  We analyze and exploit some scaling properties of the Affinity Propagation
(AP) clustering algorithm proposed by Frey and Dueck (2007). First we observe
that a divide and conquer strategy, used on a large data set hierarchically
reduces the complexity ${\cal O}(N^2)$ to ${\cal O}(N^{(h+2)/(h+1)})$, for a
data-set of size $N$ and a depth $h$ of the hierarchical strategy. For a
data-set embedded in a $d$-dimensional space, we show that this is obtained
without notably damaging the precision except in dimension $d=2$. In fact, for
$d$ larger than 2 the relative loss in precision scales like
$N^{(2-d)/(h+1)d}$. Finally, under some conditions we observe that there is a
value $s^*$ of the penalty coefficient, a free parameter used to fix the number
of clusters, which separates a fragmentation phase (for $s<s^*$) from a
coalescent one (for $s>s^*$) of the underlying hidden cluster structure. At
this precise point holds a self-similarity property which can be exploited by
the hierarchical strategy to actually locate its position. From this
observation, a strategy based on \AP can be defined to find out how many
clusters are present in a given dataset.
