Since the pioneering work of K. Pearson, R.A. Fisher, J. Neyman, C.R. Rao and other towering figures in statistics, the bulk of statistical research has focused on attributes of individual population members and scenarios in which replication is possible. In recent decades, however, it has become more obvious that the world is interdependent and interconnected, as demonstrated by recent crises that started out as local problems and turned into global crises (e.g., pandemics, political and military conflicts, economic and financial crises). Worse, many such events are unique and cannot be replicated. A promising approach to understanding and predicting such events is based on joint probability models of attributes and connections between population members. Building on the convenient mathematical platform of graphical models and leveraging additional structure in large populations, we develop scalable joint probability models of attributes and connections. The resulting models admit dependence among attributes and connections, but possess conditional independence and factorization properties that facilitate statistical computing along with statistical theory, without requiring replications in the conventional sense of the word.