Cluster analysis in multilayer networks using eigen vector centrality

Pitambar Khanra, Subrata Ghosh, Prosenjit Kundu, Chittaranjan Hens, Pinaki Pal

wrongmedium confidenceCounterexample detected

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:55 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper correctly proves that the Perron eigenvector X of the supra-adjacency A is invariant under every graph automorphism and thus is constant on each orbit (cluster). However, it then claims, with a flawed contradiction argument, that vertices in different clusters must have different X-values, concluding that clusters can be recovered exactly by grouping equal entries of X. This is false in general: connected regular graphs with trivial automorphism group (e.g., the Frucht graph) have all-one Perron vectors while clusters are singletons. The candidate solution identifies the correct part (constancy on orbits), exhibits the counterexample, and specifies a missing nondegeneracy assumption (that the stabilizer of X equals the automorphism group) under which exact recovery would follow.

Referee report (LaTeX)

\textbf{Recommendation:} reject

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

While the invariance of the Perron eigenvector under automorphisms and its constancy on orbits are correctly observed, the paper’s headline theorem—that distinct clusters necessarily yield distinct EVC values enabling single-step cluster recovery—is false in general. The provided proof contains a logical error (conflating non-commutation AP≠PA with loss of eigenvector property) and overlooks standard counterexamples. Without a corrected theorem and appropriate conditions characterizing when EVC separates orbits, the main claim does not hold.