Looking at the process of construction of permutation graph I guess that there are a few ways to describe a disorder. The first one is connected with inversion in permutation:
Let $K_m$ be the number of m-cliques, $m \in N$, in a random permutation graph $G_n$ with $n$ vertices and $\pi_n$ is the corresponding permutation representation in $S_n$. Let $K_1=n$, and $K_2$ is the number of edges in $G_n$ is the number of inversions in $\pi_n$ (denoted by $Inv(\pi_n)$).
In other words the matter of understanding is that a 1−clique is a vertex and there are n of them and the 2−clique is an edge and by construction there are $Inv(π_n)$.
The second way is to measure the average
number of incorrectly ordered components as the number ($<j>$) of diagonal lines in the permutation graph. I've known about the approach from the article about permutation glass (pls see Figure 1 at page 2).
I'm trying to understand the pros and cons of the approaches. If the ways provide identical way to describe disorders or what's the difference?
Thank you for the explanation
The connection between the number of inversions and the permutation graph is that the inversion number (the number of inversions) is equal to the number of crossings in the graph of the permutation. From your post I guess this is well-known to you.
As you mention, there are various ways to measure how 'disordered' a permutation is or, conversely, the 'sortedness' of a permutation.
The measure given in the article (i.e. the number of diagonal lines) is different from the inversion number as can clearly from the graphs that you have drawn.
You ask in particular about the pros and cons of different measures. I guess that it will always depend on the context. The inversion number is a good/simple general purpose measure. The measure given in the article would seem more appropriate when there is a reasonably high probability that $\pi(i)=i$.
Other members of this online community may be able to give you more information of specific areas where the use of a particular measure is advantageous.
The equivalence with crossing number (Please ignore this if it's already familiar to you)
$(i,j)$ with $i<j$ is an inversion if and only if $\pi(i)>\pi(j)$.
Similarly, the arrow $i-\pi(i)$ crosses the arrow $j-\pi(j)$ if and only if $\pi(i)>\pi(j)$.
Personally, I find the crossing number very much easier to consider than the number of inversions.