Magnetic monopoles have eluded experimental detection since their prediction nearly a century ago by Dirac. Recently it has been shown that classical analogues of these enigmatic particles occur as excitations out of the topological ground state of a model magnetic system, dipolar spin ice. These quasi-particle excitations do not lead to a modification of Maxwell's equations, but they do interact via Coulombs law and they are of magnetic origin. In this paper we present an experimentally measurable signature of monopole dynamics and show that magnetic relaxation measurements in spin ice materials can be interpreted entirely in terms of their diffusive motion on a diamond lattice in the grand canonical ensemble. The monopole trajectories are constrained to lie on a network of Dirac strings filling the quasi-particle vacuum. We find quantitative agreement between the time scales for relaxation in the Dirac string network and the magnetic relaxation data for the spin ice material $Dy_{2}Ti_{2}O_{7}$ . In the presence of a magnetic field the topology of the network prevents charge flow in the steady state, but transient monopole currents do occur, as well as monopole density gradients near the surface of an open system.