Weighted directed networks can exhibit stronger aggregate interactions without becoming more integrated or topologically richer. Using daily prices for 32 stock indices over 2006–2026, we construct effective transfer entropy (ETE) networks from range-based variance states and recompute GLMY path homology over a directed edge-density filtration. In 88 overlapping 120-common-date windows, total ETE is associated with a higher density required for global weak connectivity and with smaller integrated first- and second-dimensional Betti curves. These are dependence-aware descriptive associations: reduced-overlap estimates are imprecise, and controls for edge concentration and mean variance reduce the connectivity association to approximately zero. Direct coverage measures show that high-intensity windows can distribute weight broadly while their strongest edges reach nodes unevenly. Exact circular-shift tests yield empty 5% BH-FDR backbones, so individual channels are not treated as established. Null-model, rank-stability, coefficient-field, and discretization checks delimit the fixed-density result. A common absolute-weight contrast yields positive rather than negative TETE–Betti associations, showing that the negative baseline relation is specific to the fixed-density filtration. The results distinguish interaction intensity, dispersion, strong-edge coverage, and directed path-homology organization as separate network dimensions.