Large metropolitan cities bring together diverse individuals, creating opportunities for cultural and intellectual exchanges, which can ultimately lead to social and economic enrichment. In this work, we present a novel network perspective on the interconnected nature of people and places, allowing us to capture the social diversity of urban locations through the social network and mobility patterns of their visitors. We use a dataset of approximately 37K users and 42K venues in London to build a network of Foursquare places and the parallel Twitter social network of visitors through check-ins. We define four metrics of the social diversity of places which relate to their social brokerage role, their entropy, the homogeneity of their visitors and the amount of serendipitous encounters they are able to induce. This allows us to distinguish between places that bring together strangers versus those which tend to bring together friends, as well as places that attract diverse individuals as opposed to those which attract regulars. We correlate these properties with wellbeing indicators for London neighbourhoods and discover signals of gentrification in deprived areas with high entropy and brokerage, where an influx of more affluent and diverse visitors points to an overall improvement of their rank according to the UK Index of Multiple Deprivation for the area over the five-year census period. Our analysis sheds light on the relationship between the prosperity of people and places, distinguishing between different categories and urban geographies of consequence to the development of urban policy and the next generation of socially-aware location-based applications.