Spectral clustering of time-evolving networks using the inflated dynamic Laplacian for graphs

The paper’s Theorem 4 states (i) λk,a ≤ λD_k for 1≤k≤N, (ii) monotonicity in a, (iii) λk,a → λD_k for 1≤k≤N, and (iv) spatial eigenfunctions flatten in time as a→∞; the proof uses Courant–Fischer and a compactness/min–max argument in Appendix C. The candidate solution proves the same four statements via a slightly different route: it explicitly decomposes H into U⊕U⊥ (U = time-constant subspace), uses the spectral gap of Ltemp on U⊥ to show concentration in U, and applies Ky Fan for partial sums. These arguments agree with the paper’s results and assumptions (Wtemp built from a connected temporal graph so L′ has a simple zero eigenvalue), but the model additionally (and correctly) quantifies λN+1,a ≥ a2γ. Overall, both are correct and consistent; the proofs are not identical but closely aligned with the same variational backbone (Courant–Fischer), and the model makes some assumptions (temporal connectivity/spectral gap) that are implicit in the paper’s setup (Lemma 3 and Theorem 4). See Theorem 4 and its proof sketch in Sec. 3.1 and Appendix C for the paper’s version, including eqs. (29)–(30) and the convergence/flattening logic .