  In discrete contexts such as the degree distribution for a graph,
\emph{scale-free} has traditionally been \emph{defined} to be \emph{power-law}.
We propose a reasonable interpretation of \emph{scale-free}, namely, invariance
under the transformation of $p$-thinning, followed by conditioning on being
positive.
  For each $\beta \in (1,2)$, we show that there is a unique distribution which
is a fixed point of this transformation; the distribution is power-law-$\beta$,
and different from the usual Yule--Simon power law-$\beta$ that arises in
preferential attachment models.
  In addition to characterizing these fixed points, we prove convergence
results for iterates of the transformation.
