Devriendt and Lambiotte's node resistance curvature is \(p_i=1-\frac{1}{2}\sum_{j\sim i}w_{ij}R_{ij}\), where weights are electrical conductances and R is effective resistance. Equivalently, it is one minus half the expected degree in a random spanning tree whose probability is proportional to the product of its edge conductances. The expectation is taken separately within each connected component.
Arguments
- x
Network input accepted by
centrality.- ...
Additional arguments to
centrality. Withnormalized = TRUE, scores are divided by their positive maximum; negative values remain negative and the component-sum identity no longer holds. Default raw scores retain the published interpretation.
Details
Low, possibly negative, curvature characterizes tree-like junctions; larger curvature characterizes locally redundant connections. It is a geometric descriptor, not a universal ranking of influence. On a tree the score is one minus half the degree; on an unweighted clique or cycle of n vertices every node scores 1/n. Isolates score one, following the empty sum. Raw scores sum to the number of connected components.
Uses finite nonnegative edge weights as conductances when
weighted = TRUE; zero weights are absent connections. Without
weights, uses the simple undirected skeleton. Self-loops are always
removed. For weighted directed inputs, opposite arcs are added to form
undirected conductances. The simplify argument combines parallel
edges first; remaining weighted parallel edges are added. These input
projections are cograph conventions for the source's undirected domain.
mode and shortest-path weight inversion do not affect the result.
Exact dense electrical systems are solved component by component, using Cholesky factors of grounded Laplacians. Squared triangular-solve norms avoid subtracting nearly equal pseudoinverse entries. Uniform rescaling of conductances within a component leaves curvature unchanged. Extreme weight ranges can still produce numerical singularity or overflow, in which case an error is raised. Dense factorization and edge solves cost up to O(n cubed + n squared times m) per component; the measure is excluded from the default all tier and must be requested explicitly.
References
Devriendt, K., & Lambiotte, R. (2022). Discrete curvature on graphs from the effective resistance. Journal of Physics: Complexity, 3, 025008. Definition 1, equation 2; Property 2; Appendix A.1, Theorem 2. doi:10.1088/2632-072X/ac730d . https://arxiv.org/abs/2201.06385.
Examples
centrality_resistance_curvature(igraph::make_star(5, mode = "undirected"))
#> 1 2 3 4 5
#> -1.0 0.5 0.5 0.5 0.5