Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Number of Control Nodes in Boolean Networks with Degree Constraints

Liangjie Sun, Wai-Ki Ching, Tatsuya Akutsu

TL;DR

The authors address the minimum control node set problem in degree-constrained Boolean networks by developing a unified combinatorial framework that yields four nontrivial bounds (general lower, best-case upper, worst-case lower, general upper) for four BN families: $k$-$k$-XOR-BN, simple $k$-$k$-AND/OR-BN, $k$-$k$-AND-BN with negation, and $k$-$k$-NC-BN. They introduce a three-set node partitioning strategy and leverage time-to-target metrics $t^*$ to relate control size to controllability, providing concrete constructions that achieve tight or near-tight bounds in several cases and extending results from AND to OR. The paper presents explicit lower and upper bounds, including $|U| obreak o obreak n/t^*$ and $|U| obreak o obreak n-t^*+1$ for XOR-based networks, as well as specialized results such as $|U|=n/2$ for certain $2$-$2$-XOR-BNs, and $|U|=n/k$ for NC-BNs. Simulation evidence on random instances confirms that the computed minimum control sets lie within the predicted bounds and reveals instances where NC-BNs are easier or harder to control than simple AND-BNs, highlighting both the practical relevance and the remaining theoretical gaps.

Abstract

This paper studies the minimum control node set problem for Boolean networks (BNs) with degree constraints. The main contribution is to derive the nontrivial lower and upper bounds on the size of the minimum control node set through combinatorial analysis of four types of BNs (i.e., $k$-$k$-XOR-BNs, simple $k$-$k$-AND-BNs, $k$-$k$-AND-BNs with negation and $k$-$k$-NC-BNs, where the $k$-$k$-AND-BN with negation is an extension of the simple $k$-$k$-AND-BN that considers the occurrence of negation and NC means nested canalyzing). More specifically, four bounds for the size of the minimum control node set: general lower bound, best case upper bound, worst case lower bound, and general upper bound are studied. By dividing nodes into three disjoint sets, extending the time to reach the target state, and utilizing necessary conditions for controllability, these bounds are obtained, and further meaningful results and phenomena are discovered. Notably, all of the above results involving the AND function also apply to the OR function.

On the Number of Control Nodes in Boolean Networks with Degree Constraints

TL;DR

The authors address the minimum control node set problem in degree-constrained Boolean networks by developing a unified combinatorial framework that yields four nontrivial bounds (general lower, best-case upper, worst-case lower, general upper) for four BN families: --XOR-BN, simple --AND/OR-BN, --AND-BN with negation, and --NC-BN. They introduce a three-set node partitioning strategy and leverage time-to-target metrics to relate control size to controllability, providing concrete constructions that achieve tight or near-tight bounds in several cases and extending results from AND to OR. The paper presents explicit lower and upper bounds, including and for XOR-based networks, as well as specialized results such as for certain --XOR-BNs, and for NC-BNs. Simulation evidence on random instances confirms that the computed minimum control sets lie within the predicted bounds and reveals instances where NC-BNs are easier or harder to control than simple AND-BNs, highlighting both the practical relevance and the remaining theoretical gaps.

Abstract

This paper studies the minimum control node set problem for Boolean networks (BNs) with degree constraints. The main contribution is to derive the nontrivial lower and upper bounds on the size of the minimum control node set through combinatorial analysis of four types of BNs (i.e., --XOR-BNs, simple --AND-BNs, --AND-BNs with negation and --NC-BNs, where the --AND-BN with negation is an extension of the simple --AND-BN that considers the occurrence of negation and NC means nested canalyzing). More specifically, four bounds for the size of the minimum control node set: general lower bound, best case upper bound, worst case lower bound, and general upper bound are studied. By dividing nodes into three disjoint sets, extending the time to reach the target state, and utilizing necessary conditions for controllability, these bounds are obtained, and further meaningful results and phenomena are discovered. Notably, all of the above results involving the AND function also apply to the OR function.

Paper Structure

This paper contains 8 sections, 16 theorems, 56 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Boolean networks consisting of XOR functions
  4. Boolean networks consisting of simple AND/OR functions
  5. Boolean networks consisting of AND/OR functions and negations
  6. Boolean networks consisting of nested canalyzing functions
  7. Simulation results
  8. Conclusion

Key Result

Proposition 1

For each even $n$, there exists a 2-XOR-BN with a control node set of size 2.

Figures (9)

  • Figure 1: 2-2-XOR-BNs for Examples \ref{['ex:xor1']} and \ref{['ex:xor2']}.
  • Figure 2: (a) A 2-2-XOR-BN for Example \ref{['ex:xor3']}. (b) The value of nodes under different controls.
  • Figure 3: BNs constructed in the proofs of Proposition \ref{['prop:2-and-upper']} and Proposition \ref{['prop:2-2-and-upper']}.
  • Figure 4: A 2-2-AND-BN, where the nodes in the top represent those at time $t$ and the nodes in the bottom represent those at time $t+1$.
  • Figure 5: Illustration of Eq. (1).
  • ...and 4 more figures

Theorems & Definitions (37)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Example 1
  • Example 2
  • Proposition 3
  • Remark 1
  • Example 3
  • Theorem 2
  • Remark 2
  • ...and 27 more