Continuous time graph processes with known ERGM equilibria: contextual review, extensions, and synthesis

ABSTRACT

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.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ E.g., we need the change rate to not go to infinity in finite time, so that the system cannot “ergotize” out from under us.

² We also need non-simultaneous edge state transitions, i.e., all non-vanishing transition rates are on unit moves in the Hamming space of the support.

³ E.g., spontaneous deflation of a balloon is rather faster than the time required for it to spontaneously reinflate. But this seeming asymmetry is driven by the fact that we selected the balloon when it was in a very rare conformation, and the waiting time to observe such a rare conformation is very long. The “arrow” is in our selection process, not the system itself. This is merely the regression effect, much disguised.

⁴ This formulation itself appears earlier (Snijders & Duijn, 1997), but is not discussed as a general ERGM generating process.

Additional information

Funding

This work was supported by the NIH award 1R01GM144964-01 and NSF award SES-1826589. The author thanks Martina Morris, Michael Schweinberger, Chad Klumb, and Steve Goodreau for their input and helpful comments.