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Assigning Stationary Distributions to Sparse Stochastic Matrices

Nicolas Gillis, Paul Van Dooren

TL;DR

The paper tackles constructing a minimal-norm perturbation to a sparse Markov transition matrix to enforce a specified target stationary distribution under a constrained perturbation support. It presents two complementary approaches: a closed-form family of perturbations for the case $\Omega=\mathrm{supp}(G+I_n)$ with detailed optimality and rank-one conditions, and a scalable linear-optimization framework with $\ell_1$ norm and a column-generation strategy for large sparse problems. The proposed methods demonstrate near-global optimality in practice, significantly faster and sparser than naive dense perturbations, and scale to matrices with up to about $10^5$ rows/columns in minutes. These results enable practical, targeted adjustments in networked systems (e.g., road or web graphs) to achieve desired long-run behavior while preserving sparsity and interpretability.

Abstract

The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix $G$ and a target stationary distribution $\hat μ$, construct a minimum norm perturbation, $Δ$, such that $\hat G = G+Δ$ is also stochastic and has the prescribed target stationary distribution, $\hat μ$. In this paper, we revisit the TSDP under a constraint on the support of $Δ$, that is, on the set of non-zero entries of $Δ$. This is particularly meaningful in practice since one cannot typically modify all entries of $G$. We first show how to construct a feasible solution $\hat G$ that has essentially the same support as the matrix $G$. Then we show how to compute globally optimal and sparse solutions using the component-wise $\ell_1$ norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to $10^5 \times 10^5$ in a few minutes. We illustrate the proposed algorithms with several numerical experiments.

Assigning Stationary Distributions to Sparse Stochastic Matrices

TL;DR

The paper tackles constructing a minimal-norm perturbation to a sparse Markov transition matrix to enforce a specified target stationary distribution under a constrained perturbation support. It presents two complementary approaches: a closed-form family of perturbations for the case with detailed optimality and rank-one conditions, and a scalable linear-optimization framework with norm and a column-generation strategy for large sparse problems. The proposed methods demonstrate near-global optimality in practice, significantly faster and sparser than naive dense perturbations, and scale to matrices with up to about rows/columns in minutes. These results enable practical, targeted adjustments in networked systems (e.g., road or web graphs) to achieve desired long-run behavior while preserving sparsity and interpretability.

Abstract

The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix and a target stationary distribution , construct a minimum norm perturbation, , such that is also stochastic and has the prescribed target stationary distribution, . In this paper, we revisit the TSDP under a constraint on the support of , that is, on the set of non-zero entries of . This is particularly meaningful in practice since one cannot typically modify all entries of . We first show how to construct a feasible solution that has essentially the same support as the matrix . Then we show how to compute globally optimal and sparse solutions using the component-wise norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to in a few minutes. We illustrate the proposed algorithms with several numerical experiments.
Paper Structure (28 sections, 9 theorems, 64 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 28 sections, 9 theorems, 64 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution and outline of the paper
  3. Notation
  4. TSDP with $\Omega = \mathop{\mathrm{supp}}\nolimits(G+I_n)$
  5. Closed-form feasible solutions for $\Omega = \mathop{\mathrm{supp}}\nolimits(G+I_n)$
  6. How good is $\Delta(\hbox{\boldmath$\alpha$}^*)$ ?
  7. Support of $\Delta(\hbox{\boldmath$\alpha$}^*)$ and optimal rank-1 solutions
  8. Rank-1 case
  9. What $\hat{\hbox{\boldmath$\mu$}}$ can we choose to make $\Delta(\hbox{\boldmath$\alpha$}^*)$ rank-1?
  10. Optimality of the rank-1 solution $\Delta(\hbox{\boldmath$\alpha$}^*)$
  11. Ordering of $\hbox{\boldmath$\mu$}$
  12. Formulation with linear optimization
  13. Linear optimization formulation
  14. Sparsity of $\ell_1$ norm minimization
  15. Efficient algorithms via a column-generation approach
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $G$ be an irreducible stochastic matrix. Then the family of matrices is a closed convex set of stochastic matrices, and the subset ${\cal G}_{\hbox{\boldmath$\alpha$}<\hbox{\bf 1}_n}:=\{G(\hbox{\boldmath$\alpha$}) \ | \ \hbox{\bf 0}_n \le \hbox{\boldmath$\alpha$} < \hbox{\bf 1}_n\}$is the set of the irreducible stochastic matrices within${\cal G}_{\hbox{\boldmath$\a

Figures (3)

  • Figure 1: Average computational time to compute the solution $S$, that is, the solution of \ref{['eq:sparseTSDP']} with $\Omega = \mathop{\mathrm{supp}}\nolimits(G+I)$, for five queue-like transition matrices generated randomly (with about $2k$ non-zero entries per row) and $\hat{\hbox{\boldmath$\mu$}} = G^\top \hbox{\bf 1}_n/n$. This time accounts for the formulation of \ref{['eq:sparseTSDP']} and its resolution.
  • Figure 2: Average computational time to solve of \ref{['eq:sparseTSDP']} with $\Omega = \mathop{\mathrm{supp}}\nolimits(G+I)$, for 5 queue-like transition matrix generated randomly (with about $2k$ non-zero entries per row) and $\hat{\hbox{\boldmath$\mu$}} = \hbox{\bf 1}_n/n$. This time accounts only for the resolution \ref{['eq:sparseTSDP']}, not the formulation.
  • Figure 3: Average computational time to formulate and solve \ref{['eq:sparseTSDP']} with $\Omega = \mathop{\mathrm{supp}}\nolimits(G+I)$ for 5 queue-like transition matrix generated randomly (with about $2k$ non-zero entries per row) and $\hat{\hbox{\boldmath$\mu$}} = \hbox{\bf 1}_n/n$. The black curve is $\zeta \mathop{\mathrm{nnz}}\nolimits(G)^2$ with $\zeta = 9 \cdot 10^{-10}$. These are the same values as in Figure \ref{['fig:timeSall']}.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Example 2.6
  • ...and 14 more