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

TL;DR

This paper establishes bounds on the minimum control nodes needed in various degree-constrained Boolean networks, enhancing understanding of controllability in complex network models.

Contribution

It derives new combinatorial bounds on control node set sizes for specific Boolean network classes with degree constraints, including extensions to related functions.

Findings

Derived bounds for control node set sizes in four types of Boolean networks.

Extended results to networks with OR functions.

Discovered phenomena related to controllability and node division.

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.