Containment Control Approach for Steering Opinion in a Social Network

The paper’s Theorem 5 claims M-step convergence for layered (reducible/DNN) influence structures, but it is stated and used in the paper under time-varying biases and weights (see the update rule allowing λ_i(k), w_{i,j}(k) to vary over time, eq. (2), and Sec. V using time-varying coefficients while invoking Theorem 5) . The proof of Theorem 5, however, implicitly fixes the matrices (no k-dependence appears in Ā, Λ̄) and relies on a block-lower-triangular, zero-diagonal structure to conclude finite-step convergence (Eqs. (40)–(43)) . With genuinely time-varying λ_i(k) and w_{i,j}(k), the M-step stabilization claim is false; a 2-node, 1-layer counterexample oscillates forever despite satisfying the neighbor-layer conditions. The model’s solution correctly identifies this and also gives the precise strengthening—stationarity from some finite time K—under which the finite-step claim is valid by nilpotency of the block-lower-triangular gain matrix.