The identification of crucial nodes in complex networks is a fundamental problem with broad applications in graph mining, influence maximization, and other domains. Centrality measures, such as Markov centrality, quantify node importance by leveraging random walk dynamics, particularly hitting times. However, optimizing group Markov centrality, which is defined as the inverse of the expected hitting time to a node set, poses significant computational challenges due to its NP-hard nature. In this work, we propose efficient approximation algorithms based on dynamic forest sampling and Schur complement techniques to address this problem. Our algorithms exploit the supermodularity of hitting time functions and employ rooted spanning forest sampling to estimate electrical network quantities, enabling scalable and accurate node selection. Theoretical guarantees demonstrate that our methods achieve near-linear solutions with provable error bounds. Extensive experiments on diverse real-world networks validate the practical effectiveness of our approaches, demonstrating significant improvements in computational efficiency and scalability compared to conventional methods.

Online social networks have become an integral part of modern society, profoundly influencing how individuals form and exchange opinions across diverse domains ranging from politics to public health. The Friedkin-Johnsen model serves as a foundational framework for modeling opinion formation dynamics in such networks. In this paper, we address the computational task of efficiently determining the equilibrium opinion vector and associated metrics including polarization and disagreement, applicable to both directed and undirected social networks. We propose a deterministic local algorithm with relative error guarantees, scaling to networks exceeding ten million nodes. Further acceleration is achieved through integration with successive over-relaxation techniques, where a relaxation factor optimizes convergence rates. Extensive experiments on diverse real-world networks validate the practical effectiveness of our approaches, demonstrating significant improvements in computational efficiency and scalability compared to conventional methods.

Public opinion governance in social networks is critical for public health campaigns, political elections, and commercial marketing. In this paper, we addresse the problem of maximizing overall opinion in social networks by strategically modifying the internal opinions of key nodes. Traditional matrix inversion methods suffer from prohibitively high computational costs, prompting us to propose two efficient sampling-based algorithms. Furthermore, we develop a deterministic asynchronous algorithm that exactly identifies the optimal set of nodes through asynchronous update operations and progressive refinement, ensuring both efficiency and precision. Extensive experiments on real-world datasets demonstrate that our methods outperform baseline approaches. Notably, our asynchronous algorithm delivers exceptional efficiency and accuracy across all scenarios, even in networks with tens of millions of nodes.