  The operation of creating edges has been widely applied to optimize relevant
quantities of opinion dynamics. In this paper, we consider a problem of
polarization optimization for the leader-follower opinion dynamics in a noisy
social network with $n$ nodes and $m$ edges, where a group $Q$ of $q$ nodes are
leaders, and the remaining $n-q$ nodes are followers. We adopt the popular
leader-follower DeGroot model, where the opinion of every leader is identical
and remains unchanged, while the opinion of every follower is subject to white
noise. The polarization is defined as the steady-state variance of the
deviation of each node's opinion from leaders' opinion, which equals one half
of the effective resistance $\mathcal{R}_Q$ between the node group $Q$ and all
other nodes. Concretely, we propose and study the problem of minimizing
$\mathcal{R}_Q$ by adding $k$ new edges with each incident to a node in $Q$. We
show that the objective function is monotone and supermodular. We then propose
a simple greedy algorithm with an approximation factor $1-1/e$ that
approximately solves the problem in $O((n-q)^3)$ time. To speed up the
computation, we also provide a fast algorithm to compute
$(1-1/e-\eps)$-approximate effective resistance $\mathcal{R}_Q$, the running
time of which is $\Otil (mk\eps^{-2})$ for any $\eps>0$, where the $\Otil
(\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. Extensive
experiment results show that our second algorithm is both effective and
efficient.
