  Subgraph counts - in particular the number of occurrences of small shapes
such as triangles - characterize properties of random networks, and as a result
have seen wide use as network summary statistics. However, subgraphs are
typically counted globally, and existing approaches fail to describe
vertex-specific characteristics. On the other hand, rooted subgraph counts -
counts focusing on any given vertex's neighborhood - are fundamental
descriptors of local network properties. We derive the asymptotic joint
distribution of rooted subgraph counts in inhomogeneous random graphs, a model
which generalizes many popular statistical network models. This result enables
a shift in the statistical analysis of large graphs, from estimating network
summaries, to estimating models linking local network structure and
vertex-specific covariates. As an example, we consider a school friendship
network and show that local friendship patterns are significant predictors of
gender and race.
