Modular decomposition of Markov chain: detecting hierarchical organization of pervasive communities

Bayesian Modeling of Random Walker for Community Detection in Networks

“…With the aid of the joint probability distribution (10) and the mean-field approximation (24), the variational lower bound (23) can be computed as…”

“…p , p (0) , α (1) , and η (1) 2: for t = 1, 2 The original MDMC model developed by Okamoto and Qiu utilized the EM algorithm to infer the model parameters [23,24]. In the following, we derive the EM algorithm based on the results of the variational Bayesian approach, and discuss differences between the original and current formulations.…”

“…On the other hand, the contributions from the random walkers are washed away in the vanishing limit of α (t) k . The current modeling of MDMC is slightly different from the original one [23,24] in the presence of the prior distribution of π (t) dk . Since the variance of the Dirichlet distribution shrinks in large parameter regimes, the original model can be recovered by taking the infinite η limit of ours.…”

“…The estimate of the probability distribution (36), the responsibility (37), and the update equation (19) of η (t) k can form a closed set of self-consistent equations, which are equivalent to those proposed in the original MDMC [23,24]. We note that there remains some differences in the policies of updating the parameters α k are updated at the end of each time step in the current formulation, while they are updated at the end of each EM step in the original MDMC.…”

“…In return for stable and high-quality results in MDMC, however, it is necessary to solve coupled master equations, which require sophisticated schemes to optimize model parameters. Unfortunately, the EM algorithm, which was utilized in the original formulation [23,24], does not always find the optimal structures because it is sensitive to initial values of the parameters. Hence, elaboration of the optimization algorithm is highly desired to make full use of the powerful modeling of MDMC.…”