Wepropose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular 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 0 ⩽ p ⩽ 1the network percolates, yet the fraction fp of the systemthat belongs
to a percolating cluster drops sharply at pc = 1 to a finite value fp
c . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a
finitemass f > 0 p c of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p → pc that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density ϕ. Finite element methods demonstrate that, as a low-density cellular structure, the bulkmodulus Kshows a cross-over froma compression-dominated behaviour, K (ϕ) ∝ ϕκ with κ ≈ 1, at p = 0 to a bending-dominated behaviour with κ ≈ 2 at p=1
