  Scale-free percolation is a stochastic model for complex networks. In this
spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge
with probability depending on i.i.d.\ vertex weights and the Euclidean distance
$|x-y|$. Depending on the various parameters involved, we get a rich phase
diagram. We study graph distances and compare it to the Euclidean distance of
the vertices. Our main attention is on a regime where graph distances are
(poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the
logarithmic exponents. In the light tail regime, the correct exponent is
identified.
