eng
episciences.org
Discrete Mathematics & Theoretical Computer Science
1365-8050
2011-01-01
DMTCS Proceedings vol. AP,...
Proceedings
10.46298/dmtcs.2978
2978
journal article
A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks
Adrien Richard
We are interested in fixed points in Boolean networks, $\textit{i.e.}$ functions $f$ from $\{0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the hypercubes contained in $\{0,1\}^n$, and we exhibit a class $\mathcal{F}$ of Boolean networks, called even or odd self-dual networks, satisfying the following property: if a network $f$ has no subnetwork in $\mathcal{F}$, then it has a unique fixed point. We then discuss this "forbidden subnetworks theorem''. We show that it generalizes the following fixed point theorem of Shih and Dong: 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. We also show that $\mathcal{F}$ contains the class $\mathcal{F'}$ of networks whose the interaction graph is a directed cycle, but that the absence of subnetwork in $\mathcal{F'}$ does not imply the existence and the uniqueness of a fixed point.
https://dmtcs.episciences.org/2978/pdf
feedback circuit
Boolean network
fixed point
self-dual Boolean function
discrete Jacobian matrix
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
[NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG]
[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]