The weighted top candidate (WTC) method is an extension of the top candidate (TC) method designed for ranking nodes in a weighted network, such as a citation network [2]. Similar to the TC method [3], the WTC approach aims to identify a stable set of mutually reinforcing candidates, but it leverages weighted interactions to better capture node reputation. WTC assumes that nodes with higher reputation tend to nominate stronger candidates than lower-reputation nodes. The algorithm proceeds through the following steps:

  1. Initialization: all nodes are initially included in the candidate set. For each node \(i\), its reputation \(r_i\) is defined as the total weight of incoming links: \[ r_i = \sum_{j \in \mathcal{N}} w_{ji}, \] where \(w_{ji}\) is the weight of the link \((j,i)\).
  2. Weighted nomination: for each node \(i\), a neighbor \(j \in \mathcal{N}_i^{out}\) is nominated if \[ r_j \geq ω_i (1 - α), \] where \(α \in [0,1]\) controls the selectivity of nominations and \(ω_i = \max_{k \in \mathcal{N}_i^{out}} r_k\). All neighbors satisfying this condition are considered in the next iteration of the candidate set.
  3. Elimination: nodes that are not nominated by any other node are removed from the candidate set. All outgoing nominations from these removed nodes are discarded.
  4. Update: the elimination in Step 3 may cause additional nodes to lose all incoming nominations. These nodes are likewise removed, and the process is repeated until no further removals occur.
The remaining nodes form the stable candidate set . A set \(S \subseteq \mathcal{N}\) is stable if (i) every node in \(S\) is nominated by at least one other node in \(S\) and (ii) all nominees of any node in \(S\) are also contained in \(S\). The parameter \(α\) regulates the inclusiveness of the selection: smaller values of \(α\) produce a more selective set of nodes, while larger values yield a more inclusive set.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Sziklai, B. R. (2021). Ranking institutions within a discipline: The steep mountain of academic excellence. Journal of Informetrics, 15(2), 101133. doi: 10.1016/j.joi.2021.101133.
[3] Sziklai, B. (2018). How to identify experts in a community?. International Journal of Game Theory, 47(1), 155-173. doi: 10.1007/s00182-017-0582-x.