Li et al.'s weighted LeaderRank adds a ground node g. Each original directed edge and each edge from an original node to g has weight one. The edge from g to node i has weight \((k_i^{in})^{\alpha}\), using original in-degree before ground edges are added. Scores follow the stationary distribution of the row-normalized augmented matrix.
Arguments
- x
Network input accepted by
centrality.- wlr_alpha
Finite in-degree exponent, default one, a setting studied in the source rather than a universal optimum.
- ...
Additional arguments to
centrality.normalized = TRUEdivides final scores by their maximum.
Details
Raw scores retain total mass N+1 across the augmented graph, following
the all-nodes-one initialization in the original paper, section 2.
The ground score is omitted from the returned vector without redistribution.
The Zoo instead initializes the ground at zero, yielding raw scores
smaller by N/(N+1); final max-normalized scores agree. The existing
centrality_leaderrank uses a different redistribution/scale
convention, so raw equality at alpha zero is not asserted.
Directed arcs are retained; an undirected edge is treated as two opposite arcs, an explicit extension. Input weights are ignored: weighted refers to the algorithm's ground-edge weights. Loops are removed and parallel arcs count once. Mode, path inversion and cutoff do not change the result.
Alpha can be any finite number. Negative values require strictly positive original in-degree at every node. At alpha zero all ground-edge weights are one, including for zero-in-degree nodes. With positive alpha, these nodes receive no ground resource and have zero stationary score; if every in-degree is zero the ground row is undefined and all scores are NaN. Empty input returns an empty vector. These boundary conventions are explicit; no pseudocount is added to the published in-degree weights.
A native linear solve eliminates the ground variable and obtains the unique stationary distribution even when ordinary iteration is periodic. This uses O(N^3) time and O(N^2) memory. Ground transition probabilities are calculated with shifted logarithms, avoiding overflow for large exponents; extremely small probabilities may underflow to zero.
References
Li, Q., Zhou, T., Lu, L., & Chen, D. (2014). Identifying influential spreaders by weighted LeaderRank. Physica A, 404, 47-55, section 2, equations 1-2. doi:10.1016/j.physa.2014.02.041 . Author preprint: https://arxiv.org/abs/1306.5042.
Examples
centrality_weighted_leaderrank(igraph::make_ring(4, directed = TRUE))
#> 1 2 3 4
#> 0.8333333 0.8333333 0.8333333 0.8333333