Fixed-Radius, Fuzzy and Volume Local Dimensions

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.

Value

Named numeric vector, one value per node.

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.

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.

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)
#>         A         B         C         D         E 
#> 0.6666667 0.5000000 0.8000000 0.5000000 0.6666667 
centrality_fuzzy_local_dimension(path5)
#>          A          B          C          D          E 
#> 0.04895549 0.25765248 0.18702827 0.25765248 0.04895549 
centrality_local_volume_dimension(path5)
#>         A         B         C         D         E 
#> 0.7252466 0.4340194 0.4150375 0.4340194 0.7252466