A Hierarchical Random Graph Efficient Sampling Algorithm Based on Improved MCMC Algorithm

Abstract

A hierarchical random graph (HRG) model combined with a maximum likelihood approach and a Markov Chain Monte Carlo algorithm can not only be used to quantitatively describe the hierarchical organization of many real networks, but also can predict missing connections in partly known networks with high accuracy. However, the computational cost is very large when hierarchical random graphs are sampled by the Markov Chain Monte Carlo algorithm (MCMC), so that the hierarchical random graphs, which can describe the characteristics of network structure, cannot be found in a reasonable time range. This seriously limits the practicability of the model. In order to overcome this defect, an improved MCMC algorithm called two-state transitions MCMC (TST-MCMC) for efficiently sampling hierarchical random graphs is proposed in this paper. On the Markov chain composed of all possible hierarchical random graphs, TST-MCMC can generate two candidate state variables during state transition and introduce a competition mechanism to filter out the worse of the two candidate state variables. In addition, the detailed balance of Markov chain can be ensured by using Metropolis–Hastings rule. By using this method, not only can the convergence speed of Markov chain be improved, but the convergence interval of Markov chain can be narrowed as well. Three example networks are employed to verify the performance of the proposed algorithm. Experimental results show that our algorithm is more feasible and more effective than the compared schemes.