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The capacity to collect fingerprints of individuals in online media has revolutionized the way researchers explore human society. Social systems can be seen as a nonlinear superposition of a multitude of complex social networks, where nodes represent individuals and links capture a variety of different social relations. Much emphasis has been put on the network topology of social interactions, however, the multidimensional nature of these interactions has largely been ignored, mostly because of lack of data. Here, for the first time, we analyze a complete, multirelational, large social network of a society consisting of the 300,000 odd players of a massive multiplayer online game. We extract networks of six different types of one-to-one interactions between the players. Three of them carry a positive connotation (friendship, communication, trade), three a negative (enmity, armed aggression, punishment). We first analyze these types of networks as separate entities and find that negative interactions differ from positive interactions by their lower reciprocity, weaker clustering, and fatter-tail degree distribution. We then explore how the interdependence of different network types determines the organization of the social system. In particular, we study correlations and overlap between different types of links and demonstrate the tendency of individuals to play different roles in different networks. As a demonstration of the power of the approach, we present the first empirical large-scale verification of the long-standing structural balance theory, by focusing on the specific multiplex network of friendship and enmity relations. complex networks | multiplex relations | quantitative sociology doi/ 10.1073/pnas.1004008107

The remarkable robustness of many social systems has been associated with a peculiar triangular structure in the underlying social networks. Triples of people that have three positive relations (e.g., friendship) between each other are strongly overrepresented. Triples with two negative relations (e.g., enmity) and one positive relation are also overrepresented, and triples with one or three negative relations are drastically suppressed. For almost a century, the mechanism behind these very specific ("balanced") triad statistics remained elusive. Here, we propose a simple realistic adaptive network model, where agents tend to minimize social tension that arises from dyadic interactions. Both opinions of agents and their signed links (positive or negative relations) are updated in the dynamics. The key aspect of the model resides in the fact that agents only need information about their local neighbors in the network and do not require (often unrealistic) higher-order network information for their relation and opinion updates. We demonstrate the quality of the model on detailed temporal relation data of a society of thousands of players of a massive multiplayer online game where we can observe triangle formation directly. It not only successfully predicts the distribution of triangle types but also explains empirical group size distributions, which are essential for social cohesion. We discuss the details of the phase diagrams behind the model and their parameter dependence, and we comment on to what extent the results might apply universally in societies. [ABSTRACT FROM AUTHOR]

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, Dc , below which linear growth and low infection prevalence must occur. Above Dc 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 Dc ~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. [ABSTRACT FROM AUTHOR]

There are practically no quantitative tools for understanding how much stress a health care system can absorb before it loses its ability to provide care. We propose to measure the resilience of health care systems with respect to changes in the density of primary care providers. We develop a computational model on a 1-to-1 scale for a countrywide primary care sector based on patientsharing networks. Nodes represent all primary care providers in a country; links indicate patient flows between them. The removal of providers could cause a cascade of patient displacements, as patients have to find alternative providers. The model is calibrated with nationwide data from Austria that includes almost all primary care contacts over 2 y. We assign 2 properties to every provider: the "CareRank" measures the average number of displacements caused by a provider's removal (systemic risk) as well as the fraction of patients a provider can absorb when others default (systemic benefit). Below a critical number of providers, large-scale cascades of patient displacements occur, and no more providers can be found in a given region. We quantify regional resilience as the maximum fraction of providers that can be removed before cascading events prevent coverage for all patients within a district. We find considerable regional heterogeneity in the critical transition point from resilient to nonresilient behavior. We demonstrate that health care resilience cannot be quantified by physician density alone but must take into account how networked systems respond and restructure in response to shocks. The approach can identify systemically relevant providers. [ABSTRACT FROM AUTHOR]

Languages are transmitted through channels created by kinship systems. Given sufficient time, these kinship channels can change the genetic and linguistic structure of populations. In traditional societies of eastern Indonesia, finely resolved cophylogenies of languages and genes reveal persistent movements between stable speech communities facilitated by kinship rules. When multiple languages are present in a region and postmarital residence rules encourage sustained directional movement between speech communities, then languages should be channeled along uniparental lines. We find strong evidence for this pattern in 982 individuals from 25 villages on two adjacent islands, where different kinship rules have been followed. Core groups of close relatives have stayed together for generations, while remaining in contact with, and marrying into, surrounding groups. Over time, these kinship systems shaped their gene and language phylogenies: Consistently following a postmarital residence rule turned social communities into speech communities. [ABSTRACT FROM AUTHOR]

Spatial patterning often occurs in ecosystems as a result of a selforganizing process caused by feedback between organisms and the physical environment. Here, we show that the spatial patterns observable in centuries-old Balinese rice terraces are also created by feedback between farmers' decisions and the ecology of the paddies, which triggers a transition from local to globalscale control of water shortages and rice pests. We propose an evolutionary game, based on local farmers' decisions that predicts specific power laws in spatial patterning that are also seen in a multispectral image analysis of Balinese rice terraces. The model shows how feedbacks between human decisions and ecosystem processes can evolve toward an optimal state in which total harvests are maximized and the system approaches Pareto optimality. It helps explain how multiscale cooperation from the community to the watershed scale could persist for centuries, and why the disruption of this self-organizing system by the Green Revolution caused chaos in irrigation and devastating losses from pests. The model shows that adaptation in a coupled human-natural system can trigger self-organized criticality (SOC). In previous exogenously driven SOC models, adaptation plays no role, and no optimization occurs. In contrast, adaptive SOC is a self-organizing process where local adaptations drive the system toward local and global optima. [ABSTRACT FROM AUTHOR]

The contemporary global community is increasingly interdependent and confronted with systemic risks posed by the actions and interactions of actors existing beneath the level of formal institutions, often operating outside effective governance structures. Frequently, these actors are human agents, such as rogue traders or aggressive financial innovators, terrorists, groups of dissidents, or unauthorized sources of sensitive or secret information about government or private sector activities. In other instances, influential "actors" take the form of climate change, communications technologies, or socioeconomic globalization. Although these individual forces may be small relative to state governments or international institutions, or may operate on long time scales, the changes they catalyze can pose significant challenges to the analysis and practice of international relations through the operation of complex feedbacks and interactions of individual agents and interconnected systems. We call these challenges "femtorisks," and emphasize their importance for two reasons. First, in isolation, they may be inconsequential and semiautonomous; but when embedded in complex adaptive systems, characterized by individual agents able to change, learn from experience, and pursue their own agendas, the strategic interaction between actors can propel systems down paths of increasing, even global, instability. Second, because their influence stems from complex interactions at interfaces of multiple systems (e.g., social, financial, political, technological, ecological, etc.), femtorisks challenge standard approaches to risk assessment, as higher-order consequences cascade across the boundaries of socially constructed complex systems. We argue that new approaches to assessing and managing systemic risk in international relations are required, inspired by principles of evolutionary theory and development of resilient ecological systems. [ABSTRACT FROM AUTHOR]

The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there has been an ongoing controversy over whether the notion of the maximum entropy principle can be extended in a meaningful way to nonextensive, nonergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the (c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept for nonergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is to our knowledge the first exact derivation of a generalized entropy from the microscopic properties of a pathdependent random process. [ABSTRACT FROM AUTHOR]

Based on a unique dataset comprising all 325,000 Austrian patients that were under pharmaceutical treatment for diabetes during 2006 and 2007, we measured the excess risk of developing diabetes triggered by undernourishment in early life. We studied the percentage of all diabetes patients in the total population specifically for each year of birth, from 1917 to 2007. We found a massive excess risk of diabetes in people born during the times of the three major famines and immediately after, which occurred in Austria in the 20th century: 1918-1919, 1938, and 1946-1947. Depending on the region, there was an up to 40% higher chance of having diabetes when born in 1919-1921, compared with 1918 or 1922, where age-specific typical diabetes ratios are observed. The excess risk for diabetes was practically absent in those provinces of Austria that were less affected by the famines. We show that diabetes rates exhibit nontrivial, age-specific sex differences, and correlate with the economic wealth of the region. Our results might be of relevance for establishing higher awareness in the health system for those born in high-risk years, and underline the importance of ensuring sufficient nutrition in prenatal and early stages of life. [ABSTRACT FROM AUTHOR]

For statistical systems that violate one of the four Shannon-Khinchin axioms, entropy takes a more general form than the Boltzmann-Gibbs entropy. The framework of superstatistics allows one to for-mulate a maximum entropy principle with these generalized entro-pies, making them useful for understanding distribution functions of non-Markovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort prob-abilities, the other not. The two ways are connected through a dual-ity. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized log-arithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures β. so that the probability distribution is p(ϵi)∞ ∫0∞f(β)e -βϵidβ, where the "kernel" f(β) is nonnegative and normalized [∫f(β)dβ= 1]. We discuss the relation between this distribution and the generalized entropic form S = Σis(ρi). The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection, we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension. [ABSTRACT FROM AUTHOR]

REPLY TO KU'SMIERZ AND TOYOIZUMI: A network-based explanation of whymost COVID-19 infection curves are linear. As seen in figure 2A in ref. 2, the D c values for ER networks for different settings are clearly below the corresponding D c for SW networks. LETTER REPLY TO KU SMIERZ AND TOYOIZUMI: Anetwork-basedexplanationofwhymost COVID-19infectioncurvesarelinear Stefan Thurner a,b,c,1, Peter Klimek a,b , and Rudolf Hanel a,b  In a comment on our recent paper (1), the commentators point out that infection curves on small-world (SW) networks are linear only in the vicinity of the critical point (2) and that infection dynamics on complex or SW networks have already been studied, for instance,inref.3. [Extracted from the article]