Fig. 2

Principle of ShRec3D reconstruction algorithm. a Sketch of the flowchart of the algorithm. The starting point is the contact map F, with either binary or normalized components. It defines a contact network, where each edge (i,j) is given a length L _ij , equal to 1/F _ij in the original version [9] and extended into a tunable relationship \(L_{ij}= F_{ij}^{-\alpha }\) with parameter α. The shortest-path distances between the nodes, illustrated in (b), produces a complete distance matrix D. Then a metric matrix M is derived using simple algebraic formulas. Classical multidimensional scaling (MDS, step singled out by green arrows) amounts to a truncation of M into a semidefinite positive matrix G of rank 3, supported by the spectrum of M, in (c). G is the Gram matrix of the desired 3D structure. The coordinate vectors V are deduced from the 3 nonvanishing eigenvalues and corresponding eigenvectors of G. The 3D structure obtained with α=1 is presented in (d). In what follows we investigate the relationships (sketched as blue arrows) between F and either the graph distances D or the reconstructed distances R for various values of α