DK-based gravity model

The DK-based gravity model (DKGM) puts the degree k-shell index of both ends into a truncated squared-distance gravity sum: \(DKGM_i=\sum_{j\ne i,\,d(i,j)\le R}DK(i)DK(j)/d(i,j)^2\). The mass \(DK(i)=k(i)+k_s^*(i)\) adds the degree to an improved shell index \(k_s^*(i)=k_s(i)+p(i)/(\max_k q(k)+1)\), where \(p(i)\) is the removal stage inside the node's shell and \(q(k)\) the number of stages the k-level needed. The stage breaks the ties that degree and k-shell leave behind: two nodes of the same shell are separated by how late the peeling reached them.

Usage

centrality_dkgm(x, dkgm_radius = 2, ...)

Arguments

x: Network input accepted by centrality.
dkgm_radius: Nonnegative hop-distance cutoff, default two, the value used for the paper's Table 5 and one of the two the paper recommends in general. NULL or infinity includes every reachable partner. Fractional cutoffs include exactly the integer hop distances not exceeding them; values below one give zero. "auto" applies the paper's own equation 4, \(R^*\approx\langle d\rangle/2\), with cograph conventions: half the mean finite positive hop distance, rounded to the nearest integer with ties to even, minimum one. Those conventions and the treatment of disconnected graphs are cograph's, not the paper's.
...: Additional arguments to centrality.

Value

Named numeric vector in input node order.

Details

Stages restart at one inside every shell, but the denominator \(\max_k q(k)+1\) is a single global maximum. A node's mass therefore depends on the whole graph: adding a disconnected component that peels in more stages lengthens that denominator and changes every raw score. This is a property of the published definition, not a cograph choice, and it distinguishes DKGM from centrality_mixed_gravity.

The paper's Algorithm 1 says "Find all nodes in G with degree k" while its stage loop ends "until All remaining nodes in G have degree > k" and its Methods define k-shell by removing "nodes whose degree k <= 1 ... Until there are no nodes in the network with degree k <= 1". Strict equality cannot terminate on a three-node path, so cograph follows the at-most reading, which is the only one consistent with the printed termination condition and which reproduces the paper's Tables 2 to 5. Removal inside a stage is simultaneous, matching the printed two-stage two-shell. Because the level starts at one, an isolate falls in the one-shell rather than the zero-shell centrality(measures = "coreness") reports; isolates carry no edges, so no other node's shell, stage or score is affected.

Uses the simple undirected unweighted skeleton, which is the source domain: either arc creates one edge, parallel edges count once and loops are removed. This projection is a cograph convention outside that domain. Edge weights, mode, cutoff, gravity_mass and path-weight inversion are ignored. Unreachable partners contribute nothing. Isolates and singleton graphs score zero; empty graphs return no scores. Optional maximum normalization applies to the complete result over all nodes. Dense all-pairs distances cost O(n cubed) time and O(n squared) memory.

Numerical verification establishes agreement with the published equations and the printed nine-node example, not parity with author software, which was not located, nor any claim about spreading performance.

References

Li, Z. and Huang, X. (2021). Identifying influential spreaders in complex networks by an improved gravity model. Scientific Reports, 11, 22194. Equations 1-3, Algorithm 1 and Tables 2-5. doi:10.1038/s41598-021-01218-1 .

Examples

centrality_dkgm(igraph::make_ring(6))
#>      1      2      3      4      5      6 
#> 50.625 50.625 50.625 50.625 50.625 50.625 
centrality_dkgm(igraph::make_star(6), dkgm_radius = 1)
#>        1        2        3        4        5        6 
#> 77.77778 15.55556 15.55556 15.55556 15.55556 15.55556