Table 1.

Examples of the estimator functions f to be set in Eq. (1) to obtain some commonly-used centrality measures.

Undirected networks
Estimator function f	Centrality of node i	Unique contribution of node i	Corresponding metric
$f_1=\frac{K_{tot}}{N}\left(x_i+x_j-\frac{1}{N}\right)$	$x_i=\frac{k_i}{K_{tot}}$	$U{C}_i=\frac{2(N+1)k_i^2}{N^{2}TSS}$	Degree centrality
f2 = γxixj	$x_i=\frac{1}{\gamma }{\sum }_j{A}_{ij}{x}_j$	$U{C}_i=\frac{\gamma x_i^2}{TSS}(\gamma x_i^2+2\gamma )$	Eigenvector centrality
f3 = γxixj + B	$x_i=\frac{{\sum }_j{A}_{ij}{x}_j}{\gamma {\sum }_j{x}_j^2}+\frac{B{\sum }_j{x}_j}{\gamma {\sum }_j{x}_j^2}$	$U{C}_i=\frac{\gamma x_i^2}{TSS}(\gamma x_i^2-2B+2\gamma {\sum }_j{x}_j^2)$	Katz centrality

The unique contribution, which is here used to rank nodes for their centrality, is also reported. In the formulas, $K_{tot}={\sum }_i{\sum }_j{A}_{ij}$ is the total degree of the network; N is the number of nodes; $k_i={\sum }_j{A}_{ij}$ is the degree of the node i; γ and B are two parameters whose values change according to the estimator function. In case of f₂, γ equals the largest eigenvalue of A. In case of f₃, $\gamma =1/\alpha {\sum }_j{x}_j^2$ and $B=-1/{\sum }_j{x}_j$, where α is the attenuation factor of the Katz centrality. TSS is defined in the text. Further details are given in SI, Sect. 1.