  The focus of this work is on estimation of the in-degree distribution in
directed networks from sampling network nodes or edges. A number of sampling
schemes are considered, including random sampling with and without replacement,
and several approaches based on random walks with possible jumps. When sampling
nodes, it is assumed that only the out-edges of that node are visible, that is,
the in-degree of that node is not observed. The suggested estimation of the
in-degree distribution is based on two approaches. The inversion approach
exploits the relation between the original and sample in-degree distributions,
and can estimate the bulk of the in-degree distribution, but not the tail of
the distribution. The tail of the in-degree distribution is estimated through
an asymptotic approach, which itself has two versions: one assuming a power-law
tail and the other for a tail of general form. The two estimation approaches
are examined on synthetic and real networks, with good performance results,
especially striking for the asymptotic approach.
