Maybe my question is wrong or not clear so I would be grateful for any modification.
I am discovering the Gibbs-Boltzmann distribution but it seems strange for me and really hard to understand!
Generally, according to the EXPONENTIAL RANDOM GRAPHS, we can define the distribution over a graph as
$$P(G)=\frac{1}{Z} \exp \{-H(G)\},$$ where the normalizing constant $Z= \sum_{G} \exp \{-H(G) \}$ and $H(G)$ represents the graph Hamiltonian
What does it mean to define the distribution over a certain vertex? how can I also define the distribution over vertices $P(x_i)$? for example we will replace the graph Hamiltonian $H(G)$ by $H(x)$ ! but what does this mean? and also the same question for the normalizing constant $Z$?
Thanks for any correcting me if there is any misunderstanding!
The Gibbs measure $P$ in the paper is not a distribution over a graph, but rather a distribution over the set $\mathcal{G}$ of graphs. So it simply means that each graph $G\in\mathcal{G}$ is chosen with probability
$$P(G) := \frac{e^{-H(G)}}{Z},$$
where $Z$ is simply a constant so that $P$ is a probability law, i.e., $\sum_{G\in\mathcal{G}}P(G) = 1$ holds.
Up to this point, there is no relation between different elements of $\mathcal{G}$. To make sense of talking about vertices, we fix an ambient graph $(V,E)$ and let $\mathcal{G}$ be the family of its sugbraphs. Then, for example,
$$ \mathbb{P}(\text{vertex $v$ is present}) = \mathbb{P}(v \in G) = \sum_{\substack{G \in \mathcal{G} \\ v \in G}} P(G) $$
is simply the sum of all $P(G)$'s over $G\in \mathcal{G}$ having the vertex $v$.