The lobby index , or \(l\)-index, is inspired by Hirsch’s h-index, which quantifies the scientific output of a researcher [2, 3]. For a node \(i\), the lobby centrality \(c_{\mathrm{Lobby}}(i)\) is defined as the largest integer \(k\) such that \(i\) has at least \(k\) neighbors with degree at least \(k\):
\begin{equation*}
c_{\mathrm{Lobby}}(i) = \max \{ k \mid d_{i(k)} \ge k \},
\end{equation*}
where \(i(k)\) denotes the neighbor of \(i\) with the \(k\)-th largest degree.
Several extensions of the lobby index to weighted networks have been proposed:


  1. The collaboration index (\(c\)-index) computes the Hirsch h-index of the sequence formed by multiplying each neighbor’s strength by the weight of the connecting edge [4].

  2. The communication ability replaces each neighbor’s degree with the product of its weighted lobby index (also called \(h\)-degree), which is the lobby index calculated using the strengths of the incident edges, and the weight of the edge connecting it to the node [5].


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] Korn, A., Schubert, A., & Telcs, A. (2009). Lobby index in networks. Physica A: Statistical Mechanics and its Applications, 388(11), 2221-2226. doi: 10.1016/j.physa.2009.02.013.
[3] Lü, L., Zhou, T., Zhang, Q. M., & Stanley, H. E. (2016). The H-index of a network node and its relation to degree and coreness. Nature communications, 7(1), 10168. doi: 10.1038/ncomms10168.
[4] Yan, X., Zhai, L., & Fan, W. (2013). C-index: A weighted network node centrality measure for collaboration competence. Journal of Informetrics, 7(1), 223-239. doi: 10.1016/j.joi.2012.11.004.
[5] Zhai, L., Yan, X., & Zhang, G. (2013). A centrality measure for communication ability in weighted network. Physica A: Statistical Mechanics and its Applications, 392(23), 6107-6117. doi: 10.1016/j.physa.2013.07.056.