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A Discrete Perspective Towards the Construction of Sparse Probabilistic Boolean Networks

Christopher H. Fok, Chi-Wing Wong, Wai-Ki Ching

TL;DR

This work addresses constructing sparse Probabilistic Boolean Networks by decomposing a transition probability matrix $P$ into a convex combination of BN matrices: $P=\sum_{i=1}^K x_i A_i$, seeking minimal $K$ with $A_i\in B_n(P)$. It introduces the Greedy Entry Removal (GER) algorithm, augmented by GERESA for BN selection, and proves two upper-bound results for GER as well as for two SER variants; it also initiates a lower-bound theory for the decomposition length. Through extensive numerical experiments on synthetic and practical TPMs, GER consistently produces the sparsest decompositions and outperforms SER1, SER2, and MOMP in both sparsity and speed. The theoretical and empirical results together provide a practical, principled approach for sparse PBN construction with potential impact on ill-posed inverse problems and applications in biology, manufacturing, finance, and healthcare. Overall, GER emerges as a fast, effective algorithm that achieves near-optimal sparsity across a broad class of TPMs, with solid guarantees and clear pathways for future refinement and extension.

Abstract

Boolean Network (BN) and its extension Probabilistic Boolean Network (PBN) are popular mathematical models for studying genetic regulatory networks. BNs and PBNs are also applied to model manufacturing systems, financial risk and healthcare service systems. In this paper, we propose a novel Greedy Entry Removal (GER) algorithm for constructing sparse PBNs. We derive theoretical upper bounds for both existing algorithms and the GER algorithm. Furthermore, we are the first to study the lower bound problem of the construction of sparse PBNs, and to derive a series of related theoretical results. In our numerical experiments based on both synthetic and practical data, GER gives the best performance among state-of-the-art sparse PBN construction algorithms and outputs sparsest possible decompositions on most of the transition probability matrices being tested.

A Discrete Perspective Towards the Construction of Sparse Probabilistic Boolean Networks

TL;DR

This work addresses constructing sparse Probabilistic Boolean Networks by decomposing a transition probability matrix into a convex combination of BN matrices: , seeking minimal with . It introduces the Greedy Entry Removal (GER) algorithm, augmented by GERESA for BN selection, and proves two upper-bound results for GER as well as for two SER variants; it also initiates a lower-bound theory for the decomposition length. Through extensive numerical experiments on synthetic and practical TPMs, GER consistently produces the sparsest decompositions and outperforms SER1, SER2, and MOMP in both sparsity and speed. The theoretical and empirical results together provide a practical, principled approach for sparse PBN construction with potential impact on ill-posed inverse problems and applications in biology, manufacturing, finance, and healthcare. Overall, GER emerges as a fast, effective algorithm that achieves near-optimal sparsity across a broad class of TPMs, with solid guarantees and clear pathways for future refinement and extension.

Abstract

Boolean Network (BN) and its extension Probabilistic Boolean Network (PBN) are popular mathematical models for studying genetic regulatory networks. BNs and PBNs are also applied to model manufacturing systems, financial risk and healthcare service systems. In this paper, we propose a novel Greedy Entry Removal (GER) algorithm for constructing sparse PBNs. We derive theoretical upper bounds for both existing algorithms and the GER algorithm. Furthermore, we are the first to study the lower bound problem of the construction of sparse PBNs, and to derive a series of related theoretical results. In our numerical experiments based on both synthetic and practical data, GER gives the best performance among state-of-the-art sparse PBN construction algorithms and outputs sparsest possible decompositions on most of the transition probability matrices being tested.
Paper Structure (30 sections, 29 theorems, 104 equations, 25 tables, 6 algorithms)

This paper contains 30 sections, 29 theorems, 104 equations, 25 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Boolean Networks
  3. Probabilistic Boolean Networks
  4. Problem Formulation
  5. Previous Work
  6. Our Contributions
  7. Organization
  8. Some Notations and Terminologies
  9. Existing Sparse PBN Construction Algorithms and their Upper Bound Theorems
  10. On SER 1 and SER 2
  11. SER 1
  12. SER 2
  13. Termination of the SER 1 and SER 2 Algorithms
  14. New Upper Bound Theorems for SER 1 and SER 2
  15. Modified Orthogonal Matching Pursuit Algorithm
  16. ...and 15 more sections

Key Result

Proposition 2.1

Let $P = rQ$ where $r > 0$ and $Q$ is a $2^{n} \times 2^{n}$ TPM. Suppose that $x_{1}, x_{2}, \ldots, x_{K}$ are real numbers and $A_{1}, A_{2}, \ldots, A_{K}$ are BN matrices such that $P = \sum^{K}_{i = 1} x_{i} A_{i}$. Then $\sum^{K}_{i = 1} x_{i} = r$.

Theorems & Definitions (31)

  • Example 1.1
  • Example 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • ...and 21 more