Isospectral Reductions of Non-negative Matrices

TL;DR

The paper addresses the slow computation of stationary measures for large stochastic matrices when multiple eigenvalues lie near 1 by leveraging isospectral reductions to reduce dimension while preserving spectral structure. It establishes that the diameter semi-norm τ(A) decreases under isospectral reductions with λ = 1 and that Gershgorin regions shrink for positive stochastic matrices, linking structural properties to spectral concentration. A practical algorithmic scheme is proposed: choose a subset S, form the reduced matrix R_S(A,1), compute its stationary distribution v_R, and reconstruct the original stationary vector v_A via a corrective formula; randomizing S helps avoid ill-conditioning. Numerical experiments on Burr-distributed weights and Barabási–Albert graphs show faster convergence and higher accuracy for large matrices when eigenvalues near 1 are present, supporting the method's value for network analysis and centrality computations.

Abstract

Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic scheme to approximate the stationary measure of a stochastic matrix based on isospectral reductions. We run numerical experiments that indicate this scheme is advantageous when there is more than one eigenvalue near 1, precisely the case where iterative methods perform poorly. We give a partial explanation why this scheme should work well, showing that in some situations isospectral reduction improves the spectral gap.

Figures (7)

• Figure 1: Computation times and errors of nested random isospectral reduction and reconstruction of the stationary measure. In this specific example, the Mathematical default method does not converge.
• Figure 2: Computation times and errors of nested random isospectral reduction and reconstruction of stationary measure. The results are consistently more accurate.
• Figure 3: Scatter plot of $E50$ vs $T50$ for the second sparse Burr stochastic matrix.
• Figure 4: Computation time and error of nested random isospectral reduction and reconstruction of stationary measure. In this specific example the nonzero entries of the stochastic matrix are generated with uniform distribution. The dashed lines are the performance of the Mathematica default method.
• Figure 5: Fluctuation of computation times and errors for our method with reduction and reconstruction using isospectral reductions applied to a Barabási–Albert graph.

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Proposition 3.1
• proof
• Corollary 3.1
• Definition 3.3
• Proposition 3.2