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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]

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]