Truncated curvature index
The
truncated curvature index
is a simplified version of the curvature index that limits the computation to simplices of dimension \(d \leq 2\), i.e., cliques of size up to three [2]. For a node \(i\), it is defined as
\[
K_{\mathrm{trunc}}(i) = \sum_{k=0}^{2} (-1)^k \frac{V_{k-1}(i)}{k+1} =1 - \frac{d_i}{2} + \frac{t_i}{3},
\]
where \(V_k(i)\) denotes the number of \((k{+}1)\)-cliques incident to node \(i\), \(d_i\) is the degree of node \(i\) and \(t_i\) is the number of triangles passing through node \(i\).
The truncated curvature index is particularly useful for analyzing large-scale networks where computing higher-dimensional cliques is computationally expensive, yet the essential topological and curvature properties of nodes are still captured.