Graph representations are used across disciplines for the analysis and visualization of relational data. Exponential random graph models allow for a general method of modelling the underlying stochastic process that has generated the observed data conditional on observer attributes of the vertices, or nodes. Recent developments in ERGMs have introduced the notions of local dependence and the exponential random network model, or ERNM. Local dependence enforces the assumption of independence between edges that connect nodes that are, in some sense, "far apart." This is formalized by the introduction of a neighborhood structure on the graph: a partition of the vertices with the property that edges between two neighborhoods are stochastically independent of all other edge variables. This independence allows for the proof of a desirable consistency condition and a central limit theorem for statistics of the graph. The random network model allows for the joint modeling of both the graph and random attributes of the vertices. This has useful applications in network analysis, as it allows researchers to make inferences about how graphical features affect vertices and vice versa. This project combines these two developments to show that ERNMs with the local dependence property have the same useful consistency property and that a similar central limit theorem also holds.