When Dense Networks Diffuse Sparsely: A Stochastic Propagation Operator for Weighted Directed Networks

Classical resolvent centralities on weighted networks aggregate propagation over all observed links. We develop a stochastic propagation operator for weighted directed networks in which only a random subset of links is active at each round, impulses may persist, and receiving nodes may face finite capacity. In the linear regime, expected propagation is governed exactly by an effective operator that converges monotonically to the dense resolvent limit and factorizes into emitter-level retention factors, bounded by the inverse effective number of destinations, and a mass-preserving reallocation within each emitter’s links. Applied to the 2021 Colombian input–output network, which is topologically dense but concentrated in a few effective trading relationships, sparse activation preserves the broad supply-side ordering while substantially reordering the most central sectors. The reordering is not an attenuation effect: a uniformly attenuated benchmark at the same distance to criticality reproduces 3% of it, whereas retention alone reproduces 58% of the rank changes and 97% of the variation in log scores. Nor is it backbone extraction: dense resolvents on deterministic backbones reproduce at most 29%. The effective operator predicts simulated cascade reach better than Katz, PageRank, eigenvector, communicability, and Viral Centrality computed on network weights, except under strongly binding capacity. Across 79 OECD input–output tables, retention reproduces most of the reordering in every economy (median 63%), attenuation and backbones reproduce little, and the industries promoted by sparse activation are the same across countries. Dense connectivity need not imply dense effective diffusion.