Flow stability for dynamic community detection

Core definitions and properties match: the paper defines Sforw(t1,t) = P(t1) T(t1,t) P(t)^{-1} T(t1,t)^T P(t1) − p(t1)^T p(t1) and Sback(t2,t) analogously via reversed evolution; both are symmetric and properly normalized (zero row/column sums), exactly as the model uses and proves via a Gram factorization and stochasticity identities . The directed one-step reduction to Leicht–Newman modularity and the left-eigenvector identities p(t1) T T_rev = p(t1), p(t2) T_rev T = p(t2) are also in the paper and align with the model . The principal gap concerns the static-network limit: the paper states that Sforw and Sback “become equal” on a static network, but does not state the necessary matching of boundary marginals and the time-reversal mapping; in general, pointwise equality at the same t fails unless additional stationarity holds. The model provides the precise relation Sforw(t1,t) = Sback(t2, t1 + t2 − t) (under equal boundary marginals), which also yields equality of the integrated forward/backward flow objectives—this completes a missing hypothesis in the paper’s statement . The role of the jump rate λ as a resolution parameter and the exponential propagator for static networks are correctly described in the paper and consistent with the model’s elaboration .