  We identify a fundamental phenomenon of heterogeneous one dimensional random
walks: the escape (traversal) time is maximized when the heterogeneity in
transition probabilities forms a pyramid-like potential barrier. This barrier
corresponds to a distinct arrangement of transition probabilities, sometimes
referred to as the pendulum arrangement. We reduce this problem to a sum over
products, combinatorial optimization problem, proving that this unique
structure always maximizes the escape time. This general property may influence
studies in epidemiology, biology, and computer science to better understand
escape time behavior and construct intruder-resilient networks.
