centrality_local_dimension_fixed: Fixed-Radius, Fuzzy and Volume Local Dimensions

In cograph: Analysis and Visualization of Complex Networks

Fixed-Radius, Fuzzy and Volume Local Dimensions

Description

Three further members of the local-dimension family, all computed from hop counts (edge weights are ignored) with the center node counted in its own ball, as in centrality_local_dimension.

Usage

centrality_local_dimension_fixed(x, mode = "all", ld_radius = 2, ...)

centrality_fuzzy_local_dimension(x, mode = "all", ...)

centrality_local_volume_dimension(x, mode = "all", ...)

Arguments

x	Network input (matrix, igraph, network, cograph_network, tna object).
mode	For directed networks: "all" (default), "out" (distances along out-edges), or "in".
ld_radius	Radius r for local_dimension_fixed, in hops. A single number of at least 1; default 2. Anything else raises a cograph_bad_parameter error.
...	Additional arguments passed to centrality.

Details

local_dimension_fixed (Silva & Costa 2013)

The discretized estimator D_i(r) = r\, n_i(r) / B_i(r) at one radius ld_radius (default 2), where n_i(r) is the ring at distance r and B_i(r) the ball within it. A structural descriptor rather than an importance ranking; nodes with eccentricity below the radius score 0. The paper defines a curve in r and fixes r per figure; the Zoo lists this fixed-radius form separately from Pu et al.'s regression form.

fuzzy_local_dimension (Wen & Jiang 2019)

Fuzzy ball N_i(r) = \sum_{d_{ij} \le r} e^{-d_{ij}^2 / r^2} / |\{j : d_{ij} \le r\}| for r = 1, \ldots, d_{\max}(i); the measure is the slope of \log N_i(r) on \log r. Larger = more influential. Reproduces Table 1 of the paper (Krackhardt kite) and its karate-club top ten in order.

local_volume_dimension (Li & Deng 2021)

Volume V_i(l) = \sum_{d_{ij} \le l} k_j, l = 1, \ldots, ecc(i); the measure is the slope of \ln V_i(l) on \ln l. Smaller = more important. The article is closed access; the definition follows the authors' own later preprint and the Zoo entry, and no published per-node values exist to check against.

The two regression measures return NaN for a node with fewer than two radii.

Value

Named numeric vector, one value per node.

References

Silva, F. N., & Costa, L. da F. (2013). Local dimension of complex networks. arXiv:1209.2476.

Wen, T., & Jiang, W. (2019). Identifying influential nodes based on fuzzy local dimension in complex networks. Chaos, Solitons & Fractals, 119, 332-342.

Li, H., & Deng, Y. (2021). Local volume dimension: A novel approach for important nodes identification in complex networks. International Journal of Modern Physics B, 35(5), 2150069.

See Also

centrality_local_dimension, centrality_local_information_dimension.

Examples

path5 <- matrix(0, 5, 5)
path5[cbind(1:4, 2:5)] <- 1; path5 <- path5 + t(path5)
rownames(path5) <- colnames(path5) <- LETTERS[1:5]
centrality_local_dimension_fixed(path5)
centrality_fuzzy_local_dimension(path5)
centrality_local_volume_dimension(path5)

cograph documentation built on Sept. 30, 2026, 5:08 p.m.