  Reconstructing weighted networks from partial information is necessary in
many important circumstances, e.g. for a correct estimation of systemic risk.
It has been shown that, in order to achieve an accurate reconstruction, it is
crucial to reliably replicate the empirical degree sequence, which is however
unknown in many realistic situations. More recently, it has been found that the
knowledge of the degree sequence can be replaced by the knowledge of the
strength sequence, which is typically accessible, complemented by that of the
total number of links, thus considerably relaxing the observational
requirements. Here we further relax these requirements and devise a procedure
valid when even the the total number of links is unavailable. We assume that,
apart from the heterogeneity induced by the degree sequence itself, the network
is homogeneous, so that its (global) link density can be estimated by sampling
subsets of nodes with representative density. We show that the best way of
sampling nodes is the random selection scheme, any other procedure being biased
towards unrealistically large, or small, link densities. We then introduce our
core technique for reconstructing both the topology and the link weights of the
unknown network in detail. When tested on real economic and financial data
sets, our method achieves a remarkable accuracy and is very robust with respect
to the sampled subsets, thus representing a reliable practical tool whenever
the available topological information is restricted to small portions of nodes.
