Exact Rank and Reversal Structure of Horizon-Specific Network Exposure: Mathematical Extensions to “Social Networks and the Propagation of Household Shocks to Asset Prices”

Quaiyyum [1] shows that, when response decay differs across network patterns, a county’s financial importance generically changes with the horizon, so that no single centrality measure summarizes exposure at all horizons. That result is stated as a “generic” non-proportionality claim, and the ranking reversals in the paper’s Figure 3 are documented only empirically. This note makes both statements exact. First, the county-by-horizon matrix of pricing loadings has a closed-form rank: under a single diffusion channel it equals the number of distinct Laplacian eigenspaces on which the financial weights have support (a Krylov dimension), so a single score suffices if and only if nonnegative weights are proportional to square-root degree. Second, the best low-rank summaries are controlled by an explicit singular-value decay bound that depends on the horizon length, the diffusion rate and the active spectral width. Third, the number of ranking reversals between any two counties is bounded by the number of sign changes in a spectral coefficient sequence; the asymptotic ordering is the degree ordering; reversals are guaranteed for pairs whose initial and degree orderings disagree; and the last reversal time is bounded in terms of the spectral gap. All results are proved and checked numerically on synthetic graphs. Code available upon request.