What is the probability of an edge?

Question

We call a graph $G$ $(p, α)$-jumbled if, for every induced subgraph holds; here p and α are real numbers with $0<p<1≤α$, and $e(H)$ is the number of edges in H. We show that a (p, α)-jumbled graph behaves in many ways like a random graph with edge probability $p$, and some aspects of this similarity are examined.

What is the probability of an edge from a graph? How do I calculate it? Please give me an example. Thank you.

Answer 1

A random graph with edge probability $p$ means a graph with a given number of vertices, where:

For any two vertices the probability that there is an edge connecting them is $p$, independently of the other edges.

You have therefore a set $\Omega$ made of all possible simple graphs on your set of $n$ vertices, and the probability of each graph is $p^k(1-p)^{N-k}$ where $k$ is the number of edges and $N$ is the number of potential edges $n\choose 2$.