The development of models for multiple heterogeneous network data is of critical importance both in statistical network theory and across multiple application domains. Although single-graph inference is well-studied, multiple graph inference is largely unexplored, in part because of the challenges inherent in appropriately modeling graph differences and yet retaining sufficient model simplicity to render estimation feasible. The common subspace independent-edge (COSIE) multiple random graph model addresses this gap, by describing a heterogeneous collection of networks with a shared latent structure on the vertices but potentially different connectivity patterns for each graph. The COSIE model is both flexible to account for important graph differences, and tractable to allow for accurate spectral inference. In particular, a joint spectral embedding leads to a simultaneous consistent estimation of the common invariant subspace, and asymptotically normal estimates of the individual graph parameters. Performance is demonstrated on a dataset of connectomes, showing an accurate classification and a meaningful determination of heterogeneity across different subjects.