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

The grounded Laplacian matrix L-s of a graph G = (V,E) with n = | v | nodes and m = |E| edges is a (n -s) x(n -s) submatrix of its Laplacian matrix L, obtained from L by deleting rows and columns corresponding to s = | s | << n ground nodes forming set s subset of V. The smallest eigenvalue of L-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 << n edges among all the nonexistent edges forming the candidate edge set Q = (v x v)\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 x (V \S))\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(2)). 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 (O) over tilde (km), where (O) over tilde (center dot) suppresses the poly(log..) factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.