cograph (version 2.7.2)

centrality_local_dimension: Local Dimension

Description

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", ...)

Value

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

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.

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.

See Also

centrality_local_information_dimension for the entropy-weighted variant, centrality_distance_entropy.

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)