6 problems

Let $H$) be a fixed graph containing $t$ triangles and $s$ two stars. Under the same conditions as in the two-star conditional central limit theorem, let $F_n$ denote the number of…

Let $\mu$ be an ERGM measure on graphs with vertex set $[n]$, and let $e_1,e_2,e_3$ be edges spanning a triangle in $K_n$. Let $e$ be an arbitrary edge. Triangle moment conjecture.…

Let two random graphs be sampled under a coupling of an ERGM and a corresponding Erdős–Rényi model. Localization conjecture. For certain ERGM specifications, under some coupling wi…

Let $X$ be sampled from the ERGM measure $\mu$ on graphs with vertex set $[n]$, let $e$ be any edge in $\binom{[n]}{2}$, and let $p^*$ denote the corresponding Erdős–Rényi edge pro…

Let $S_n$ be the number of non-zero elements of the random adjacency matrix under $\mathbb{P}_{n;\alpha,h}$, let $(\alpha_c,h_c)$ be the critical point, and let $u^*(\alpha_c,h_c)$…

Constant-community approximation conjecture. There is a constant $c$ independent of $N$, but dependent on the weights $\beta_i$, such that every $X\in\mathcal{X}_f$ is $o(n)$-close…