  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 {\em
approximately} the same way (but their behavior still may not be identical). A
Markov chain with this structure is called {\em quasi-lumpable}. This property
allows the {\em 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.
