Topological Flow Data Analysis for Transient Flow Patterns: A Graph-Based Approach

The paper cleanly constructs a Markov transition matrix M from TFDA transition counts between node labels defined by the corner-symbol sequences ci+ to the right of the common root, and analyzes the spectrum: Eq. (9) gives a standard spectral expansion under distinct eigenvalues, with unit-circle eigenvalues yielding persistent/periodic components and subunit-modulus eigenvalues yielding decaying transients. This is used to explain the period‑five regime (pentagonal spectrum on the unit circle at Re=14000) and the aperiodic regime (eigenvalues scattered inside the disk at Re=16000) . By contrast, the candidate solution introduces a key “rigidity lemma” asserting that a row‑stochastic matrix can have peripheral eigenvalues (|λ|=1, λ≠1) only if each row has a single positive entry (i.e., is a permutation). That claim is incorrect in general Markov chain theory and is not stated in the paper. The rest of the candidate’s spectral expansion matches Eq. (9), but several scenario‑classifications (e.g., “two directed 3‑cycles linked through a degenerate node”) are speculative and not supported by the paper’s analysis. Hence the paper’s argument is correct within its stated scope, while the model contains a critical false lemma and unsupported extensions.