I have a general question as to which methods are considered "standard" or "best practice" in asssessing the following type of hypothesis.
I am running simulations where I generate random graphs according to some prescription. I am interested in a certain graph property which is just an integer, call it $p$.
The one parameter in my simulations is the expected graph size $N$. I run simulations for a range of values of $N$, say $N=2^k$ for $k=1,2,\ldots20$. For each value of $N$ I do, say, 100 runs, obtaining 100 values for $p$.
I have a hypothesis about the distribution of $p$: I believe $p\sim Poisson(\alpha \sqrt N)$, for some $\alpha\in\mathbb R$.
Now I want to test that hypothesis.
I've done a number of things (maximum-likelihood estimation of $\lambda$ for each individual dataset against $Poisson(\lambda)$ and then a visual check how $\lambda$ goes with $N$, maximum-likelihood estimation of the constant $\alpha$ assuming the relationship given above, each one of these followed by different distribution fit test...) but I would be interested to know whether there is a "canonical" or "best practice" approach to this. What would a reader like to see as evidence for the hypothesis? What numbers or plots would be considered appropriate?