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Abstract: Motivated by a simple model of sequential decision making, we study small random perturbations of a dynamical system in a neighborhood of a heteroclinic network, that is, a collection of hyperbolic equilibrium points and corresponding connecting trajectories. Based on a detailed study of the exit problem from a neighborhood of a hyperbolic equilibrium, we show that the probability of tracing any particular path in the network decays at most polynomially in the size of the noise and establish sharp asymptotics on the time required to complete these journeys. Using these results, we describe the metastable hierarchy that emerges on polynomially long timescales.