In social systems, interactions frequently influence individual behavior and beliefs which can, in turn, impact interaction. From the perspective of complex networks, such phenomena have been usefully conceptualized as co-evolving networks, in which both the links between nodes and certain characteristics (or attributes) of nodes evolve over time, each in a way that impacts the other. To date, however, there has generally been far less attention given to statistical modeling and analysis for co-evolving networks. We develop a class of dynamic latent space network models with node attraction and edge persistence (DLSNM-ap) for characterizing social behaviors such as flocking, polarization and homophily in co-evolving networks. These models are potentially quite rich -- we initially focus on statistical inference and prediction for simplified variants of the model. We present an MCMC method for statistical inference and prediction tasks such as estimating parameters and inferring/predicting latent positions in the model with a given observed network time series.