Opinion Maximization in Social Networks via Leader Selection

Xiaotian Zhou, Zhongzhi Zhang

Abstract: We study a leader selection problem for the DeGroot model of opinion dynamics in a social network with n nodes and m edges, in the presence of s0 = O(1) leaders with opinion 0. Concretely, we consider the problem of maximizing the average opinion in equilibrium by selecting k = O(1) leaders with opinion 1 from the remaining n − s0 nodes, which was previously proved to be NP-hard. A deterministic greedy algorithm was also proposed to approximately solve the problem, which has an approximation factor (1 − 1/e) and time complexity O(n3), and thus does not apply to large networks. In this paper, we first give an interpretation for the opinion of each node in equilibrium and the disagreement of the model from the perspective of resistor networks. We then develop a fast randomized greedy algorithm to solve the problem. To this end, we express the average opinion in terms of the pseudoinverse and Schur complement of Laplacian matrix for . The key ingredients of our randomized algorithm are Laplacian solvers and node sparsifiers, where the latter can preserve pairwise effective resistance by viewing Schur complement as random walks with average length l. For any error parameter ϵ > 0, at each iteration, the randomized algorithm selects a node that deviates from the local optimum marginal gain at most ϵ. The time complexity of the fast algorithm is O(mkllog nϵ− 2). Extensive experiments on various real networks show that the effectiveness of our randomized algorithm is similar to that of the deterministic algorithm, both of which are better than several baseline algorithms, and that our randomized algorithm is more efficient and scalable to large graphs with more than one million nodes.