The leader score: global vs community-conditioned NCE • lcdaGRASP

Is the NCE measuring the right thing?

The paper’s Node-Connection Entropy uses the global connection probability $p_1(v) = k_v / n$ . But a hub whose degree is spread across all communities can maximise this score while being a poor community leader. Since a score is a simplification, it is worth checking what it actually rewards — which motivates a community-conditioned alternative:

$I_{\text{local}}(v) = H_b\!\left(\frac{|\Gamma(v) \cap C(v)|}{|C(v)| - 1}\right),$

exposed in the package as nce_local_score(). We measure how often the designated leader changes when re-picked under the local score.

d$results |>
  dplyr::group_by(graph) |>
  dplyr::summarise(
    communities       = round(mean(n_communities), 1),
    H_global          = round(mean(H_paper_global), 3),
    H_local           = round(mean(H_paper_local), 3),
    pct_leaders_changed = round(mean(pct_leaders_changed), 1),
    .groups = "drop") |>
  knitr::kable(caption = "Mean leader-score values and the fraction of leaders that change identity under the local NCE.")

Mean leader-score values and the fraction of leaders that change identity under the local NCE.
graph	communities	H_global	H_local	pct_leaders_changed
Dolphins	5.0	0.581	0.741	80.0
Karate	3.0	0.840	0.598	100.0
Polbooks	5.0	0.535	0.736	80.0
SBM_5	5.1	0.343	0.764	33.7

library(ggplot2)
ggplot(d$results, aes(graph, pct_leaders_changed, fill = graph)) +
  geom_boxplot(alpha = 0.6, show.legend = FALSE) +
  geom_hline(yintercept = 10, linetype = 2, colour = "grey40") +
  labs(title = "Leaders reassigned when switching global -> community-conditioned NCE",
       subtitle = "Dashed line = 10%: above it, the NCE choice materially changes who is a leader",
       x = NULL, y = "% of leaders changed") +
  theme_minimal(base_size = 10)

Reading. If the change fraction is non-trivial (above ~10%), the choice of NCE definition is not an interpretive detail — it materially decides who is crowned the leader. Both scores are exported (nce_score() and nce_local_score()) precisely so this comparison is part of the contribution rather than buried.

Reproducibility and data provenance

The comparison above is read from a dataset shipped with the package; nothing is re-run at build time. Each dataset records the package version that generated it (not necessarily the version you installed: the generators are re-run only when the algorithms change), the release it first shipped in, and a SHA-256 checksum matching inst/extdata/SHA256SUMS:

lcda_provenance("nce_alternatives")
#>            dataset generated_by generated_on first_release shipped_in
#> 1 nce_alternatives        0.3.1   2026-05-31         0.3.1      0.3.2
#>                                                             sha256
#> 1 19271bb3e20da57a7bf0bd51045f7fc92f925b2993d11d29ec0763a1c72f92a3

Regenerate with data-raw/80_nce.R (single documented seed).

Honest reading. The global-vs-community-conditioned NCE comparison is a structural contrast of two exported scores, not a downstream-utility claim; relatedly, the lexicographic $H$ tie-break is empirically never decisive (it fired in $0\%$ of iterations across the pool-sensitivity sweep), so $H$ acts as a tie-breaking safeguard rather than a search driver.

References

Ospina, R., Silva, G., Matos Junior, F. J., Leite, A., & Ochi, L. S. (2026). A GRASP Framework for Community and Leader Detection in Complex Networks. Preprint.
Akachar, E., Bougteb, Y., Ouhbi, B., & Frikh, B. (2025). LeaDCD: Leadership concept-based method for community detection. Information Sciences, 686, 121341. doi:10.1016/j.ins.2024.121341