Phase transitions in the edge/concurrent vertex model

ABSTRACT

Although it is well known that some exponential family random graph model (ERGM) families exhibit phase transitions (in which small parameter changes lead to qualitative changes in graph structure), the behavior of other models is still poorly understood. Recently, Krivitsky and Morris have reported a previously unobserved phase transition in the edge/concurrent vertex family (a simple starting point for models of sexual contact networks). Here, we examine this phase transition, showing it to be a first-order transition with respect to an order parameter associated with the fraction of concurrent vertices. This transition stems from weak cooperativity in the recruitment of vertices to the concurrent phase, which may not be a desirable property in some applications.

KEYWORDS:

• ERGMs
• phase transitions
• edge/concurrent vertex model
• concurrency
• partition function

Notes

¹ It should be noted that the case of ${\theta }_c>0$, pendant avoidance, is also interesting in contexts for which relying on a single partner is penalized. The most obvious example would be negative exchange networks (Willer, 1999), in which pendant nodes are vulnerable to being placed in Bertrand competition.

² This should not be confused with the ultra-dense regimes that arise, e.g., from the edge-triangle model. Where ${\theta }_e\ll 0$, these graphs may be quite sparse in an absolute sense. They are, however, dense enough to have mean degree near or exceeding 2.

³ Note that it is possible for temperature to be negative; this is a common feature of systems for which the maximum energy per degree of freedom is bounded, and is not peculiar to ERGMs.

⁴ Or better, if we observe such a system at a random time, we are more likely to find it in low energy states than high energy states.

Additional information

Funding

This work was supported by NSF awards [DMS-1361425 and SES-1826589].