  What do generic networks that have certain properties look like? We define
Relative Canonical Network ensembles as the ensembles that realize a property R
while being as indistinguishable as possible from a generic network ensemble.
This allows us to study the most generic features of the networks giving rise
to the property under investigation. To test the approach we apply it first to
the network measure "small-world-ness", thought to characterize small-world
networks. We find several phase transitions as we go to less and less generic
networks in which cliques and hubs emerge. Such features are not shared by
typical small-world networks, showing that high "small-world-ness" does not
characterize small-world networks as they are commonly understood. On the other
hand we see that for embedded networks, the average shortest path length and
total Euclidean link length are better at characterizing small-world networks,
with hubs that emerge as a defining feature at low genericity. We expect the
overall approach to have wide applicability for understanding network
properties of real world interest.
