Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks

Abstract

We are interested in fixed points in Boolean networks, {\em i.e.} functions $f$ from $\{0,1{\}}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{0,1{\}}^n$, and we characterizes a class $\mathcal{F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every $x$ in $\{0,1{\}}^n$ there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. Then, denoting by ${\mathcal{C}}^{+}$ (resp. ${\mathcal{C}}^{-}$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{F}$ are exactly the networks of ${\mathcal{C}}^{+}\cup {\mathcal{C}}^{-}$; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in ${\mathcal{C}}^{+}$ (resp. ${\mathcal{C}}^{-}$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetwork in ${\mathcal{C}}^{+}$.

arXiv comment40 pages