Resistance curvature
Resistance curvature
is a discrete curvature measure based on the effective resistance between nodes in a network [2]. The resistance curvature of node \(i\), denoted \(c_{RC}(i)\), is defined as
\[
c_{RC}(i) = 1 - \frac{1}{2} \sum_{j \in \mathcal{N}(i)} ω_{ij} w_{ij},
\]
where \(w_{ij}\) is the weight of edge \((i,j)\), and \(ω_{ij}\) is the effective resistance between nodes \(i\) and \(j\), computed using the pseudoinverse Laplacian \(Q^{\dagger}\):
\[
ω_{ij} = (e_i - e_j)^T Q^{\dagger} (e_i - e_j),
\]
with \(e_i\) denoting the \(i\)th unit vector.
Intuitively, the effective resistance \(ω_{ij}\) measures the voltage difference between nodes \(i\) and \(j\) in the network. A redundant link, which connects nodes within a densely interconnected cluster, has low relative resistance, as its removal has little effect on overall connectivity. Conversely, a link with high relative resistance is critical for maintaining connectivity between its endpoints. Hence, if the neighborhood of node \(i\) is tree-like with few short cycles, local relative resistances are high and the curvature \(c_{RC}(i)\) is small. In contrast, in densely connected regions with many short cycles, resistances are lower and \(c_{RC}(i)\) is larger, consistent with the notion of curvature.