All networks look the same to me: Testing for homogeneity in networks

How can researchers test for heterogeneity in the local structure of a network? In this paper, we present a framework that utilizes random sampling to give subgraphs which are then used in a goodness of fit test to test for heterogeneity. We illustrate how to use the goodness of fit test for an analytically derived distribution as well as an empirical distribution. To demonstrate our framework, we consider the simple case of testing for edge probability heterogeneity. We examine the significance level, power and computation time for this case with appropriate examples. Finally we outline how to apply our framework to other heterogeneity problems.