  We propose a statistical model defined on the three-dimensional diamond
network where the splitting of randomly selected nodes leads to a spatially
disordered network, with decreasing degree of connectivity. The terminal state,
that is reached when all nodes have been split, is a dense configuration of
self-avoiding walks on the diamond network. Starting from the crystallographic
diamond network, each of the four-coordinated nodes is replaced with
probability p by a pair of two edges, each connecting a pair of the adjacent
vertices. For all values of 0 <= p <= 1 the network percolates, yet the
fraction fp of the system that belongs to a percolating cluster drops sharply
at pc=1 to a finite value. This transition has the signature of a phase
transition with scaling exponents for p close to pc that are different from the
critical exponents of the second-order phase transition of standard percolation
models. As is the case for percolation transitions, this transition
significantly affects the mechanical properties of linear-elastic realisations,
obtained by replacing edges with solid circular struts to give an effective
density phi. Finite element methods demonstrate that, as a low-density cellular
structure, the bulk modulus K shows a cross-over from a compression-dominated
behaviour (with K proportional to phi) at p=0 to a bending-dominated behaviour
(with quadratic dependence of K on phi) at p=1.
