Is this the correct time complexity of my algorithm?
For each of the n people in the list you do $|E|\cdot log(n)$ plus the time of getting it from the list, which is $log(n)$. So the time you start with is complexity: $O(n(log(n)+(log(n)|\cdot|E|)))=O(n\cdot|E|\cdot log(n))$

Assuming E could be as big as n, you could simply say the time complexity is $O(log(n)\cdot n^2)$.