  In this Letter we find numerically the strong finite-size effects in the
critical behavior of Erd\H{o}s-R\'enyi (ER) networks supplemented with chemical
potentials for some motifs, in particular 2-stars and triangles. For the 2-star
model above the critical value of the chemical potential a ground state looks
as star-like graph with the finite set of hubs at ER parameter $p<0.5$ or as
the single cluster at $p>0.5$. It is found that there exists the critical value
of number of nodes $N^{*}(p)$ when the ground state undergoes clear-cut
crossover and at $N>N^{*}(p)$ the network flows via a cluster evaporation to
the state involving the small star in the ER environment. The similar
evaporation of the cluster takes place at $N>N^{*}(p)$ in the Strauss model. We
suggest that the entropic trap mechanism is relevant for microscopic mechanism
behind the crossover regime. The possible analogies concerning the strong
entropic finite-size effects in the holographic description of matrix black
hole (BH) formation and evaporation are mentioned.
