Given a Graph G with vertex set $V$ and edge set $E$ and degree distribution $P(k) = k^{- \lambda}$ where $\lambda \in Z$ how can we find the expected amount of degree of nodes for $G$?
I started by using the expected value formula $ \int^{X}_{0} k \cdot k^{-\lambda}$ but I am not quite sure if this is the right approach. How would I approach this?
Nice question. In reality things are a bit more complicated than what I suggest down here, have a look at does a power law degree distribution imply graphs are sparse?.
If $P(k)$ is a probability distribution, you could use the expectation for a discrete random variable so so the expected degree of nodes should be $$\sum_{k=1}^{\infty} k^{1-\lambda}$$
But $P(k)$ has to be a probability (mass) distribution, which means that you should have $$\sum_{k} P(k) = 1$$
