  We analyze the 3-parameter family of random networks which are uniform on
networks with fixed number of edges, triangles, and nodes (between 33 and 66).
We find precursors of phase transitions which are known to be present in the
asymptotic node regime as the edge and triangle numbers are varied, and focus
on one of the discontinuous ones. By use of a natural edge flip dynamics we
determine nucleation barriers as a random network crosses the transition, in
analogy to the process a material undergoes when frozen or melted, and
characterize some of the stochastic properties of the network nucleation.
