Previous formulations of the gravity kernel in the literature have considered either density-dependent (1 ¼ 0) or density-independent transmission (1 ¼ 1). Figure 2 compares the fit (expressed by the posterior deviance) of these two formulations and with that from the model where the degree of density dependence, 1, is estimated. This comparison is made for models assuming no linear or a fitted power-dependence of spatial coupling on both source and destination city population size. In figure 2a , w is estimated, whereas in b it is fixed at 1. See model components in the electronic supplementary material for further comparisons.