We present an exact particle tracking algorithm for advective-dispersive transport in a network of one-dimensional bonds connected at volume-less nodes. We derive an analytical solution for the transit probability of a particle that has just arrived at a node to first arrive at a nearest-neighbor node. This solution is compared to traditional random walk simulations in a network where particle motion is simulated in small time steps with empirical mixing rules at nodes. We demonstrate that this latter approach gives a good representation of the cumulative arrival time probability distribution, but that the fraction of the total flux arriving at each node is in error by more than 1% when compared to the analytical probability unless 105 or more time steps are used to simulate transport between nodes. Finally, we show how to obtain semianalytical solutions for tracer transport in a network and discuss how these results can be used for more complex settings where there is incomplete mixing at nodes, or where the bonds are no longer one-dimensional.