On Reduction of Graphs and Markov Chain Models

This paper introduces a new method for reducing large directed graphs to simpler graphs with fewer nodes. The reduction is carried out through node and edge aggregation, where the simpler graph is representative of the original large graph. Representativeness is measured using a metric defined herein, which is motivated by thermodynamic free energy and vector quantization problems in the data compression literature. The resulting aggregation scheme is largely based on the maximum entropy principle. The proposed algorithm is general in the sense that it can accommodate a large class of functions for characterizing distance between the nodes. As a special case, we show that this method applies to the Markov chain model-reduction problem, providing a soft-clustering approach that enables better aggregation of state-transition matrices than existing methods. Simulation results are provided to illustrate the theoretical results.