K-Independent Boolean Networks

TL;DR

This paper introduces the independence number for Boolean networks, linking it to combinatorial designs and analyzing how the interaction graph's structure influences the network's fixed points.

Contribution

It defines the independence number for Boolean networks, relates it to combinatorial designs, and provides conditions and constructions for $k$-independent networks.

Findings

A condition on in-degree for $k$-independence.

All regulatory networks are at most $n/2$-independent.

Constructs $k$-independent networks for all $k$ in monotone complete graphs.

Abstract

This paper proposes a new parameter for studying Boolean networks: the independence number. We establish that a Boolean network is $k$-independent if, for any set of $k$ variables and any combination of binary values assigned to them, there exists at least one fixed point in the network that takes those values at the given set of $k$ indices. In this context, we define the independence number of a network as the maximum value of $k$ such that the network is $k$-independent. This definition is closely related to widely studied combinatorial designs, such as "$k$-strength covering arrays", also known as Boolean sets with all $k$-projections surjective. Our motivation arises from understanding the relationship between a network's interaction graph and its fixed points, which deepens the classical paradigm of research in this direction by incorporating a particular structure on the set of fixed points, beyond merely observing their quantity. Specifically, among the results of this paper, we highlight a condition on the in-degree of the interaction graph for a network to be $k$-independent, we show that all regulatory networks are at most $n/2$-independent, and we construct $k$-independent networks for all possible $k$ in the case of monotone networks with a complete interaction graph.