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The abundance of data about social relationships allows the human behavior to be analyzed as any other natural phenomenon. Here we focus on balance theory, stating that social actors tend to avoid establishing cycles with an odd number of negative links. This statement, however, can be supported only after a comparison with a benchmark. Since the existing ones disregard actors' heterogeneity, we extend Exponential Random Graphs to signed networks with both global and local constraints and employ them to assess the significance of empirical unbalanced patterns. We find that the nature of balance crucially depends on the null model: while homogeneous benchmarks favor the weak balance theory, according to which only triangles with one negative link should be under-represented, heterogeneous benchmarks favor the strong balance theory, according to which also triangles with all negative links should be under-represented. Biological networks, instead, display strong frustration under any benchmark, confirming that structural balance inherently characterizes social networks. According to balance theory, social actors avoid establishing cycles with an odd number of negative links. This statement can be supported only after a comparison with a benchmark. The authors find that the level of balance depends on the null-model employed: homogeneous ones favor the weak balance theory; heterogeneous ones favor the strong balance theory. [ABSTRACT FROM AUTHOR]

The asymptotic equivalence of canonical and microcanonical ensembles is a central concept in statistical physics, with important consequences for both theoretical research and practical applications. However, this property breaks down under certain circumstances. The most studied violation of ensemble equivalence requires phase transitions, in which case it has a ‘restricted’ (i.e. confined to a certain region in parameter space) but ‘strong’ (i.e. characterized by a difference between the entropies of the two ensembles that is of the same order as the entropies themselves) form. However, recent research on networks has shown that the presence of an extensive number of local constraints can lead to ensemble nonequivalence (EN) even in the absence of phase transitions. This occurs in a ‘weak’ (i.e. leading to a subleading entropy difference) but remarkably ‘unrestricted’ (i.e. valid in the entire parameter space) form. Here we look for more general manifestations of EN in arbitrary ensembles of matrices with given margins. These models have widespread applications in the study of spatially heterogeneous and/or temporally nonstationary systems, with consequences for the analysis of multivariate financial and neural time-series, multi-platform social activity, gene expression profiles and other big data. We confirm that EN appears in ‘unrestricted’ form throughout the entire parameter space due to the extensivity of local constraints. Surprisingly, at the same time it can also exhibit the ‘strong’ form. This novel, simultaneously ‘strong and unrestricted’ form of nonequivalence is very robust and imposes a principled choice of the ensemble. We calculate the proper mathematical quantities to be used in real-world applications. [ABSTRACT FROM AUTHOR]

For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (‘hard constraint’), while the canonical ensemble requires the constraint to be met only on average (‘soft constraint’). It is known that breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-zero specific relative entropy of the two ensembles. In this paper we analyse a formula for the relative entropy of generic discrete random structures recently put forward by Squartini and Garlaschelli. We consider the case of a random graph with a given degree sequence (configuration model), and show that in the dense regime this formula correctly predicts that the specific relative entropy is determined by the scaling of the determinant of the matrix of canonical covariances of the constraints. The formula also correctly predicts that an extra correction term is required in the sparse regime and in the ultra-dense regime. We further show that the different expressions correspond to the degrees in the canonical ensemble being asymptotically Gaussian in the dense regime and asymptotically Poisson in the sparse regime (the latter confirms what we found in earlier work), and the dual degrees in the canonical ensemble being asymptotically Poisson in the ultra-dense regime. In general, we show that the degrees follow a multivariate version of the Poisson-Binomial distribution in the canonical ensemble. [ABSTRACT FROM AUTHOR]

Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers. In real-world multiplexes, it is expected that the observed inter-layer overlap may result partly from spurious correlations arising from the heterogeneity of nodes, and partly from true inter-layer dependencies. It is therefore important to consider rigorous ways to disentangle these two effects. In this paper, we introduce an unbiased maximum entropy model of multiplexes with controllable intra-layer node degrees and controllable inter-layer overlap. The model can be mapped to a generalized Ising model, where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that node heterogeneity favors the splitting of critical points characterizing different pairs of nodes, leading to link-specific phase transitions that may, in turn, increase the overlap. By quantifying how the overlap can be increased by increasing either the intra-layer node heterogeneity (spurious correlation) or the strength of the inter-layer coupling (true correlation), the model allows us to disentangle the two effects. As an application, we show that the empirical overlap observed in the International Trade Multiplex genuinely requires a nonzero inter-layer coupling in its modeling, as it is not merely a spurious result of the correlation between node degrees across different layers. [ABSTRACT FROM AUTHOR]

Crowded trades by similarly trading peers influence the dynamics of asset prices, possibly creating systemic risk. We propose a market clustering measure using granular trading data. For each stock, the clustering measure captures the degree of trading overlap among any two investors in that stock, based on a comparison with the expected crowding in a null model where trades are maximally random while still respecting the empirical heterogeneity of both stocks and investors. We investigate the effect of crowded trades on stock price stability and present evidence that market clustering has a causal effect on the properties of the tails of the stock return distribution, particularly the positive tail, even after controlling for commonly considered risk drivers. Reduced investor pool diversity could thus negatively affect stock price stability. [ABSTRACT FROM AUTHOR]

Due to the interconnectedness of financial entities, estimating certain key properties of a complex financial system, including the implied level of systemic risk, requires detailed information about the structure of the underlying network of dependencies. However, since data about financial linkages are typically subject to confidentiality, network reconstruction techniques become necessary to infer both the presence of connections and their intensity. Recently, several 'horse races' have been conducted to compare the performance of the available financial network reconstruction methods. These comparisons were based on arbitrarily chosen metrics of similarity between the real network and its reconstructed versions. Here we establish a generalized maximum-likelihood approach to rigorously define and compare weighted reconstruction methods. Our generalization uses the maximization of a certain conditional entropy to solve the problem represented by the fact that the density-dependent constraints required to reliably reconstruct the network are typically unobserved and, therefore, cannot enter directly, as sufficient statistics, in the likelihood function. The resulting approach admits as input any reconstruction method for the purely binary topology and, conditionally on the latter, exploits the available partial information to infer link weights. We find that the most reliable method is obtained by 'dressing' the best-performing binary method with an exponential distribution of link weights having a properly density-corrected and link-specific mean value and propose two safe (i.e. unbiased in the sense of maximum conditional entropy) variants of it. While the one named CReMA is perfectly general (as a particular case, it can place optimal weights on a network if the bare topology is known), the one named CReMB is recommended both in case of full uncertainty about the network topology and if the existence of some links is certain. In these cases, the CReMB is faster and reproduces empirical networks with highest generalized likelihood among the considered competing models. [ABSTRACT FROM AUTHOR]

Recent crises have shown that the knowledge of the structure of input–output networks, at the firm level, is crucial when studying economic resilience from the microscopic point of view of firms that try to rewire their connections under supply and demand constraints. Unfortunately, empirical inter-firm network data are protected by confidentiality, hence rarely accessible. The available methods for network reconstruction from partial information treat all pairs of nodes as potentially interacting, thereby overestimating the rewiring capabilities of the system and the implied resilience. Here, we use two big data sets of transactions in the Netherlands to represent a large portion of the Dutch inter-firm network and document its properties. We, then, introduce a generalized maximum-entropy reconstruction method that preserves the production function of each firm in the data, i.e. the input and output flows of each node for each product type. We confirm that the new method becomes increasingly more reliable in reconstructing the empirical network as a finer product resolution is considered and can, therefore, be used as a realistic generative model of inter-firm networks with fine production constraints. Moreover, the likelihood of the model directly enumerates the number of alternative network configurations that leave each firm in its current production state, thereby estimating the reduction in the rewiring capability of the system implied by the observed input–output constraints. [ABSTRACT FROM AUTHOR]

In the context of agent based modeling and network theory, we focus on the problem of recovering behavior-related choice information from origin-destination type data, a topic also known under the name of network tomography. As a basis for predicting agents' choices we emphasize the connection between adaptive intelligent behavior, causal entropy maximization, and self-organized behavior in an open dynamic system. We cast this problem in the form of binary and weighted networks and suggest information theoretic entropy-driven methods to recover estimates of the unknown behavioral flow parameters. Our objective is to recover the unknown behavioral values across the ensemble analytically, without explicitly sampling the configuration space. In order to do so, we consider the Cressie-Read family of entropic functionals, enlarging the set of estimators commonly employed to make optimal use of the available information. More specifically, we explicitly work out two cases of particular interest: Shannon functional and the likelihood functional. We then employ them for the analysis of both univariate and bivariate data sets, comparing their accuracy in reproducing the observed trends. [ABSTRACT FROM AUTHOR]

Ecologists have long debated the properties that confer stability to complex, species-rich ecological networks. Species-level soil food webs are large and structured networks of central importance to ecosystem functioning. Here, we conducted an analysis of the stability properties of an up-to-date set of theoretical soil food web models that account both for realistic levels of species richness and the most recent views on the topological structure (who is connected to whom) of these food webs. The stability of the network was best explained by two factors: strong correlations between interaction strengths and the blocked, nonrandom trophic structure of the web. These two factors could stabilize our model food webs even at the high levels of species richness that are typically found in soil, and that would make random systems very unstable. Also, the stability of our soil food webs is well-approximated by the cascade model. This result suggests that stability could emerge from the hierarchical structure of the functional organization of the web. Our study shows that under the assumption of equilibrium and small perturbations, theoretical soil food webs possess a topological structure that allows them to be complex yet more locally stable than their random counterpart. In particular, results strongly support the general hypothesis that the stability of rich and complex soil food webs is mostly driven by correlations in interaction strength and the organization of the soil food web into functional groups. The implication is that in real-world food web, any force disrupting the functional structure and distribution pattern of interaction strengths (i.e., energy fluxes) of the soil food webs will destabilize the dynamics of the system, leading to species extinction and major changes in the relative abundances of species. [ABSTRACT FROM AUTHOR]

Many real networks feature the property of nestedness, i.e. the neighbours of nodes with a few connections are hierarchically nested within the neighbours of nodes with more connections. Despite the abstract simplicity of this notion, various mathematical definitions of nestedness have been proposed, sometimes giving contrasting results. Moreover, there is an ongoing debate on the statistical significance of nestedness, since random networks where the number of connections (degree) of each node is fixed to its empirical value are typically as nested as real ones. By using only ergodic and unbiased null models, we propose a clarification that exploits the recent finding that random networks where the degrees are enforced as hard constraints (microcanonical ensembles) are thermodynamically different from random networks where the degrees are enforced as soft constraints (canonical ensembles). Indeed, alternative definitions of nestedness can be negatively correlated in the microcanonical one, while being positively correlated in the canonical one. This result disentangles distinct notions of nestedness captured by different metrics and highlights the importance of making a principled choice between hard and soft constraints in null models of ecological networks. [ABSTRACT FROM AUTHOR]

The concepts of economic fitness and complexity, based on iterative and interdependent definitions of the quality of exporting countries and exported products, have led to novel insights into the dynamics of production and trade. A key step in the calculation of these quantities is the preliminary identification of statistically relevant country-product pairs.In this paper, we propose a method that could improve the current practice of filtering based on the revealed comparative advantage, by employing the maximum-entropy principle to construct an unbiased link weight probability distribution that, unlike the traditional thresholding method, allows for the statistical assessment of empirical trade volumes. The result is an adjusted geometric distribution for trade links that refines the revealed comparative advantage approach. This allows us to define the statistical significance of each trade link weight, leading to statistically supported trade link filtering decisions. Using this statistically justified filtering method, we have obtained results that are similar in nature to those that were found without this method, even though there are significant deviations in the details. In addition, the statistical information thus obtained on each trade link allows us to perform a spectral analysis of the export portfolio of individual economies. [ABSTRACT FROM AUTHOR]

Neural systems are organized in a modular way, serving multiple functionalities. This multiplicity requires that both positive (e.g. excitatory, phase-coherent) and negative (e.g. inhibitory, phase-opposing) interactions take place across brain modules. Unfortunately, most methods to detect modules from time series either neglect or convert to positive, any measured negative correlation. This may leave a significant part of the sign-dependent functional structure undetected. Here we present a novel method, based on random matrix theory, for the identification of sign-dependent modules in the brain. Our method filters out both local (unit-specific) noise and global (system-wide) dependencies that typically obfuscate the presence of such structure. The method is guaranteed to identify an optimally contrasted functional ‘signature’, i.e. a partition into modules that are positively correlated internally and negatively correlated across. The method is purely data-driven, does not use any arbitrary threshold or network projection, and outputs only statistically significant structure. In measurements of neuronal gene expression in the biological clock of mice, the method systematically uncovers two otherwise undetectable, negatively correlated modules whose relative size and mutual interaction strength are found to depend on photoperiod. [ABSTRACT FROM AUTHOR]

Abstract The study of social, economic and biological systems is often (when not always) limited by the partial information about the structure of the underlying networks. An example of paramount importance is provided by financial systems: information on the interconnections between financial institutions is privacy-protected, dramatically reducing the possibility of correctly estimating crucial systemic properties such as the resilience to the propagation of shocks. The need to compensate for the scarcity of data, while optimally employing the available information, has led to the birth of a research field known as network reconstruction. Since the latter has benefited from the contribution of researchers working in disciplines as different as mathematics, physics and economics, the results achieved so far are still scattered across heterogeneous publications. Most importantly, a systematic comparison of the network reconstruction methods proposed up to now is currently missing. This review aims at providing a unifying framework to present all these studies, mainly focusing on their application to economic and financial networks. [ABSTRACT FROM AUTHOR]

For animals living in temperate latitudes, seasonal changes in day length are an important cue for adaptations of their physiology and behavior to the altered environmental conditions. The suprachiasmatic nucleus (SCN) is known as the central circadian clock in mammals, but may also play an important role in adaptations to different photoperiods. The SCN receives direct light input from the retina and is able to encode day-length by approximating the waveform of the electrical activity rhythm to the duration of daylight. Changing the overall waveform requires a reorganization of the neuronal network within the SCN with a change in the degree of synchrony between the neurons; however, the underlying mechanisms are yet unknown. In the present study we used PER2::LUC bioluminescence imaging in cultured SCN slices to characterize network dynamics on the single-cell level and we aimed to provide evidence for a role of modulations in coupling strength in the photoperiodic-induced phase dispersal. Exposure to long photoperiod (LP) induced a larger distribution of peak times of the single-cell PER2::LUC rhythms in the anterior SCN, compared to short photoperiod. Interestingly, the cycle-to-cycle variability in single-cell period of PER2::LUC rhythms is also higher in the anterior SCN in LP, and is positively correlated with peak time dispersal. Applying a new, impartial community detection method on the time series data of the PER2::LUC rhythm revealed two clusters of cells with a specific spatial distribution, which we define as dorsolateral and ventromedial SCN. Post hoc analysis of rhythm characteristics of these clusters showed larger cycle-to-cycle single-cell period variability in the dorsolateral compared to the ventromedial cluster in the anterior SCN. We conclude that a change in coupling strength within the SCN network is a plausible explanation to the observed changes in single-cell period variability, which can contribute to the photoperiod-induced phase distribution. [ABSTRACT FROM AUTHOR]

In recent years, in Italy, the trend of the electricity demand and the need to connect a large number of renewable energy power generators to the power-grid, developed a novel type of energy transmission/distribution infrastructure. The Italian Transmission System Operator (TSO) and the Distribution System Operator (DSO), worked on a new infrastructural model, based on electronic meters and information technology. In pursuing this objective it is crucial importance to understand how even more larger shares of renewable energy can be fully integrated, providing a constant and reliable energy background over space and time. This is particularly true for intermittent sources as photovoltaic installations due to the fine-grained distribution of them across the Country. In this work we use an over-simplified model to characterize the Italian power grid as a graph whose nodes are Italian municipalities and the edges cross the administrative boundaries between a selected municipality and its first neighbours, following a Delaunay triangulation. Our aim is to describe the power flow as a diffusion process over a network, and using open data on the solar irradiation at the ground level, we estimate the production of photovoltaic energy in each node. An attraction index was also defined using demographic data, in accordance with average per capita energy consumption data. The available energy on each node was calculated by finding the stationary state of a generation-attraction model. [ABSTRACT FROM AUTHOR]

We address a fundamental problem that is systematically encountered when modeling real-world complex systems of societal relevance: the limitedness of the information available. In the case of economic and financial networks, privacy issues severely limit the information that can be accessed and, as a consequence, the possibility of correctly estimating the resilience of these systems to events such as financial shocks, crises and cascade failures. Here we present an innovative method to reconstruct the structure of such partially-accessible systems, based on the knowledge of intrinsic node-specific properties and of the number of connections of only a limited subset of nodes. This information is used to calibrate an inference procedure based on fundamental concepts derived from statistical physics, which allows to generate ensembles of directed weighted networks intended to represent the real system-so that the real network properties can be estimated as their average values within the ensemble. We test the method both on synthetic and empirical networks, focusing on the properties that are commonly used to measure systemic risk. Indeed, the method shows a remarkable robustness with respect to the limitedness of the information available, thus representing a valuable tool for gaining insights on privacy-protected economic and financial systems. [ABSTRACT FROM AUTHOR]

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in gener [ABSTRACT FROM AUTHOR]

A problem typically encountered when studying complex systems is the limitedness of the information available on their topology, which hinders our understanding of their structure and of the dynamical processes taking place on them. A paramount example is provided by financial networks, whose data are privacy protected: Banks publicly disclose only their aggregate exposure towards other banks, keeping individual exposures towards each single bank secret. Yet, the estimation of systemic risk strongly depends on the detailed structure of the interbank network. The resulting challenge is that of using aggregate information to statistically reconstruct a network and correctly predict its higher-order properties. Standard approaches either generate unrealistically dense networks, or fail to reproduce the observed topology by assigning homogeneous link weights. Here, we develop a reconstruction method, based on statistical mechanics concepts, that makes use of the empirical link density in a highly nontrivial way. Technically, our approach consists in the preliminary estimation of node degrees from empirical node strengths and link density, followed by a maximum-entropy inference based on a combination of empirical strengths and estimated degrees. Our method is successfully tested on the international trade network and the interbank money market, and represents a valuable tool tor gaining insights on privacy-protected or partially accessible systems. [ABSTRACT FROM AUTHOR]

The mesoscopic organization of complex systems, from financial markets to the brain, is an intermediate between the microscopic dynamics of individual units (stocks or neurons, in the mentioned cases), and the macroscopic dynamics of the system as a whole. The organization is determined by “communities” of units whose dynamics, represented by time series of activity, is more strongly correlated internally than with the rest of the system. Recent studies have shown that the binary projections of various financial and neural time series exhibit nontrivial dynamical features that resemble those of the original data. This implies that a significant piece of information is encoded into the binary projection (i.e. the sign) of such increments. Here, we explore whether the binary signatures of multiple time series can replicate the same complex community organization of the financial market, as the original weighted time series. We adopt a method that has been specifically designed to detect communities from cross-correlation matrices of time series data. Our analysis shows that the simpler binary representation leads to a community structure that is almost identical with that obtained using the full weighted representation. These results confirm that binary projections of financial time series contain significant structural information. [ABSTRACT FROM AUTHOR]

The article discusses the application of complexity theory and financial regulations in monitoring and managing the financial systems. It refers to a study which shows that standard analysis using a homogenous network model could lead to late detection of the 2008 crisis. It notes that generic empirical quantitative indicators of resilience may be used across complex systems to detect tipping points.

Reconstructing patterns of interconnections from partial information is one of the most important issues in the statistical physics of complex networks. A paramount example is provided by financial networks. In fact, the spreading and amplification of financial distress in capital markets are strongly affected by the interconnections among financial institutions. Yet, while the aggregate balance sheets of institutions are publicly disclosed, information on single positions is mostly confidential and, as such, unavailable. Standard approaches to reconstruct the network of financial interconnection produce unrealistically dense topologies, leading to a biased estimation of systemic risk. Moreover, reconstruction techniques are generally designed for monopartite networks of bilateral exposures between financial institutions, thus failing in reproducing bipartite networks of security holdings (e.g., investment portfolios). Here we propose a reconstruction method based on constrained entropy maximization, tailored for bipartite financial networks. Such a procedure enhances the traditional capital-asset pricing model (CAPM) and allows us to reproduce the correct topology of the network. We test this enhanced CAPM (ECAPM) method on a dataset, collected by the European Central Bank, of detailed security holdings of European institutional sectors over a period of six years (2009-2015). Our approach outperforms the traditional CAPM and the recently proposed maximum-entropy CAPM both in reproducing the network topology and in estimating systemic risk due to fire sales spillovers. In general, ECAPM can be applied to the whole class of weighted bipartite networks described by the fitness model. [ABSTRACT FROM AUTHOR]

Real-world multilayer networks feature nontrivial dependencies among links of different layers. Here we argue that if links are directed, then dependencies are twofold. Besides the ordinary tendency of links of different layers to align as the result of "multiplexity," there is also a tendency to antialign as a result of what we call "multireciprocity," i.e., the fact that links in one layer can be reciprocated by opposite links in a different layer. Multireciprocity generalizes the scalar definition of single-layer reciprocity to that of a square matrix involving all pairs of layers. We introduce multiplexity and multireciprocity matrices for both binary and weighted multiplexes and validate their statistical significance against maximum-entropy null models that filter out the effects of node heterogeneity. We then perform a detailed empirical analysis of the world trade multiplex (WTM), representing the import-export relationships between world countries in different commodities. We show that the WTM exhibits strong multiplexity and multireciprocity, an effect which is, however, largely encoded into the degree or strength sequences of individual layers. The residual effects are still significant and allow us to classify pairs of commodities according to their tendency to be traded together in the same direction and/or in opposite ones. We also find that the multireciprocity of the WTM is significantly lower than the usual reciprocity measured on the aggregate network. Moreover, layers with low (high) internal reciprocity are embedded within sets of layers with comparably low (high) mutual multireciprocity. This suggests that, in the WTM, reciprocity is inherent to groups of related commodities rather than to individual commodities. We discuss the implications for international trade research focusing on product taxonomies, the product space, and fitness and complexity metrics. [ABSTRACT FROM AUTHOR]

A response from the authors of the article "Complexity theory and financial regulation" which was published in the February 19, 2016 issue is presented.

In economic and financial networks, the strength of each node has always an important economic meaning, such as the size of supply and demand, import and export, or financial exposure. Constructing null models of networks matching the observed strengths of all nodes is crucial in order to either detect interesting deviations of an empirical network from economically meaningful benchmarks or reconstruct the most likely structure of an economic network when the latter is unknown. However, several studies have proved that real economic networks and multiplexes topologically differ from configurations inferred only from node strengths. Here we provide a detailed analysis of the world trade multiplex by comparing it to an enhanced null model that simultaneously reproduces the strength and the degree of each node. We study several temporal snapshots and almost 100 layers (commodity classes) of the multiplex and find that the observed properties are systematically well reproduced by our model. Our formalism allows us to introduce the (static) concept of extensive and intensive bias, defined as a measurable tendency of the network to prefer either the formation of extra links or the reinforcement of link weights, with respect to a reference case where only strengths are enforced. Our findings complement the existing economic literature on (dynamic) intensive and extensive trade margins. More generally, they show that real-world multiplexes can be strongly shaped by layer-specific local constraints. [ABSTRACT FROM AUTHOR]