Local Dimension

Growth exponent of the ball around a node (Silva & Costa 2013; Pu et al. 2014). Let $B_i(r)$ be the number of nodes within $r$ hops of $i$, the node itself included. The local dimension is the slope of $\ln B_i(r)$ on $\ln r$ over $r = 1, \ldots, d_{\max}(i)$: $$D_i = \frac{d \ln B_i(r)}{d \ln r}.$$ A node that reaches most of the network in a few hops has a small exponent, so lower values mark more influential nodes. When a node has a single radius (it reaches every other node in one hop) the regression is undefined and the discretized derivative $r\, n_i(r) / B_i(r)$ at $r = 1$ is reported, where $n_i(r)$ counts the nodes at distance exactly $r$.

Usage

centrality_local_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".
...: Additional arguments passed to centrality.

Value

Named numeric vector, one value per node. NaN for a node that reaches no other node.

Details

The implementation reproduces the worked example in Wen & Jiang (2019), which reports 0.9231 for ring sizes 4, 5, 4, 4. Distances are hop counts; edge weights are ignored.

References

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

Pu, J., Chen, X., Wei, D., Liu, Q., & Deng, Y. (2014). Identifying influential nodes based on local dimension. EPL, 107(1), 10010.

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

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_local_dimension(star5)
#>        A        B        C        D        E 
#> 0.800000 1.321928 1.321928 1.321928 1.321928