The knowledge of the structure of a financial network gives valuable information for the estimation of systemic risk and other important properties. However, since financial data are typically subject to confidentiality, network reconstruction techniques become necessary to infer both the presence of connections (i.e. the topology) and their intensity (i.e. the link weights) from partial information. Recently, various horse races have been conducted to compare the performance of these methods. These comparisons were based on arbitrarily chosen metrics of network similarity. Here we establish a generalised likelihood approach to the rigorous definition, estimation and comparison of methods of reconstruction of weighted networks. The crucial ingredient is the possibility of separately constraining the topology and the link weights to follow certain "tight" empirical relationships that connect them to the observed marginals. We find that a statistically rigorous estimation of the probability of each link weight should incorporate the whole distribution of reconstructed network topologies, and we provide a general procedure to do this. Our results indicate that the best method is obtained by "dressing" the best-performing method available to date with a certain exponential distribution of link weights. The method is fast, unbiased and accurate and reproduces the empirical networks with highest generalised likelihood.Network reconstruction is an active field of research within the broader field of complex networks [20]. Addressing the network reconstruction problem means facing the double challenge represented by the estimation of topology and link weights. The task at hand consists in determining both binary and weighted ensemble distributions, and to understand the interplay between them. Among the methods proposed so far, some assume that the binary and weighted constraints jointly determine the final configuration in terms of both topology and weights while others attribute weights to the binary configuration using a completely separate methodology [17,21]. Amidst the former ones, a special mention is deserved by the Enhanced Configuration Model [15] . This is defined by simultaneously constraining the degrees and the strengths of nodes which jointly affect the estimation of the two sets of quantities, the linkage probabilities and the weight estimates. Since these are jointly determined on the basis of the same information (i.e. constraints), this implies the impossibility to include purely topological additional information. Examples of algorithms belonging to the second group are those iteratively adjusting the link weights (e.g. via the RAS recipe [19]) on top of some previously-determined topological structure, in such a way to satisfy the constraints concerning strengths a posteriori. This approach has encountered critiques in [16]. It is important to notice that this kind of procedure assigns weights deterministically, and therefore the likelihood of observing any real matrix is exactly zero, assuming conti...

Link-prediction is an active research field within network theory, aiming at uncovering missing connections or predicting the emergence of future relationships from the observed network structure. This paper represents our contribution to the stream of research concerning missing links prediction. Here, we propose an entropy-based method to predict a given percentage of missing links, by identifying them with the most probable non-observed ones. The probability coefficients are computed by solving opportunely defined null-models over the accessible network structure. Upon comparing our likelihood-based, local method with the most popular algorithms over a set of economic, financial and food networks, we find ours to perform best, as pointed out by a number of statistical indicators (e.g. the precision, the area under the ROC curve, etc.). Moreover, the entropy-based formalism adopted in the present paper allows us to straightforwardly extend the link-prediction exercise to directed networks as well, thus overcoming one of the main limitations of current algorithms. The higher accuracy achievable by employing these methods -together with their larger flexibilitymakes them strong competitors of available link-prediction algorithms.

In this paper a new approach to Coordinated Universal Time (UTC) calculation is presented by means of the Kalman filter. An ensemble of atomic clocks participating in UTC is selected for analyzing and testing the potentiality of this new method.

Introduction. Network reconstruction is an active field of research. Among the methods proposed so far, some assume that the binary and weighted constraints jointly determine the reconstruction output; others consider the weights estimation step as completely unrelated to the binary one. Amidst the former ones, a special mention is deserved by the Enhanced Configuration Model; the algorithms of the second group, instead, are those iteratively adjusting the link weights on top of some previouslydetermined topology.Methods and Results. Here we develop a theoretical framework that provides an analytical, unbiased procedure to estimate the weighted structure of a network, once its topology has been determined, thus extending the Exponential Random Graph (ERG) recipe. Our approach treats the information about the topological structure as a priori; together with the proper weighted constraints, it represents the input of our generalized reconstruction procedure. The probability distribution describing link weights is, then, determined by maximizing the key quantity of our algorithm, i.e., the conditional entropy under a properly-defined set of constraints. This algorithm returns a conditional probability distribution depending on a vector of unknown parameters; in alignment with previous results, their estimation is carried out by considering a generalized likelihood function. In our work, we compare two possible specifications of this framework.Conclusions. The knowledge of the structure of a financial network gives valuable information for estimating the systemic risk. However, since financial data 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 these methods. Here, we establish a generalised likelihood approach to rigorously define and compare methods for reconstructing weighted networks: the best one is obtained by "dressing" the bestperforming, available binary method with an exponential distribution of weights.