s-shell Index

Liu, Tang, Do and Hui's (2017) strength-based generalization of k-shell for identifying spreaders. Each link is given an asymmetric weight from the topology alone, $$w_{ij} = 1 + (k_i \, k^{out}_j)^a,$$ where $k^{out}_j$ is the number of $j$'s neighbors that lie outside $i$'s closed neighborhood (links that lead a spreading process to new territory), and each node's strength is $s_i = \sum_{j \in N(i)} w_{ij}$. The graph is then peeled like a k-shell but by strength: the minimum remaining strength is the threshold, everything at or below it is removed (neighbors lose the corresponding $w_{ji}$), removals cascade until the threshold holds, and the removed nodes receive the next shell index. Higher index = more central. With $a = 0$ the shells are the dense ranks of the k-core numbers.

Usage

centrality_s_shell(x, s_shell_a = 0.5, ...)

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
s_shell_a: Exponent $a$ of the link weights. A single non-negative number; default 0.5. Anything else raises a cograph_bad_parameter error.
...: Additional arguments passed to centrality.

Value

Named integer vector of shell indices, one per node.

Details

The index is an ordinal counter (1 = outermost shell), not a strength value, so it is not comparable across graphs. Isolates form shell 1 on their own, shifting every other shell up by one, as the paper's rule implies. Direction, edge weights and self-loops are ignored. The paper's robust default is $a = 0.5$.

Validated against the shell peeled at each threshold being exactly the complement of the maximal subgraph in which every node keeps strength above the threshold (brute force over all vertex subsets), and against k-core dense ranks at $a = 0$.

References

Liu, Y., Tang, M., Do, Y., & Hui, P. M. (2017). Accurate ranking of influential spreaders in networks based on dynamically asymmetric link weights. Physical Review E, 96(2), 022323.

Examples