We model transport of individuals across a heterogeneous scale-free network where a few weakly connected nodes exhibit heavy-tailed residence times. Such power laws are consistent with the Axiom of Cumulative Inertia, an empirical law stating that the rate at which people leave a place decreases with the associated residence time. We show numerically and analytically that "cumulative inertia" overpowers highly connected nodes in attracting network individuals. Our result, confirmed by empirical evidence, challenges the classical view that individuals will accumulate in highly connected nodes. ; Comment: Clarification added to the discussion of theory and data. More explanations included for better reading