Balanced centrality (SWIPD)
Balanced centrality , also known as SWIPD, is a hybrid centrality measure that combines multiple classical centrality indices into a single vector [2]. The centrality of node \(i\), denoted \(c_{\text{SWIPD}}(i)\), is defined as\[c_{\text{SWIPD}}(i) = γ_1 A^2 D^{-1} A u \; + \;γ_2 (D - α D A)^{-1} A u \; + \;γ_3 A D^{-2} u \; + \;γ_4 A D^{-1} A u,\]where the four terms correspond, respectively, to square centrality , walk centrality , power-like centrality , and degree-weighted centrality . Specifically, \(D\) is the degree diagonal matrix, \(u\) is the \(N \times 1\) vector of ones, and the coefficients \(γ_i\) control the relative contribution of each component. The attenuation factor in the walk centrality is \(α = \frac{1}{d_{\max}+1}\), where \(d_{\max}\) is the maximum node degree in the network. Each of these terms captures a distinct aspect of node importance, which can be summarized as follows:
- Square centrality: emphasizes the influence of two-hop neighbors.
- Walk centrality: incorporates attenuated walks starting from each node.
- Power-like centrality: highlights nodes with strong direct connections while penalizing high-degree nodes.
- Degree-weighted centrality: captures the influence of a node based on the degree-weighted connectivity of its neighbors.