Abstract: Spin-glasses are natural Gibbs distributions that have been studied in theoretical computer science for many decades. Recently, they have been gaining renewed attention from the community as they emerge naturally in neural computation and learning, network inference, optimisation and many other areas. Here we consider the Edwards-Anderson spin-glass distribution at inverse temperature \beta when the underlying graph is an instance of G(n, d/n). This is a random graph on n vertices such that each edge appears independently with probability d/n, where the expected degree d is constant. We study the problem of efficiently sampling from the aforementioned distribution using the well-known Markov chain called Glauber dynamics. For a certain range of \beta, that depends only on the expected degree d of the graph, and for typical instances of the Edwards-Anderson model on G(n, d/n), we show that the corresponding (single-site) Glauber dynamics exhibits mixing time O(n^2). The range of \beta for which we obtain our rapid-mixing result corresponds to the expected influence being smaller than 1/d. This bound is very natural, and we conjecture that it is the best possible for rapid mixing. As opposed to the mean-field spin-glasses, where the Glauber dynamics has been studied before, less is known for the diluted cases like the one we consider here. The latter problems are more challenging to work with because the corresponding instances involve two levels of randomness, i.e., the random graph instance and the random Gibbs distribution. We establish our results by utilising the well-known path-coupling technique. In the standard setting of Glauber dynamics on G(n, d/n) one has to deal with the so-called effect of high degree vertices. Here, with the spin-glasses, rather than considering vertex-degrees, it is more natural to use a different measure on the vertices of the graph, that we call aggregate influence. We build on the block-construction approach proposed by [Dyer, Flaxman, Frieze and Vigoda:2006] to circumvent the problem with the high degrees in the path-coupling analysis. Specifically, to obtain our results, we first establish rapid mixing for an appropriately defined block-dynamics. We design this dynamics such that vertices of large aggregate influence are placed deep inside their blocks. Then, we obtain rapid mixing for the (single-site) Glauber dynamics by utilising a comparison argument.