  We consider a system of composite bosons given by strongly bound fermion
pairs tunneling through sites that form a low-dimensional network. It has been
shown that the ground state of this system can have condensate-like properties
in the very dilute regime for two-dimensional lattices but displays
fermionization for one-dimensional lattices. Studying graphs with fractal
dimensions, we explore intermediate situations between these two cases and
observe a correlation between increasing dimension and increasing
condensate-like character. However, this is only the case for graphs for which
the average path length grows with power smaller than 1 in the number of sites,
and which have an unbounded circuit rank. We thus conjecture that these two
conditions are relevant for condensation of composite bosons in arbitrary
networks, and should be considered jointly with the well-established criterion
of high entanglement between constituents.
