Assuming one has a directed weighted graph with n nodes and let us denote the weight going from node $i$ to node $j$ by $e_{ij}$. Then one can simply just sum all weights $e_{ij}$ up.
But if we are now in the set-up of a directed weighted random Erdos-Renyi graph $G(N,p)$ where $N$ stands for the number of nodes and $p$ for the probability having a link between two edges (just pick for simplicity one direction by probability $1/2$, as we are in a directed graph), how can I calculate the expected value of $\sum_{i=1}^N e_{ij}$ assuming that the weights $e_{ij}$ follow a given distribution? (constant/uniform/normal ect)
Best Answer
Given a vertex $i,$ let $K_i$ be a random variable whose value is the number of edges incident from $i.$ By linearity of expectation, the expected value of the sum the weights on these edges is just $K_i\mu,$ where $\mu$ is the mean of the distribution of the weights. Therefore, if $p_k=\Pr(K_i=k),$ then the expected value of the sum of the weights is $$\sum_{k=1}^{n-1}kp_k\mu=\mu E(K_i)={\mu(n-1)p\over2}$$