  In this paper we study the difficulty of counting nodes in a synchronous
dynamic network where nodes share the same identifier, they communicate by
using a broadcast with unlimited bandwidth and, at each synchronous round,
network topology may change. To count in such setting, it has been shown that
the presence of a leader is necessary. We focus on a particularly interesting
subset of dynamic networks, namely \textit{Persistent Distance} - ${\cal
G}($PD$)_{h}$, in which each node has a fixed distance from the leader across
rounds and such distance is at most $h$. In these networks the dynamic diameter
$D$ is at most $2h$. We prove the number of rounds for counting in ${\cal
G}($PD$)_{2}$ is at least logarithmic with respect to the network size $|V|$.
Thanks to this result, we show that counting on any dynamic anonymous network
with $D$ constant w.r.t. $|V|$ takes at least $D+ \Omega(\text{log}\, |V| )$
rounds where $\Omega(\text{log}\, |V|)$ represents the additional cost to be
payed for handling anonymity. At the best of our knowledge this is the fist non
trivial, i.e. different from $\Omega(D)$, lower bounds on counting in anonymous
interval connected networks with broadcast and unlimited bandwith.
