On reduction of graphs and Markov chain models

Abstract

Abstract-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.

Aggregation of Markov chains: an analysis of deterministic annealing based methods

“…In prior work, [9], [14], [15], we've studied the problem of graph clustering via aggregating nodes with similar edge connections. More specifically, for a general weighted directed graph we construct a reduced-order representative graph, such that a well-motivated dissimilarity measure between the two graphs is minimized.…”

“…For a given a Markov chain X with n states whose transition probability matrix is P we would like to find a loworder Markov chain Y with m states and transition matrix Q such that the dissimilarity between X and Y is minimized; this dissimilarity is given by [9], [14], [15] ν(P, Q) = min…”

“…Further, for large graph models, the identification of underlying coarse connectivity structure is frequently of primary interest. In this context, clustering and aggregation methods play an important role in terms of both tractability, for example in aggregation of large Markov chain models, and identification of underlying network structures.Recently, clustering based algorithms have been proposed for the purpose of determining reduced dimension graphbased models [9]. As such, an optimal aggregation of nodes in the graph is sought, where the optimality is evaluated based on a distance measure quantifying similarity in connectivity.…”

“…Recently, clustering based algorithms have been proposed for the purpose of determining reduced dimension graphbased models [9]. As such, an optimal aggregation of nodes in the graph is sought, where the optimality is evaluated based on a distance measure quantifying similarity in connectivity.…”

“…In this paper, we analyze aggregation algorithms previously proposed in [9], [14], [15] for reducing the size of directed weighted graphs and Markov chains. These algorithms are applied using an original formulation for a dissimilarity or distance function that explicitly measures how close two graphs or chains are, as determined by the connectivity between nodes or states.…”