# On fitting a Poisson distribution to make sense of data

Hi guys I am working with a regular network which has the shape of a square grid and contains 100x100=10000 nodes. The edges (links) between these nodes simply follow the shape of a chess table: each node which is not placed in the corner or along the boundary has 4 connections only, all of them involving its nearest neighbors. Accordingly, the nodes on the boundary have 3 connections only, while the nodes in the four corners have 2.

Now, this brings me to my problem. If you plot such data you will figure out you have 4 nodes with links=2, 98*4 nodes with links=3, and 10000-(98*4)-4 with links=4. This translates into three tuples to plot: (x,y)=(2,4),(3,392),(4,9604). The resulting histogram is heavily skewed toward the right hand side:

My question is: what kind of distribution do you think would fit this dataset? I was thinking of a Poisson distribution (the x-axis values are discrete and not continuous) skewed to the right. I will appreciate any kind of help/guidance. Thank you!

• What do you mean by "make sense of"? What are you using the distribution for? Oct 19 '15 at 12:20
• This distribution will be useful for describing the degree distribution (sorry for repeating myself) of a regular, 2D lattice. Oct 19 '15 at 12:22

## 1 Answer

Unlike other graphs, the degree distribution is a function of N. Specifically, the number of nodes with k=2 is constant (4), the number of nodes with k=3 grows linearly with the length of one side $4l-4$, while the number of nodes with k=4 grows exponentially with the length of a side, $l^2-4l-4$. In the large $l$ (or $N$) limit, the degree distribution is 4.

You can get the exact distribution by normalizing this piecewise defined function (divide by $l^2=N$).

• This will only rescale (normalize) my sample histogram. My question was more like "what distribution could fit this very dataset, based on the histogram shape?". Based on the shape of this histogram I am not able to recall a distribution with which to fit the data and try to give them an analytical interpretation. Oct 19 '15 at 12:25
• Got it. Would you mind elaborating on your desired analytical properties? It's not clear to me what you are trying to do with the distribution. Oct 19 '15 at 12:33
• This is part of a task concerning the analysis of some specific kinds of networks (regular, random, scale-free, etc.). For the random and scale-free cases we have some pretty analytical models that describe their distribution (the degree distribution is a Poisson distribution for a random network, while the scale-free network follows a power law distribution). For the regular network I failed to find something, so I decided to work this out myself by building one and plotting its histogram. But now I am stuck at that precise question I re-formulated before... Oct 19 '15 at 12:38
• So, the properties I am after are simply those of an analytical curve, provided it exists, that fits these data. Oct 19 '15 at 12:40
• I learned stats in a traditional academic environment, so I don't know much about moocs. I have heard good things about Think Stats if you are already familiar with coding. Good luck! :) Oct 19 '15 at 13:31