This book is a comprehensive introduction to quantitative approaches to complex adaptive systems. Practically all areas of life on this planet are constantly confronted with complex systems, be it ecosystems, societies, traffic, financial markets, opinion formation, epidemic spreading, or the internet and social media. Complex systems are systems composed of many elements that interact with each other, which makes them extremely rich dynamical systems showing a huge range of phenomena. Properties of complex systems that are of particular importance are their efficiency, robustness, resilience, and proneness to collapse. The quantitative tools and concepts needed to understand the co-evolutionary nature of networked systems and their properties are challenging. The intention of the book is to give a self-contained introduction to these concepts so that the reader will be equipped with a conceptual and mathematical toolset that allows her to engage in the science of complex systems. Topics covered include random processes of path-dependent processes, co-evolutionary dynamics, the statistics of driven nonequilibrium systems, dynamics of networks, the theory of scaling, and approaches from statistical mechanics and information theory. The book extends well beyond the early classical literature in the field of complex systems and summarizes the methodological progress over the past twenty years in a clear, structured, and comprehensive way. The book is intended for natural scientists and graduate students.

Many countries have passed their first COVID-19 epidemic peak. Traditional epidemiological models describe this as a result of nonpharmaceutical interventions pushing the growth rate below the recovery rate. In this phase of the pandemic many countries showed an almost linear growth of confirmed cases for extended time periods. This new containment regime is hard to explain by traditional models where either infection numbers grow explosively until herd immunity is reached or the epidemic is completely suppressed. Here we offer an explanation of this puzzling observation based on the structure of contact networks. We show that for any given transmission rate there exists a critical number of social contacts, $D_c$, below which linear growth and low infection prevalence must occur. Above $D_c$ traditional epidemiological dynamics take place, e.g., as in susceptible–infected–recovered (SIR) models. When calibrating our model to empirical estimates of the transmission rate and the number of days being contagious, we find $D_c\sim 7.2$. Assuming realistic contact networks with a degree of about 5, and assuming that lockdown measures would reduce that to household size (about 2.5), we reproduce actual infection curves with remarkable precision, without fitting or fine-tuning of parameters. In particular, we compare the United States and Austria, as examples for one country that initially did not impose measures and one that responded with a severe lockdown early on. Our findings question the applicability of standard compartmental models to describe the COVID-19 containment phase. The probability to observe linear growth in these is practically zero. Significance For many countries a plain-eye inspection of the COVID-19 infection curves reveals a remarkable linear growth over extended time periods. This observation is practically impossible to understand with traditional epidemiological models. These, to make them expressible in compact mathematical form, typically ignore the structure of real contact networks that are essential in the characteristic spreading dynamics of COVID-19. Here we show that by properly taking some relevant network features into account, linear growth can be naturally explained. Further, the effect of nonpharmaceutical interventions (NPIs), like national lockdowns, can be modeled with a remarkable degree of precision without fitting or fine-tuning of parameters.

Democratic societies are built around the principle of free and fair elections, and that each citizen’s vote should count equally. National elections can be regarded as large-scale social experiments, where people are grouped into usually large numbers of electoral districts and vote according to their preferences. The large number of samples implies statistical consequences for the polling results, which can be used to identify election irregularities. Using a suitable data representation, we find that vote distributions of elections with alleged fraud show a kurtosis substantially exceeding the kurtosis of normal elections, depending on the level of data aggregation. As an example, we show that reported irregularities in recent Russian elections are, indeed, well-explained by systematic ballot stuffing. We develop a parametric model quantifying the extent to which fraudulent mechanisms are present. We formulate a parametric test detecting these statistical properties in election results. Remarkably, this technique produces robust outcomes with respect to the resolution of the data and therefore, allows for cross-country comparisons.

An interrelation between cancer and thrombosis is known, but population-based studies on the risk of both arterial thromboembolism (ATE) and venous thromboembolism (VTE) have not been performed. International Classification of Disease 10th Revision (ICD-10) diagnosis codes of all publicly insured persons in Austria (0–90 years) were extracted from the Austrian Association of Social Security Providers dataset covering the years 2006–07 (n = 8 306 244). Patients with a history of cancer or active cancer were defined as having at least one ICD-10 ‘C’ diagnosis code, and patients with ATE and/or VTE as having at least one of I21/I24 (myocardial infarction), I63/I64 (stroke), I74 (arterial embolism), and I26/I80/I82 (venous thromboembolism) diagnosis code. Among 158 675 people with cancer, 8559 (5.4%) had an ATE diagnosis code and 7244 (4.6%) a VTE diagnosis code. In contrast, among 8 147 569 people without cancer, 69 381 (0.9%) had an ATE diagnosis code and 29 307 (0.4%) a VTE diagnosis code. This corresponds to age-stratified random-effects relative risks (RR) of 6.88 [95% confidence interval (CI) 4.81–9.84] for ATE and 14.91 (95% CI 8.90–24.95) for VTE. ATE proportion was highest in patients with urinary tract malignancies (RR: 7.16 [6.74–7.61]) and lowest in patients with endocrine cancer (RR: 2.49 [2.00–3.10]). The corresponding VTE proportion was highest in cancer of the mesothelium/soft tissue (RR: 19.35 [17.44–21.47]) and lowest in oropharyngeal cancer (RR: 6.62 [5.61–7.81]). The RR of both ATE and VTE are significantly higher in persons with cancer. Our population-level meta-data indicate a strong association between cancer, ATE and VTE, and support the concept of shared risk factors and pathobiology between these diseases. Relative risk of ATE and VTE in persons with a cancer diagnosis code versus persons without a cancer diagnosis code. Relative risk of ATE and VTE in persons with a cancer diagnosis code versus persons without a cancer diagnosis code