Transforming Norm-based To Graph-based Spatial Representation for Spatio-Temporal Epidemiological Models

The paper asserts—without proof—that under the average-of-others walk L_j(t+1) = (∑_i L_i(t) − L_j(t))/(n−1) and β_{i,j}(t) = 1/|L_i − L_j|, the smallest pairwise distance converges to zero, so no fixed discretization can stay at least as fine as that distance indefinitely. This is stated correctly but omits the necessary assumption n ≥ 3 and provides no derivation; the graph-based well-mixed-node formalization is also essential to the conclusion (paper, Sec. 2.3 and 3) . The candidate solution supplies a complete, rigorous proof: centroid invariance, exact contraction of all pairwise differences by 1/(n−1), d_min(t) = (1/(n−1))^t d_min(0), the n = 2 exception, and a precise argument showing why any fixed finite graph discretization with well-mixing cannot exactly preserve β_{i,j}(t) for all initial conditions and all times.