  The study of random graphs has become very popular for real-life network
modeling such as social networks or financial networks. Inhomogeneous
long-range percolation (or scale-free percolation) on the lattice $\mathbb
Z^d$, $d\ge1$, is a particular attractive example of a random graph model
because it fulfills several stylized facts of real-life networks. For this
model various geometric properties such as the percolation behavior, the degree
distribution and graph distances have been analyzed. In the present paper we
complement the picture about graph distances. Moreover, we prove continuity of
the percolation probability in the phase transition point.
