Probability spaces over graphs: which area has focus on them? Suppose a simple graph $G$. Now consider probability space $G(v;p)$ where $0\leq p\leq 1$ and $v$ vertices. I want to calculate globally-determined properties of $G(v;p)$ such as connectivity and expected value of indicator function $$\mathbb E_{p\sim [0,1]^n}(\phi(G))$$ in terms of st-connectedness where $p$ follows let say uniform distribution. 
I want to understand which area investigates such structures. Extremal graph theory? Probabilistic Method? Random Graphs? Or some other? 
Which area has a focus on probability spaces over graphs?

I start the thread with an observation:

  
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*The Probabilistic Method book by Noga Alon has a treament of random graphs on page 155 where they calculate the expected value and the threshold function for graph-theoretic property such as connectedness
  

