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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]

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]

Real-world social and economic networks typically display a number of particular topological properties, such as a giant connected component, a broad degree distribution, the small-world property and the presence of communities of densely interconnected nodes. Several models, including ensembles of networks, also known in social science as Exponential Random Graphs, have been proposed with the aim of reproducing each of these properties in isolation. Here, we define a generalized ensemble of graphs by introducing the concept of graph temperature, controlling the degree of topological optimization of a network. We consider the temperature-dependent version of both existing and novel models and show that all the aforementioned topological properties can be simultaneously understood as the natural outcomes of an optimized, low-temperature topology. We also show that seemingly different graph models, as well as techniques used to extract information from real networks are all found to be particular low-temperature cases of the same generalized formalism. One such technique allows us to extend our approach to real weighted networks. Our results suggest that a low graph temperature might be a ubiquitous property of real socio-economic networks, placing conditions on the diffusion of information across these systems. [ABSTRACT FROM AUTHOR]

In order to detect patterns in real networks, randomized graph ensembles that preserve only part of the topology of an observed network are systematically used as fundamental null models. However, the generation of them is still problematic. Existing approaches are either computationally demanding and beyond analytic control or analytically accessible but highly approximate. Here, we propose a solution to this long-standing problem by introducing a fast method that allows one to obtain expectation values and standard deviations of any topological property analytically, for any binary, weighted, directed or undirected network. Remarkably, the time required to obtain the expectation value of any property analytically across the entire graph ensemble is as short as that required to compute the same property using the adjacency matrix of the single original network. Our method reveals that the null behavior of various correlation properties is different from what was believed previously, and is highly sensitive to the particular network considered. Moreover, our approach shows that important structural properties (such as the modularity used in community detection problems) are currently based on incorrect expressions, and provides the exact quantities that should replace them. [ABSTRACT FROM AUTHOR]

We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in economic space determines particular symmetries of the network, while the observed evolution of reciprocity is consistent with a symmetry breaking taking place in production space. Our results show that networks can be strongly affected by symmetry-breaking phenomena occurring in embedding spaces, and that stochastic network symmetries can successfully suggest, or rule out, possible underlying mechanisms. [ABSTRACT FROM AUTHOR]

In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques. [ABSTRACT FROM AUTHOR]

We introduce the weighted random graph (WRG) model, which represents the weighted counterpart of the Erdos-Renyi random graph and provides fundamental insights into more complicated weighted networks. We find analytically that the WRG is characterized by a geometric weight distribution, a binomial degree distribution and a negative binomial strength distribution. We also characterize exactly the percolation phase transitions associated with edge removal and with the appearance of weighted subgraphs of any order and intensity. We find that even this completely null model displays a percolation behaviour similar to what is observed in real weighted networks, implying that edge removal cannot be used to detect community structure empirically. By contrast, the analysis of clustering successfully reveals different patterns between the WRG and real networks. [ABSTRACT FROM AUTHOR]

The structure of ecological communities is usually represented by food webs[SUP1-3]. In these webs, we describe species by means of vertices connected by links representing the predations. We can therefore study different webs by considering the shape (topology) of these networks[SUP4,5]. Comparing food webs by searching for regularities is of fundamental importance, because universal patterns would reveal common principles underlying the organization of different ecosystems. However, features observed in small food webs[SUP1-3,6] are different from those found in large ones[SUP7-15] . Furthermore, food webs (except in isolated cases[SUP16,17]) do not share[SUP18,19] general features with other types of network (including the Internet, the World Wide Web and biological webs). These features are a small-world character[SUP4,5] and a scale-free (power-law) distribution of the degree[SUP4,5] (the number of links per vertex). Here we propose to describe food webs as transportation networks[SUP20] by extending to them the concept of allometric scaling[SUP20-22] (how branching properties change with network size). We then decompose food webs in spanning trees and loop-forming links. We show that, whereas the number of loops varies significantly across real webs, spanning trees are characterized by universal scaling relations. [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]

Although Camacho and Arenas raise potentially interesting points, we believe that some of their arguments are flawed or undermined by poor statistics, and therefore that they do not invalidate our results. [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]

The financial crisis clearly illustrated the importance of characterizing the level of 'systemic' risk associated with an entire credit network, rather than with single institutions. However, the interplay between financial distress and topological changes is still poorly understood. Here we analyze the quarterly interbank exposures among Dutch banks over the period 1998-2008, ending with the crisis. After controlling for the link density, many topological properties display an abrupt change in 2008, providing a clear - but unpredictable - signature of the crisis. By contrast, if the heterogeneity of banks' connectivity is controlled for, the same properties show a gradual transition to the crisis, starting in 2005 and preceded by an even earlier period during which anomalous debt loops could have led to the underestimation of counter-party risk. These early-warning signals are undetectable if the network is reconstructed from partial bank-specific data, as routinely done. We discuss important implications for bank regulatory policies. [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 article focuses the use of network theory in understanding the nature arbuscular mycorrhizal (AM) fungi-plant interactions. It discusses Chagnon and colleagues and Montesinos-Navarro and colleagues' research and arguments concerning bipartite networks. It also recall the basic theory on interaction matrix and the basic facts implying that co-occurrence ≠ interaction.

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]

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]

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]

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]

Network topology plays a key role in many phenomena, from the spreading of diseases to that of financial crises. Whenever the whole structure of a network is unknown, one must resort to reconstruction methods that identify the least biased ensemble of networks consistent with the partial information available. A challenging case, frequently encountered due to privacy issues in the analysis of interbank flows and Big Data, is when there is only local (node-specific) aggregate information available. For binary networks, the relevant ensemble is one where the degree (number of links) of each node is constrained to its observed value. However, for weighted networks the problem is much more complicated. While the na�ve approach prescribes to constrain the strengths (total link weights) of all nodes, recent counter-intuitive results suggest that in weighted networks the degrees are often more informative than the strengths. This implies that the reconstruction of weighted networks would be significantly enhanced by the specification of both strengths and degrees, a computationally hard and bias-prone procedure. Here we solve this problem by introducing an analytical and unbiased maximum-entropy method that works in the shortest possible time and does not require the explicit generation of reconstructed samples. We consider several real-world examples and show that, while the strengths alone give poor results, the additional knowledge of the degrees yields accurately reconstructed networks. Information-theoretic criteria rigorously confirm that the degree sequence, as soon as it is non-trivial, is irreducible to the strength sequence. Our results have strong implications for the analysis of motifs and communities and whenever the reconstructed ensemble is required as a null model to detect higher-order patterns. [ABSTRACT FROM AUTHOR]

An outstanding open problem is whether collective social phenomena occurring over short timescales can systematically reduce cultural heterogeneity in the long run, and whether offline and online human interactions contribute differently to the process. Theoretical models suggest that short-term collective behavior and long-term cultural diversity are mutually excluding, since they require very different levels of social influence. The latter jointly depends on two factors: the topology of the underlying social network and the overlap between individuals in multidimensional cultural space. However, while the empirical properties of social networks are intensively studied, little is known about the large-scale organization of real societies in cultural space, so that random input specifications are necessarily used in models. Here we use a large dataset to perform a high-dimensional analysis of the scientific beliefs of thousands of Europeans. We find that interopinion correlations determine a nontrivial ultrametric hierarchy of individuals in cultural space. When empirical data are used as inputs in models, ultrametricity has strong and counterintuitive effects. On short timescales, it facilitates a symmetry-breaking phase transition triggering coordinated social behavior. On long timescales, it suppresses cultural convergence by restricting it within disjoint groups. Moreover, ultrametricity implies that these results are surprisingly robust to modifications of the dynamical rules considered. Thus the empirical distribution of individuals in cultural space appears to systematically optimize the coexistence of short-term collective behavior and long-term cultural diversity, which can be realized simultaneously for the same moderate level of mutual influence in a diverse range of online and offline settings. [ABSTRACT FROM AUTHOR]