Heavy-tailed distribution of cyber-risks

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(IDloss ≥ V) ~ 1/Vb, with b = 0.7 ± 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 ≃ t1/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 ~S1.3 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 los

Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)\theta}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < \theta <1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.Comment: 40 pages, 8 figure

Fractal Weyl law for Linux Kernel Architecture

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be $\nu \approx 0.63$ that corresponds to the fractal dimension of the network $d \approx 1.2$. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension $d<2$.Comment: RevTex 6 pages, 7 figs, linked to arXiv:1003.5455[cs.SE]. Research at http://www.quantware.ups-tlse.fr/, Improved version, changed forma

Variational Principle underlying Scale Invariant Social Systems

MaxEnt's variational principle, in conjunction with Shannon's logarithmic information measure, yields only exponential functional forms in straightforward fashion. In this communication we show how to overcome this limitation via the incorporation, into the variational process, of suitable dynamical information. As a consequence, we are able to formulate a somewhat generalized Shannonian Maximum Entropy approach which provides a unifying "thermodynamic-like" explanation for the scale-invariant phenomena observed in social contexts, as city-population distributions. We confirm the MaxEnt predictions by means of numerical experiments with random walkers, and compare them with some empirical data

Unravelling the size distribution of social groups with information theory on complex networks

The minimization of Fisher's information (MFI) approach of Frieden et al. [Phys. Rev. E {\bf 60} 48 (1999)] is applied to the study of size distributions in social groups on the basis of a recently established analogy between scale invariant systems and classical gases [arXiv:0908.0504]. Going beyond the ideal gas scenario is seen to be tantamount to simulating the interactions taking place in a network's competitive cluster growth process. We find a scaling rule that allows to classify the final cluster-size distributions using only one parameter that we call the competitiveness. Empirical city-size distributions and electoral results can be thus reproduced and classified according to this competitiveness, which also allows to correctly predict well-established assessments such as the "six-degrees of separation", which is shown here to be a direct consequence of the maximum number of stable social relationships that one person can maintain, known as Dunbar's number. Finally, we show that scaled city-size distributions of large countries follow the same universal distribution

Heavy-Tailed Distribution of Cyber-Risks

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.Comment: 9 pages, 3 figure

Treatment of MOG-IgG-associated disorder with rituximab: An international study of 121 patients

OBJECTIVE: To assess the effect of anti-CD20 B-cell depletion with rituximab (RTX) on relapse rates in myelin oligodendrocyte glycoprotein antibody-associated disorder (MOGAD). METHODS: Retrospective review of RTX-treated MOGAD patients from 29 centres in 13 countries. The primary outcome measure was change in relapse rate after starting rituximab (Poisson regression model). RESULTS: Data on 121 patients were analysed, including 30 (24.8%) children. Twenty/121 (16.5%) were treated after one attack, of whom 14/20 (70.0%) remained relapse-free after median (IQR) 11.2 (6.3-14.1) months. The remainder (101/121, 83.5%) were treated after two or more attacks, of whom 53/101 (52.5%) remained relapse-free after median 12.1 (6.3-24.9) months. In this 'relapsing group', relapse rate declined by 37% (95%CI=19-52%, p<0.001) overall, 63% (95%CI=35-79%, p = 0.001) when RTX was used first line (n = 47), and 26% (95%CI=2-44%, p = 0.038) when used after other steroid-sparing immunotherapies (n = 54). Predicted 1-year and 2-year relapse-free survival was 79% and 55% for first-line RTX therapy, and 38% and 18% for second-/third-line therapy. Circulating CD19+B-cells were suppressed to <1% of total circulating lymphocyte population at the time of 45/57 (78.9%) relapses. CONCLUSION: RTX reduced relapse rates in MOGAD. However, many patients continued to relapse despite apparent B-cell depletion. Prospective controlled studies are needed to validate these results