Table 3 Degeneracy at λ₀ and λ₋₁

Mathematically in a network, duplication of nodes yields the same neighbors for two nodes in the corresponding adjacency matrices (Kitano 2004; Yadav and Jalan 2015). It has been shown that duplication of nodes leads to lowering of the rank of the corresponding matrix, hence contributing one additional zero eigenvalue in the spectra (Banerjee and Jost J 2007; Yadav and Jalan 2015; Shinde and et al. 2015). For the adjacency matrix of size N and rank r, the matrix has exactly N−r zero eigenvalues (Poole 2006). We discuss here three possible cases when the rank can lower in an adjacency matrix:

(i) two rows (columns) have exactly same entries, it is termed as complete row (column) duplication R₁=R₂,

(ii) the partial duplication of rows (columns) where R₁=R₂+R₃, where, R_i denotes ith row of the adjacency matrix (Fig. 3). This condition is computationally exhaustive as to find this state in the matrix has large possibilities (Yadav and Jalan 2015),

(iii) if there is an isolated node in the network, the row and column corresponding to it has zero entries and thus the rank of the matrix is lowered by one. For a connected network, the number of zero eigenvalues (λ₀) provides an exact measure of (i) and (ii) conditions (Yadav and Jalan 2015; Golub and Van-Loan C 2012). Similarly, the calculations for identification of degenerate − 1 eigenvalues is given in (Marrec and Jalan 2017) for A + I matrices.