  Motivated by recent physics papers describing the formation of biological
transport networks we study a discrete model proposed by Hu and Cai consisting
of an energy consumption function constrained by a linear system on a graph.
For the spatially two-dimensional rectangular setting we prove the rigorous
continuum limit of the constrained energy functional as the number of nodes of
the underlying graph tends to infinity and the edge lengths shrink to zero
uniformly. The proof is based on reformulating the discrete energy functional
as a sequence of integral functionals and proving their $\Gamma$-converge
towards a continuum energy functional.
