  Network topology plays a key role in many phenomena, from the spreading of
diseases to that of financial crises. Whenever the whole structure of a network
is unknown, one must resort to reconstruction methods that identify the least
biased ensemble of networks consistent with the partial information available.
A challenging case, frequently encountered due to privacy issues in the
analysis of interbank flows and Big Data, is when there is only local
(node-specific) aggregate information available. For binary networks, the
relevant ensemble is one where the degree (number of links) of each node is
constrained to its observed value. However, for weighted networks the problem
is much more complicated. While the naive approach prescribes to constrain the
strengths (total link weights) of all nodes, recent counter-intuitive results
suggest that in weighted networks the degrees are often more informative than
the strengths. This implies that the reconstruction of weighted networks would
be significantly enhanced by the specification of both strengths and degrees, a
computationally hard and bias-prone procedure. Here we solve this problem by
introducing an analytical and unbiased maximum-entropy method that works in the
shortest possible time and does not require the explicit generation of
reconstructed samples. We consider several real-world examples and show that,
while the strengths alone give poor results, the additional knowledge of the
degrees yields accurately reconstructed networks. Information-theoretic
criteria rigorously confirm that the degree sequence, as soon as it is
non-trivial, is irreducible to the strength sequence. Our results have strong
implications for the analysis of motifs and communities and whenever the
reconstructed ensemble is required as a null model to detect higher-order
patterns.
