2016/07/29 by Ahmad Askarian, Rupei Xu, András Faragó · 1 citation
Computer Science · Mathematics · #Algorithm #Artificial intelligence #Bayesian Modeling and Causal Inference #Cluster analysis #Computer science #Convergence (economics) #Data mining #Limit (mathematics) #Machine learning #Markov Chains and Monte Carlo Methods #Markov chain #Markov chain Monte Carlo #Mathematical optimization #Mathematics #Measure (data warehouse) #Monte Carlo method #Rate of convergence #State space #Statistics #Stochastic processes and statistical mechanics #cs.DS #math.PR
paper · pdf · doi:10.3390/a9030050
published in Algorithms 9(3), 50 (Multidisciplinary Digital Publishing Institute)
openalex publication_date 2016/07/29 · openalex created_date 2016/08/23 · arxiv created 2018/09/18 · arxiv updated 2020/09/01 · openalex updated_date 2026/08/05
We consider the problem of estimating the measure of subsets in very large networks. A prime tool for this purpose is the Markov Chain Monte Carlo (MCMC) algorithm. This algorithm, while extremely useful in many cases, still often suffers from the drawback of very slow convergence. We show that in a special, but important case, it is possible to obtain significantly better bounds on the convergence rate. This special case is when the huge state space can be aggregated into a smaller number of clusters, in which the states behave approximately the same way (but their behavior still may not be identical). A Markov chain with this structure is called quasi-lumpable. This property allows the aggregation of states (nodes) into clusters. Our main contribution is a rigorously proved bound on the rate at which the aggregated state distribution approaches its limit in quasi-lumpable Markov chains. We also demonstrate numerically that in certain cases this can indeed lead to a significantly accelerated way of estimating the measure of subsets. The result can be a useful tool in the analysis of complex networks, whenever they have a clustering that aggregates nodes with similar (but not necessarily identical) behavior.