Socioeconomic segregation is one of the main factors behind the emergence of large-scale inequalities in urban areas. We propose two non-parametric measures of spatial variance and local spatial diversity, based on the statistical properties of the trajectories of random walks on graphs. We analyse the segregation of synthetic systems and large metropolitan areas in the US and the UK and our results confirm that the spatial variance and local diversity as measured through simple diffusion on graphs provides meaningful insights about the spatial organisation of ethnicities across a city.