Topic: Toric ideals on Expected value of Structure Functions in Random Graphs

Goal: to understand the toric ideal where the exponents $h_i$ and $s_j$ are st-vertex-cuts of a digraph

\begin{equation} f_{ij}(p)\in \prod_i x_i^{h_i} -\prod_i x_j^{s_j}. \end{equation}

where

\begin{equation} f_{ij}(p) = \mathbb E\left[(\phi(G')\mid j\not\in V(G')\right] - \mathbb E\left[\phi(G')\mid i\not\in V(G')\right]. \end{equation}

Definitions:

$G'=(V,A)$ is a random digraph

$f_{ij}(p)$ is a polynomial where $p$ is a probability and each of $i,j$ refer to a vertex of $G'$

$\mathbb E$ is expected value

$\phi$ is the structure function of $G'$

vertex-cut is a vertex partition $V_{\text{outside cut}}\cup V_{\text{cut}}=V$

st-vertex-cut is a vertex cut where vertices $s$ and $t$ are not connected by any path in $G'-V_{\text{cut}}$ where $-$ refers to vertex deletion