"net-position" of a node in directed Erdos-Renyi graph

Question

In a directed weighted random Erdos-Renyi graph $G(N,p)$ with only positive weights, let $e_{ij}$ denote the weight going from node $i$ to node $j$ and assume all $e_{ij}$'s are normally distributed. How can one calculate the expected value of the "net-position" of a node $i$?

i.e. is there a simple way to get the expected value of $\sum_{j=1}^n(e_{ij}-e_{ji})$?

or even better, to get the expected value of all the positive net-positions in the graph?

Thanks for any help.

Answer 1

Assuming the $e_{ij}$ are identical and independent normal distributions, any linear combination will also be normally distributed. Since then the expected value of $e_{ij} - e_{ji}$ is zero, the expected value of the "net-position" is always $0$.