| Summary: | We propose 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 $0\leqslant p\leqslant 1$ the network percolates, yet the fraction f _p of the system that belongs to a percolating cluster drops sharply at p _c = 1 to a finite value $f_{p}^{c}$ . This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass $f_{p}^{c}\gt 0$ of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for $p\to {{p}_{c}}$ 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 bulk modulus K shows a cross-over from a compression-dominated behaviour, $K(\phi )\propto {{\phi }^{\kappa }}$ with $\kappa \approx 1$ , at p = 0 to a bending-dominated behaviour with $\kappa \approx 2$ at p = 1.
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