Detection of Functional Communities in Networks of Randomly Coupled Oscillators Using the Dynamic-Mode Decomposition

The paper defines Cojl = |∑n ξ̂n,j ξ̂* n,ℓ| with each mode vector ξn normalized across nodes, then asserts via Cauchy–Schwarz that 0 ≤ Cojl ≤ 1 and uses this to justify choosing a threshold Ccr ∈ (0,1). However, with only per-mode normalization, Cojl can exceed 1; a two-mode, one-node example gives Co11 = 2, contradicting the stated bound (definition and claim appear explicitly in the paper’s Section III ). The model correctly identifies this flaw and explains how an additional normalization (e.g., column-wise across modes or an averaging factor) would repair the bound. On the other two points—symmetry of A(md) and that the time-evolution graph Gti,tf is a forest—the model’s arguments are correct and supply the proof details that the paper only states informally (construction of Gti,tf and the ‘forest’ claim are in the paper’s Section III ; symmetry of Co and A(md) is also stated there ).