Community detection in directed graphs using stationary distribution and hitting times methods

Community detection has been extensively developed using various algorithms. One of the most powerful algorithms for undirected graphs is Walktrap, which determines the distance between vertices by employing random walk and evaluates clusters using modularity based on vertex degrees. Although several directions have been explored to extend this method to directed graphs, none of them have been effective. In this paper, we investigate the Walktrap algorithm (Pons and Latapy in J Graph Algorithms Appl 10:191–218, 2006) and the spectral method (Newman in Phys Rev E 88:042822, 2013) and extend them to directed graphs. We propose a novel approach in which the distance between vertices is defined using hitting time, and modularity is computed based on the stationary distribution of a random walk. These definitions are highly effective, as algorithms for hitting time and stationary distribution have been developed, allowing for good computational complexity. Our proposed method is particularly useful for directed graphs, with the well-known results for undirected graphs being special cases. Additionally, we utilize the spectral method for these problems, and we have implemented our algorithms to demonstrate their plausibility and effectiveness.