Heavy-tailed distribution of cyber-risks

Abstract

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, Pr(ID lossThis 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 ≃ 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 ∼ S 1.3 with the organization size S. The small value b ≃ 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. * Electronic address: tmaillart@ethz.ch † Electronic address: dsornette@ethz.ch 1

Online Promiscuity: Prophylactic Patching and the Spread of Computer Transmitted Infections

“…We tabulated the total number of attacks on each entity to find the top ten targets of observed attacks. These attacks have a heavy tail distribution ( Figure 14, and seem to confirm Maillart and Sornette's work [24], but lower ranked entities still create large jitters in the overall behavior.…”

Cumulative Advantage: Conductors and Insulators of Heavy‐Tailed Productivity Distributions and Productivity Stars

“…A set of values x follows a power law if it fits the following probability distribution (Clauset, Shalizi, & Newman, ): where α is the scaling exponent (also called scaling parameter), which is a constant (Maillart & Sornette, ). The scaling exponent is calculated using MLE based on running a semiparametric Monte Carlo bootstrap calculation 1,000 times—specifically, the Hill estimator (Hill, ).…”

“…where α is the scaling exponent (also called scaling parameter), which is a constant (Maillart & Sornette, 2010). The scaling exponent is calculated using MLE based on running a semiparametric Monte Carlo bootstrap calculation 1,000 times-specifically, the Hill estimator (Hill, 1975).…”

Insurance mechanisms in information security risk management

“…Due to this dependence on the tail structure, it is relevant to analyze the cumulative distribution of cyber-risk with distribution functions that have parameters for the shape of their tails. Simulation experiments and empirical tests for data on cyber-attacks can be found in [BK06], who model correlated risk with Student-t copulas and estimate from honeynet data, and [MS09], who study the tail structure of data released through breach disclosure laws in the US.…”