How to prove that adding incoming edges to source node doesn't alter the max flow
Let $G=(V,E)$ be your input graph. Now consider a maximum flow $f$ on $G$. 


  Let $f$ be a flow in $G$ such that the residual network $G_R$ has no s-t path, then $f$ is a maximum flow.


Let's define $G'=(V,E')$ to be your graph  with $E'=E \cup \{(u_i,s)\}$  for $i=1,...,N$.   

Since $f$ is a maximum flow on $G$, then $G_R$ has no s-t paths.
Now you can show that given $G'$, its residual network $G'_R$ does not have any s-t path too. Thus, $f$ is a max flow also for $G'$.