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intermediate updated. section 2 w.i.p.; 2 figures though the example might change. Section 2 still needs more text. Rest unchanged :(

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Tom Zuidberg
2026-05-14 14:45:52 +02:00
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@@ -21,6 +21,9 @@
% some packages which may be useful when drafting, but should be removed before submission
\usepackage{todonotes}
\usepackage{tikz}
% You can define your own commands. This is useful for shorthands, for example
\newcommand{\EE}{\mathbb{E}}
\newcommand{\RR}{\mathbb{R}}
@@ -54,7 +57,7 @@
\begin{document}
\title{Influence of update schemes on gene regulatory networks}
\title{Influence of update schemes on boolean networks}
% author information
% COMMENT OUT THESE LINES FOR YOUR CONFERENCE SUBMISSION!
@@ -81,7 +84,162 @@ Possible points to mention here:
\section{Boolean networks}
Explain Boolean network and what an update scheme is using the synchronous/parallel scheme. Also use figure, e.g. \cref{fig:boolean_example} to further visualize and create a better understanding of boolean networks. Also explain what chaotic behavior is due its relevance for the update schemes.
A boolean network consists of nodes $x_i(t)$ that have a boolean state, either $0$ or $1$, at a point in time $t$. Each node has a corresponding update function $x_i(t+1) = f_i(x_1(t), x_2(t), \ldots, x_n(t))$ expressing the new state of $x_i(t+1)$. The state of the boolean network can be describe as a boolean number $x_1 x_2\ldots x_n$ where each node is replaced with the corresponding state.
\begin{figure}
\centering
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\draw (111.33,207.8) node [anchor=north west][inner sep=0.75pt] {$ \begin{array}{l}
f_{1} =\neg x_{2}\\
f_{2} =x_{1}\\
f_{3} =x_{1} \oplus x_{4}\\
f_{4} =x_{3}
\end{array}$};
\end{tikzpicture}
\caption{Example of a boolean network with four nodes. Each vertex indicates that a node is part of the update function of the other node. For instance $x_1$ is part of $f_2$ and $f_3$.
}
\label{fig:bn_example}
\end{figure}
Consider the boolean network shown in \cref{fig:bn_example}. Assume the current state is $0011$, meaning that $x_1, x_2$ are $0$ and $x_3, x_4$ are $1$. The next state when applying the update function all at once is $1011$. Updating the network multiple times creates a trajectory, which is the sequence of the states starting from the initial state. With our example, the trajectory is $0011 \rightarrow 1011 \rightarrow 1101 \rightarrow 0100 \rightarrow \ldots$. Each trajectory in a reasonably small boolean network eventually reaches one of two scenarios. Either it falls into a state that doesn't change when updated, called attractor, or it falls into a cycle of states. In the example, there are 2 cycles of length 8 visible in \cref{fig:bn_ex_state_graph}.
\begin{figure}
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\centering
\caption{State graph of the boolean network shown in \cref{fig:bn_example} using synchronous update scheme.}
\label{fig:bn_ex_state_graph}
\end{figure}
\subsection{Notation}
Define clear notation used throughout the paper. Position of this subsection could change to be part of the Introduction instead.
@@ -93,16 +251,21 @@ Explain different update schemes including characteristics for behavior especial
\subsection{Synchronous scheme}
The Synchronous (also known as Parallel) update scheme assumes that every node is updated at once.
\subsection{Sequential scheme}
close to synchronous. the nodes update in a specific order and take into account the updated input node if that node had been updated before/is positioned earlier in the sequence
\subsection{Block-sequential scheme}
mix of synchronous and sequential. predefined blocks update sequential, inside a block the update follows the synchronous scheme
\subsection{Asynchronous deterministic}
one node is updated per tick following a specific sequence
\subsection{Asynchronous generalized}
same as asynchronous deterministic with the slight change that within the sequence nodes may appear multiple times.
\section{Relevance for Gene Regulatory Networks}
\label{sec:relevance_grn}
Tie the update schemes and their different outcomes or behavior to GRN.
@@ -116,18 +279,6 @@ References: \cite{schwab2020concepts}\cite{aracena2009robustness}\cite{bornholdt
\begin{figure} % The starred version uses both columns; unstarred only one column
\centering
\includegraphics[width=3in]{4bit BN.png}
% TIP: Ensure the original image file has approximately the right dimensions
% (if using matplotlib, specify correct figure size) so that the image is not rescaled too brutally.
\caption{Boolean example (image will be changed!).
% NB: The '~' inserts a non-breaking space, ensuring 'Ref.' is never separated from its number
Simple boolean network example; a graph of the network and most likely a table as well for the updated states $x(t) \rightarrow x(t+1)$ (currently missing)
}
\label{fig:boolean_example}
\end{figure}
\begin{figure*} % The starred version uses both columns; unstarred only one column
\centering