim S^{1.3}$ with the organization size $S$. The small value $b \simeq 0.7$ of the power law distribution of ID losses is explained by the interplay between Zipf's law and the size effect. We also infer that compromised entities exhibit basically the same probability to incur a small or large loss.
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