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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]

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]

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]

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]

Community detection techniques are widely used to infer hidden structures within interconnected systems. Despite demonstrating high accuracy on benchmarks, they reproduce the external classification for many real-world systems with a significant level of discrepancy. A widely accepted reason behind such outcome is the unavoidable loss of non-topological information (such as node attributes) encountered when the original complex system is converted to a network. In this article we systematically show that the observed discrepancies may also be caused by a different reason: the external classification itself. For this end we use scientific publication data which (i) exhibit a well defined modular structure and (ii) hold an expert-made classification of research articles. Having represented the articles and the extracted scientific concepts both as a bipartite network and as its unipartite projection, we applied modularity optimization to uncover the inner thematic structure. The resulting clusters are shown to partly reflect the author-made classification, although some significant discrepancies are observed. A detailed analysis of these discrepancies shows that they may carry essential information about the system, mainly related to the use of similar techniques and methods across different (sub)disciplines, that is otherwise omitted when only the external classification is considered. [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]

In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the world trade web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models. [ABSTRACT FROM AUTHOR]

The interplay between topology and dynamics in complex networks is a fundamental but widely unexplored problem. Here, we study this phenomenon on a prototype model in which the network is shaped by a dynamical variable. We couple the dynamics of the Bak–Sneppen evolution model with the rules of the so-called fitness network model for establishing the topology of a network; each vertex is assigned a ‘fitness’, and the vertex with minimum fitness and its neighbours are updated in each iteration. At the same time, the links between the updated vertices and all other vertices are drawn anew with a fitness-dependent connection probability. We show analytically and numerically that the system self-organizes to a non-trivial state that differs from what is obtained when the two processes are decoupled. A power-law decay of dynamical and topological quantities above a threshold emerges spontaneously, as well as a feedback between different dynamical regimes and the underlying correlation and percolation properties of the network. [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]

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]

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]

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]