Graph regularization centrality (GRC)
Graph regularization centrality
(GRC) is a centrality measure derived from graph signal processing theory [2]. It evaluates node importance by examining how a delta signal centered on each node spreads across the graph under a regularization constraint.
For node \(i\), the GRC centrality \(c_{GRC}(i)\) is defined as
\[
c_{GRC}(i) = \frac{1}{s_i(i)},
\]
where \(s_i\) is obtained by solving the regularized optimization problem
\[
s_i = \arg\min_g \left( \|g - δ_i\|^2 + γ g^T L g \right).
\]
Here, \(g \in \mathbb{R}^{N \times 1}\) is an \(N\)-dimensional graph signal over all nodes, \(δ_i\) is an \(N \times 1\) delta signal with \(δ_i(k) = 1\) if \(k = i\) and \(δ_i(k) = 0\) otherwise, \(L\) is the Laplacian matrix of the graph \(G\), and \(γ\) is a regularization parameter controlling the spread of the signal across the network. Larger values of \(γ\) allow the delta signals to diffuse further along the graph. The optimization balances two objectives: the signal should remain close to the original delta (locality) while spreading smoothly over the network according to the Laplacian (global influence). Nodes whose signals diffuse more widely have smaller \(s_i(i)\) values and thus higher GRC centrality, capturing both local and global network characteristics.
When \(γ = 0\), \(s_i = δ_i\), so \(s_i(i) = 1\) and all nodes have centrality equal to one. According to Dal Col and Petronetto [2], setting \(γ = 1\) produces centrality values that balance local and global network characteristics without additional parameter tuning.