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In the study of economic networks, econometric approaches interpret the traditional Gravity Model specification as the expected link weight coming from a probability distribution whose functional form can be chosen arbitrarily, while statistical-physics approaches construct maximum-entropy distributions of weighted graphs, constrained to satisfy a given set of measurable network properties. In a recent, companion paper, we integrated the two approaches and applied them to the World Trade Web, i.e. the network of international trade among world countries. While the companion paper dealt only with discrete-valued link weights, the present paper extends the theoretical framework to continuous-valued link weights. In particular, we construct two broad classes of maximum-entropy models, namely the integrated and the conditional ones, defined by different criteria to derive and combine the probabilistic rules for placing links and loading them with weights. In the integrated models, both rules follow from a single, constrained optimization of the continuous Kullback–Leibler divergence; in the conditional models, the two rules are disentangled and the functional form of the weight distribution follows from a conditional, optimization procedure. After deriving the general functional form of the two classes, we turn each of them into a proper family of econometric models via a suitable identification of the econometric function relating the corresponding, expected link weights to macroeconomic factors. After testing the two classes of models on World Trade Web data, we discuss their strengths and weaknesses. • We consider the Minimum Discrimination Information Principle. • We derive each network model solving a constrained K-L divergence minimization. • These models are informed by structural constraints and economic factors. • The structural information encoded into the degrees cannot be sacrificed. • Weighted information, instead, can be accounted for by purely economic factors. [ABSTRACT FROM AUTHOR]

Networks of financial exposures are the key propagators of risk and distress among banks, but their empirical structure is not publicly available because of confidentiality. This limitation has triggered the development of methods of network reconstruction from partial, aggregate information. Unfortunately, even the best methods available fail in replicating the number of directed cycles, which on the other hand play a crucial role in determining graph spectra and hence the degree of network stability and systemic risk. Here we address this challenge by exploiting the hypothesis that the statistics of higher-order cycles is strongly constrained by that of the shortest ones, i.e. by the amount of dyads with reciprocated links. First, we provide a detailed analysis of link reciprocity on the e-MID dataset of Italian banks, finding that correlations between reciprocal links systematically increase with the temporal resolution, typically changing from negative to positive around a timescale of up to 50 days. Then, we propose a new network reconstruction method capable of enforcing, only from the knowledge of aggregate interbank assets and liabilities, both a desired sparsity and a desired link reciprocity. We confirm that the addition of reciprocity dramatically improves the prediction of several structural and spectral network properties, including the largest real eigenvalue and the eccentricity of the elliptical distribution of the other eigenvalues in the complex plane. These results illustrate the importance of correctly addressing the temporal resolution and the resulting level of reciprocity in the reconstruction of financial networks. [ABSTRACT FROM AUTHOR]