19 problems

Dense-limit conjecture. Asymptotic distributions can be derived in the dense setting as well, which would further strengthen the results.

Motif-constraint testing conjecture. Formulating null hypotheses that constrain several motif counts simultaneously could yield tests for coefficient nullity in ERGMs.

Let $(\alpha,h)\in\mathcal{U}_{\alpha,h}^{rs}\setminus\{(\alpha_c,h_c)\}$, and let $u^*_{\alpha,h}$ denote the relevant solution associated with the edge-triangle model. Define … T…

Consider the edge-triangle exponential random graph model in the phase coexistence case, with two optimal densities $p \in M_\beta$, and let $c_p^\eta(n)$ denote the mixture coeffi…

Let the edge-triangle exponential random graph model be at its critical point, and let $\zeta$ denote the appropriately normalized fluctuation of the edge count. Bianchi–De Launey'…

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 $G_n$ denote the sparse exponential random graph model under consideration, and let $a$ be the parameter appearing in the asymptotic number of triangles. Consider also the mode…

Central limit conjecture for the triangle count. If $(\alpha,h)\in\mathcal{U}^{rs}\setminus\{(\alpha_c,h_c)\}$, then

The two-phase limiting-law conjecture. For all $(\alpha,h)\in\mathcal{M}^{rs}$,

One-half condition for the GHS inequality. In the general setting, the GHS inequality holds if and only if

Consider the edge-triangle model in the regime where Theorem gives a convex combination of delta measures, with limiting maximizers $u_1^*$ and $u_2^*$. Let $\kappa$ denote the coe…

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 $u^*(\alpha_c,h_c)$…

An exponential random graph model (ERGM) is fitted using a complete set of constraints up to order $r'$, meaning that all subgraph-count constraints through that order are included…

Growth conjecture for $\tilde{\gamma}_n$. As $n$ grows,

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…

Let $\mathcal{G}_{n,0}$ be the set of triangle-free graphs, let $T(n,2)$ denote a Turán graph with two parts, and let $\mathbb Q_{n,\beta_1}$ be the probability distribution on tri…