  We study the dynamics of the Bak-Sneppen model on small-world networks. For
each site in the network, we define a ``connectance,'' which measures the
distance to all other sites. We find radically different patterns of activity
for different sites, depending on their connectance and also on the topology of
the network. For a given network, the site with the minimal connectance shows
long periods of stasis interrupted by much smaller periods of activity. In
contrast, the activity pattern for the maximally connected site appears uniform
on the same time scale. We discuss the significance of these results for
speciation events.
