Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition

The grounded Laplacian matrix $\LL{-S}$ of a graph $\calG=(V,E)$ with $n=|V|$ nodes and $m=|E|$ edges is a $(n-s)\times (n-s)$ submatrix of its Laplacian matrix $\LL$, obtained from $\LL$ by deleting rows and columns corresponding to $s=|S| \ll n $ ground nodes forming set $S\subset V$. The smallest eigenvalue of $\LL{-S}$ plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding $k \ll n$ edges among all the nonexistent edges forming the candidate edge set $Q = (V\times V)\backslash E$, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set $Q$ to be $(S\times (V\backslash S))\backslash E$, and prove that it has the same optimal solution as the original problem, although the size of set $Q$ is reduced from $O(n^2)$ to $O(n)$. Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio $(1-e^{{-\alpha\gamma})/\alpha$} and time complexity $O(kn^4)$, where $\gamma$ and $\alpha$ are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time $\tilde{O}(km)$, where $\tilde{O}(\cdot)$ suppresses the ${\rm poly} (\log n)$ factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.