Modelling maximum cyber incident losses of German organisations: an empirical study and modified extreme value distribution approach.

Cyber incidents are among the most critical business risks for organisations and can lead to large financial losses. However, previous research on loss modelling is based on unassured data sources because the representativeness and completeness of op-risk databases cannot be assured. Moreover, there is a lack of modelling approaches that focus on the tail behaviour and adequately account for extreme losses. In this paper, we introduce a novel 'tempered' generalised extreme value (GEV) approach. Based on a stratified random sample of 5000 interviewed German organisations, we model different loss distributions and compare them to our empirical data using graphical analysis and goodness-of-fit tests. We differentiate various subsamples (industry, size, attack type, loss type) and find our modified GEV outperforms other distributions, such as the lognormal and Weibull distributions. Finally, we calculate losses for the German economy, present application examples, derive implications as well as discuss the comparison of loss estimates in the literature.

The scaling of human interactions with city size.

The size of cities is known to play a fundamental role in social and economic life. Yet, its relation to the structure of the underlying network of human interactions has not been investigated empirically in detail. In this paper, we map society-wide communication networks to the urban areas of two European countries. We show that both the total number of contacts and the total communication activity grow superlinearly with city population size, according to well-defined scaling relations and resulting from a multiplicative increase that affects most citizens. Perhaps surprisingly, however, the probability that an individual's contacts are also connected with each other remains largely unaffected. These empirical results predict a systematic and scale-invariant acceleration of interaction-based spreading phenomena as cities get bigger, which is numerically confirmed by applying epidemiological models to the studied networks. Our findings should provide a microscopic basis towards understanding the superlinear increase of different socioeconomic quantities with city size, that applies to almost all urban systems and includes, for instance, the creation of new inventions or the prevalence of certain contagious diseases.

Measuring degree-degree association in networks.

The Pearson correlation coefficient is commonly used for quantifying the global level of degree-degree association in complex networks. Here, we use a probabilistic representation of the underlying network structure for assessing the applicability of different association measures to heavy-tailed degree distributions. Theoretical arguments together with our numerical study indicate that Pearson's coefficient often depends on the size of networks with equal association structure, impeding a systematic comparison of real-world networks. In contrast, Kendall-Gibbons' τ{b} is a considerably more robust measure of the degree-degree association.