probability that node has d links is

**binomial**$$\frac{(n-1)!}{d!(n-d-1)!} p^d (1-p)^{n-d-1}$$Large n, small p, this is approximately a

**Poisson**distribution $$\frac{(n-1)^d}{d!} p^d e^{-(n-1)p}$$why Poisson? If you want to approximate this if you want to approximate this formula(

*binomial formula*) for large n and relatively small p.- $(1-p)^{n-d-1} \rightarrow e^{-(n-1)p}$
- $\frac{(n-1)!}{d!(n-d-1)!} \rightarrow \frac{(n-1)^d}{d!}$
- $\frac{(n-1)!}{(n-d-1)!} \rightarrow (n-1)^d$

**Note:**

- many isolated nodes
- several components
- no component has more than a small fraction of the nodes, just starting to see one large one emerge

- More high and low degree nodes than predicted at random
- Citation Networks - too many with 0 citations, too many with high numbers of of citations to have citations drawn at random
- "Fat tails" compared to random network

$P(d) = cd^{-a}$

$log(P(d)) = log(c) - alog(d)$