Continuous Time Graph Processes with Known ERGM Equilibria: Contextual Review, Extensions, and Synthesis

Graph processes that unfold in continuous time are of obvious theoretical and practical interest. Particularly useful are those whose long-term behavior converges to a graph distribution of known form. Here, we review some of the conditions for such convergence, and provide examples of novel and/or known processes that do so. These include subfamilies of the well-known stochastic actor oriented models, as well as continuum extensions of temporal and separable temporal exponential family random graph models. We also comment on some related threads in the broader work on network dynamics, which provide additional context for the continuous time case. Graph processes that unfold in continuous time are natural models for social network dynamics: able to directly represent changes in structure as they unfold (rather than, e.g. as snapshots at discrete intervals), such models not only offer the promise of capturing dynamics at high temporal resolution, but are also easily mapped to empirical data without the need to preselect a level of granularity with respect to which the dynamics are defined. Although relatively few general frameworks of this type have been extensively studied, at least one (the stochastic actor-oriented models, or SAOMs) is arguably among the most successful and widely used families of models in the social sciences (see, e.g., Snijders (2001); Steglich et al. (2010); Burk et al. (2007); Sijtsema et al. (2010); de la Haye et al. (2011); Weerman (2011); Schaefer and Kreager (2020) among many others). Work using other continuous time graph processes has also found applications both within (Koskinen and Snijders, 2007; Koskinen et al., 2015; Stadtfeld et al., 2017; Hoffman et al., 2020) and beyond (Grazioli et al., 2019; Yu et al., 2020) the social sciences, suggesting the potential for further advances.