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Due to the interconnectedness of financial entities, estimating certain keyproperties of a complex financial system (e.g. the implied level of systemicrisk) requires detailed information about the structure of the underlyingnetwork. However, since data about financial linkages are typically subject toconfidentiality, network reconstruction techniques become necessary to inferboth the presence of connections and their intensity. Recently, several "horseraces" have been conducted to compare the performance of the availablefinancial network reconstruction methods. These comparisons, however, werebased on arbitrarily-chosen similarity metrics between the real and thereconstructed network. Here we establish a generalised maximum-likelihoodapproach to rigorously define and compare weighted reconstruction methods. Ourgeneralization maximizes the conditional entropy to solve the problemrepresented by the fact that the density-dependent constraints required toreliably reconstruct the network are typically unobserved. The resultingapproach admits as input any reconstruction method for the purely binarytopology and, conditionally on the latter, exploits the available partialinformation to infer link weights. We find that the most reliable method isobtained by "dressing" the best-performing binary method with an exponentialdistribution of link weights having a properly density-corrected andlink-specific mean value and propose two unbiased (in the sense of maximumconditional entropy) variants of it. While the one named CReMA is perfectlygeneral (as a particular case, it can place optimal weights on a network whosetopology is known), the one named CReMB is recommended both in case of fulluncertainty about the network topology and if the existence of some links iscertain. In these cases, the CReMB is faster and reproduces empirical networkswith highest generalised likelihood.

A faster horse on a safer trail: generalized inference for the efficient reconstruction of weighted networks