On Boolean Control Networks with Maximal Topological Entropy

TL;DR

This paper characterizes when Boolean control networks achieve maximum topological entropy, providing a necessary and sufficient condition and demonstrating the NP-hardness of the problem.

Contribution

It introduces a condition based on algebraic state-space representation for maximal entropy and proves the computational complexity of verifying this condition.

Findings

Maximal topological entropy condition derived

Verification of the condition is exponential in complexity

Determined the problem is NP-hard for certain BCNs

Abstract

Boolean control networks (BCNs) are discrete-time dynamical systems with Boolean state-variables and inputs that are interconnected via Boolean functions. BCNs are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs. The topological entropy of a BCN with m inputs is a nonnegative real number in the interval [0,m*log 2]. Roughly speaking, a larger topological entropy means that asymptotically the control is "more powerful". We derive a necessary and sufficient condition for a BCN to have the maximal possible topological entropy. Our condition is stated in the framework of Cheng's algebraic state-space representation of BCNs. This means that verifying this condition incurs an exponential time-complexity. We also show that the problem of determining whether a BCN with n state variables and m=n inputs has a maximum topological entropy is NP-hard, suggesting that this problem cannot be solved in general using a polynomial-time algorithm.