Xu and Wang's adaptive LeaderRank computes original node H-indices, then adds a ground node with H-index one, joined bidirectionally to every original node. Each augmented arc from j to i has weight \(a_{ji}h_i\). Row-normalized weights define the resource transition matrix. Raw stationary scores retain total augmented mass N, following initial score one on ordinary nodes and zero on ground. The ground score is omitted without redistribution. H-indices are not recomputed after ground edges are added.
Arguments
- x
Network input accepted by
centrality.- alr_h_mode
Original H-index convention: all (default), out or in.
- ...
Additional arguments to
centrality.
Details
The H-index is the largest integer h for which at least h original
neighbors have degree at least h. The focal node is excluded from that
neighbor list. This differs from cograph's existing closed-neighborhood
centrality_lobby convention.
The paper evaluates directed and undirected networks but does not pin
a directed H-index convention. alr_h_mode makes that choice
explicit. Default "all" computes H-indices on the simple
undirected skeleton, merging reciprocal arcs. "out" uses outgoing
neighbors' out-degrees; "in" uses incoming neighbors' in-degrees.
These directed H-index choices are explicit cograph conventions, not
claims of the authors' directed-software behavior. In every case, resource
flow retains the original directed arcs. Undirected edges become opposite
arcs, and all H-index modes then coincide.
Input weights are ignored; the algorithm generates its own destination
weights. Loops are removed and parallel arcs count once. The generic
mode, inversion and cutoff arguments are ignored. Original nodes
with H-index zero receive zero stationary score. If every H-index is zero,
the ground transition row is undefined and all scores are NaN. This can
occur on edgeless inputs or some directed inputs in in/out H-index modes.
Empty input returns an empty vector. No H-index pseudocount is added.
A native ground-elimination solve obtains the unique stationary solution in O(N^3) time and O(N^2) memory, including periodic chains for which ordinary iteration need not converge. Optional final max normalization acts on the returned ordinary-node scores. Numerical definition agreement does not establish superior spreading predictions or author-code parity.
References
Xu, S., & Wang, P. (2017). Identifying important nodes by adaptive LeaderRank. Physica A, 469, 654-664, section 2.2, equation 3 and algorithm steps 1-4. doi:10.1016/j.physa.2016.11.034 .
Examples
centrality_adaptive_leaderrank(igraph::make_ring(4))
#> 1 2 3 4
#> 0.8333333 0.8333333 0.8333333 0.8333333