“…Curved ERGMs with geometrically weighted model terms are well-posed as long as θ 3 ≥ 0; note that θ 3 ∈ [− log 2, 0) implies that the added value of the m-th triangle either decreases or increases, depending on the sign of θ 2 and whether m is even or odd, and that θ 3 ∈ (−∞, − log 2) implies a form of model near-degeneracy when |N| is large (Schweinberger, 2011). In practice, curved ERGMs with GWESP terms and other geometrically weighted model terms have turned out to be considerably better-behaved than the triangle model: selected applications can be found in Snijders et al (2006), Hunter and Handcock (2006), Hunter (2007), Hunter, Goodreau and, Goodreau, Kitts and Morris (2009), Gile and Handcock (2006), Handcock and Gile (2010), Koskinen, Robins and Pattison (2010), Simpson, Hayasaka and Laurienti (2011), Suesse (2012), Rolls et al (2013), Wang et al (2013), Almquist and Bagozzi (2015), Obando andDe Vico Fallani (2017), andGondal (2018). We apply curved ERGMs to human brain network data in Section 8, demonstrating that curved ERGMs outperform both Bernoulli random graphs and latent space models.…”