# Tikz: Randomly drop connections in neural network

I used Tikz to draw a full connected neural network. Now I would like to drop randomly a certain proportion of arrows. How can I do that and is it possible to use my code for that? Here is my code and an example output:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}

\begin{document}

\def\layersep{2cm}
\def\hsep{1cm}
\def\ilsize{8}
\def\hlsize{8}
\def\olsize{8}
\def\rootlrp{6}
\def\neuronsize{4mm}

\tikzset{>=latex}

\begin{figure}
\centering

\begin{tikzpicture}[shorten >=0pt, ->, draw=black!100, node distance=\layersep]
\tikzstyle{every pin edge}=[<-,shorten <=1pt]
\tikzstyle{neuron}=[circle, draw, fill=black!100, minimum size=\neuronsize,inner sep=0pt]
\tikzstyle{input neuron}=[neuron, fill=black!0]
\tikzstyle{hidden neuron}=[neuron, fill=black!0]
\tikzstyle{output neuron}=[neuron, fill=black!0]

%%%%%%%%%%%%
% DRAW NODES
%%%%%%%%%%%%
% Draw the input layer nodes
\foreach \name / \y in {1,...,\ilsize}
\node[input neuron] (In-\name) at (0.0cm+\hsep,-\y cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H0-\name) at (1.5cm+\hsep,-\y cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H1-\name) at (3.0cm+\hsep,-\y cm) {};
% Draw the output layer nodes
\foreach \name / \y in {1,...,\olsize}
\node[hidden neuron] (Out-\name) at (4.5cm+\hsep,-\y cm) {};

%%%%%%%%%%%%%%%%%%
% DRAW CONNECTIONS
%%%%%%%%%%%%%%%%%%
% Connect every node in the input layer with every node in the hidden layer.
\foreach \source in {1,...,\ilsize}
\foreach \dest in {1,...,\hlsize}
\path (In-\source) edge (H0-\dest);
% Connect first with second hidden layer
\foreach \source in {1,...,\hlsize}
\foreach \dest in {1,...,\hlsize}
\path (H0-\source) edge (H1-\dest);
% Connect every node from the last hidden layer with the output layer
\foreach \source in {1,...,\hlsize}
\foreach \dest in {1,...,\olsize}
\path (H1-\source) edge (Out-\dest);

\end{tikzpicture}
\end{figure}

\end{document}


• I don't know much about neural networks; what do you mean by "randomly dropping a certain proportion of connections"? Do you mean randomly selecting arrows until a certain percentage is reached, and remove them from the picture? – steve Apr 23 at 19:01
• @ABlueChameleon Yes, I want to drop arrows randomly. – recipe_for_disaster Apr 23 at 20:32

Here a \cutoff gets introduced. It is between 0 and 1. If you choose it closer to 1, more connections get dropped, if you take it closer to 0, less get dropped.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}

\begin{document}
\def\layersep{2cm}
\def\hsep{1cm}
\def\ilsize{8}
\def\hlsize{8}
\def\olsize{8}
\def\rootlrp{6}
\def\neuronsize{4mm}

\tikzset{>=latex}

\begin{figure}
\centering

\begin{tikzpicture}[shorten >=0pt, ->, draw=black!100, node distance=\layersep,
every pin edge/.style={<-,shorten <=1pt},
neuron/.style={circle, draw, fill=black!100, minimum size=\neuronsize,inner sep=0pt},
input neuron/.style={neuron, fill=black!0},
hidden neuron/.style={neuron, fill=black!0},
output neuron/.style={neuron, fill=black!0}]
\pgfmathsetmacro{\iyshift}{0.5*\ilsize-0.5*\hlsize}
\pgfmathsetmacro{\oyshift}{0.5*\olsize-0.5*\hlsize}
%%%%%%%%%%%%
% DRAW NODES
%%%%%%%%%%%%
% Draw the input layer nodes
\foreach \name / \y in {1,...,\ilsize}
\node[input neuron] (In-\name) at (0.0cm+\hsep,-\y cm+\iyshift cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H0-\name) at (1.5cm+\hsep,-\y cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H1-\name) at (3.0cm+\hsep,-\y cm) {};
% Draw the output layer nodes
\foreach \name / \y in {1,...,\olsize}
\node[hidden neuron] (Out-\name) at (4.5cm+\hsep,-\y cm+\oyshift cm) {};

%%%%%%%%%%%%%%%%%%
% DRAW CONNECTIONS
%%%%%%%%%%%%%%%%%%
\pgfmathsetmacro{\cutoff}{0.5}
% Connect every node in the input layer with every node in the hidden layer.
\foreach \source in {1,...,\ilsize}
{\foreach \dest in {1,...,\hlsize}
{\pgfmathparse{int(sign(rnd-\cutoff))}
\ifnum\pgfmathresult=1
\path (In-\source) edge (H0-\dest);
\fi}}
\pgfmathsetmacro{\cutoff}{0.3}
% Connect first with second hidden layer
\foreach \source in {1,...,\hlsize}
{\foreach \dest in {1,...,\hlsize}
{\pgfmathparse{int(sign(rnd-\cutoff))}
\ifnum\pgfmathresult=1
\path (H0-\source) edge (H1-\dest);
\fi}}
\pgfmathsetmacro{\cutoff}{0.7}
% Connect every node from the last hidden layer with the output layer
\foreach \source in {1,...,\hlsize}
{\foreach \dest in {1,...,\olsize}
{\pgfmathparse{int(sign(rnd-\cutoff))}
\ifnum\pgfmathresult=1
\path (H1-\source) edge (Out-\dest);
\fi}}

\end{tikzpicture}
\end{figure}

\end{document}


This is a version that replaces all these \defs by pgf keys. You can use it as

\begin{tikzpicture}[every pin edge/.style={<-,shorten <=1pt}]
\pic{neural network={inputs=7,outputs=6,
cutoff 1=0.5,cutoff 2=1.1,cutoff 3=0.2}};
\end{tikzpicture}


All the keys can be set on the spot, and if you have several of these networks, you things will become much easier. If you set a cutoff to a value larger than 1, all connections will be suppressed, if you set it to 0 or smaller, none of them.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\tikzset{pics/neural network/.style={code={
\tikzset{neural network/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/neural network/##1}}%
\pgfmathsetmacro{\iyshift}{0.5*\pv{inputs}-0.5*\pv{hidden}}
\pgfmathsetmacro{\oyshift}{0.5*\pv{outputs}-0.5*\pv{hidden}}
%%%%%%%%%%%%
% DRAW NODES
%%%%%%%%%%%%
% Draw the input layer nodes
\foreach \y in {1,...,\pv{inputs}}
\node[/tikz/neural network/input neuron] (In-\y) at (0.0cm,-\y cm+\iyshift cm) {};
% Draw the hidden layer nodes
\foreach \y in {1,...,\pv{hidden}}
\node[/tikz/neural network/hidden neuron] (H0-\y) at (2cm,-\y cm) {};
% Draw the hidden layer nodes
\foreach \y in {1,...,\pv{hidden}}
\node[/tikz/neural network/hidden neuron] (H1-\y) at (4cm,-\y cm) {};
% Draw the output layer nodes
\foreach \name / \y in {1,...,\pv{outputs}}
\node[/tikz/neural network/hidden neuron] (Out-\name) at (6cm,-\y cm+\oyshift cm) {};
%%%%%%%%%%%%%%%%%%
% DRAW CONNECTIONS
%%%%%%%%%%%%%%%%%%
% Connect every node in the input layer with every node in the hidden layer.
\foreach \source in {1,...,\pv{inputs}}
{\foreach \dest in {1,...,\pv{hidden}}
{\pgfmathparse{int(sign(rnd-\pv{cutoff 1}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (In-\source) edge (H0-\dest);
\fi}}
% Connect first with second hidden layer
\foreach \source in {1,...,\pv{hidden}}
{\foreach \dest in {1,...,\pv{hidden}}
{\pgfmathparse{int(sign(rnd-\pv{cutoff 2}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (H0-\source) edge (H1-\dest);
\fi}}
% Connect every node from the last hidden layer with the output layer
\foreach \source in {1,...,\pv{hidden}}
{\foreach \dest in {1,...,\pv{outputs}}
{\pgfmathparse{int(sign(rnd-\pv{cutoff 3}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (H1-\source) edge (Out-\dest);
\fi}}
}},neural network/.cd,inputs/.initial=6,outputs/.initial=6,
hidden/.initial=8,size/.initial=8mm,edge/.style={draw,->},
neuron/.style={circle, draw, fill=black!100,
minimum size=\pgfkeysvalueof{/tikz/neural network/size},inner sep=0pt},
input neuron/.style={/tikz/neural network/neuron, fill=black!0},
hidden neuron/.style={/tikz/neural network/neuron, fill=black!0},
output neuron/.style={/tikz/neural network/neuron, fill=black!0},
cutoff 1/.initial=0,
cutoff 2/.initial=0,
cutoff 3/.initial=0,}

\begin{document}
\tikzset{>=latex}

\begin{figure}
\centering
\begin{tikzpicture}[every pin edge/.style={<-,shorten <=1pt}]
\pic{neural network={inputs=7,outputs=6,
cutoff 1=0.5,cutoff 2=1.1,cutoff 3=0.2}};
\end{tikzpicture}
\end{figure}
\end{document}


In order to make things visually more appealing, you may let the probability depend on the distance between the neurons, and suppress connections to more distant neurons more strongly.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\tikzset{pics/neural network/.style={code={
\tikzset{neural network/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/neural network/##1}}%
\pgfmathsetmacro{\iyshift}{0.5*\pv{inputs}-0.5*\pv{hidden}}
\pgfmathsetmacro{\oyshift}{0.5*\pv{outputs}-0.5*\pv{hidden}}
%%%%%%%%%%%%
% DRAW NODES
%%%%%%%%%%%%
% Draw the input layer nodes
\foreach \y in {1,...,\pv{inputs}}
\node[/tikz/neural network/input neuron] (In-\y) at (0.0cm,-\y cm+\iyshift cm) {};
% Draw the hidden layer nodes
\foreach \y in {1,...,\pv{hidden}}
\node[/tikz/neural network/hidden neuron] (H0-\y) at (2cm,-\y cm) {};
% Draw the hidden layer nodes
\foreach \y in {1,...,\pv{hidden}}
\node[/tikz/neural network/hidden neuron] (H1-\y) at (4cm,-\y cm) {};
% Draw the output layer nodes
\foreach \name / \y in {1,...,\pv{outputs}}
\node[/tikz/neural network/hidden neuron] (Out-\name) at (6cm,-\y cm+\oyshift cm) {};
%%%%%%%%%%%%%%%%%%
% DRAW CONNECTIONS
%%%%%%%%%%%%%%%%%%
% Connect every node in the input layer with every node in the hidden layer.
\foreach \source in {1,...,\pv{inputs}}
{\foreach \dest in {1,...,\pv{hidden}}
{\pgfmathparse{int(sign(rnd-abs(\source-\pv{inputs}/2-\dest+\pv{hidden}/2)*\pv{cutoff 1}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (In-\source) edge (H0-\dest);
\fi}}
% Connect first with second hidden layer
\foreach \source in {1,...,\pv{hidden}}
{\foreach \dest in {1,...,\pv{hidden}}
{\pgfmathparse{int(sign(rnd-abs(\source-\pv{hidden}/2-\dest+\pv{hidden}/2)*\pv{cutoff 2}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (H0-\source) edge (H1-\dest);
\fi}}
% Connect every node from the last hidden layer with the output layer
\foreach \source in {1,...,\pv{hidden}}
{\foreach \dest in {1,...,\pv{outputs}}
{\pgfmathparse{int(sign(rnd-abs(\source-\pv{hidden}/2-\dest+\pv{outputs}/2)*\pv{cutoff 3}))}
\ifnum\pgfmathresult=1
\path[/tikz/neural network/edge] (H1-\source) edge (Out-\dest);
\fi}}
}},neural network/.cd,inputs/.initial=6,outputs/.initial=6,
hidden/.initial=8,size/.initial=8mm,edge/.style={draw,->},
neuron/.style={circle, draw, fill=black!100,
minimum size=\pgfkeysvalueof{/tikz/neural network/size},inner sep=0pt},
input neuron/.style={/tikz/neural network/neuron, fill=black!0},
hidden neuron/.style={/tikz/neural network/neuron, fill=black!0},
output neuron/.style={/tikz/neural network/neuron, fill=black!0},
cutoff 1/.initial=0,
cutoff 2/.initial=0,
cutoff 3/.initial=0,}

\begin{document}
\tikzset{>=latex}

\begin{figure}
\centering
\begin{tikzpicture}[every pin edge/.style={<-,shorten <=1pt}]
\pic{neural network={inputs=7,outputs=6,
cutoff 1=0.2,cutoff 2=0.25,cutoff 3=0.3}};
\end{tikzpicture}
\end{figure}
\end{document}


• Is it possible to reset the random seed? Or how can I drop the same arrows between the first two layers and last two layers? – recipe_for_disaster Apr 23 at 20:56
• @random9 Does replacing \pgfmathsetmacro{\cutoff}{0.3} by \pgfmathsetmacro{\cutoff}{1.1} give you what you want? – user194703 Apr 23 at 21:12
• @random9 I added a version which is much easier to handle (in my opinion) and also addresses the \def problem. – user194703 Apr 23 at 21:20
• What an answer! It could actually be adapted to have a very pedagogic way of explaining the dynamic of prices on stock Exchanges. Question to follow soon ! – JeT Apr 23 at 22:22

Now, because apparently I don't know when to stop, this is a version that will draw exactly \percentage% of the total number of possible connections, never more, never less (which is the one disadvantage I see in @Schrödinger's cat's otherwise much nicer answer).

The basic idea in this approach is to assign to each possible connection a number, and then randomly choose numbers to draw with a for loop, with recursion used to avoid duplicates.

Now, personally, I'm looking at this as more of a proof of concept than anything; I don't really want to spend time with styling details after this.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{calc}

\makeatletter
\def\drawconnection{
\pgfmathrandominteger{\rand}{1}{\totalnumberofconnections}
\@ifundefined{pgf@sh@ns@\rand}{ % https://tex.stackexchange.com/a/37713/170958
\node (\rand) at (0,0) {}; % we define these nodes to keep track of which \rand's we've already drawn
\ifnum\rand<\first
\pgfmathtruncatemacro{\source}{ceil(\rand/\ilsize)}
\pgfmathtruncatemacro{\dest}{Mod(\rand,\hlsize)+1}
\path (In-\source) edge (H0-\dest);
\else
\ifnum\rand<\second
\pgfmathtruncatemacro{\source}{ceil((\rand-\first+1)/\hlsize)}
\pgfmathtruncatemacro{\dest}{Mod((\rand-\first+1),\hlsize)+1}
\path (H0-\source) edge (H1-\dest);
\else
\pgfmathtruncatemacro{\source}{ceil((\rand-\second+1)/\ilsize)}
\pgfmathtruncatemacro{\dest}{Mod((\rand-\second+1),\olsize)+1}
\path (H1-\source) edge (Out-\dest);
\fi
\fi
}{% If the connection already exists, start from the beginning
\drawconnection
}
}
\makeatother

\begin{document}

\def\layersep{2cm}
\def\hsep{1cm}
\def\ilsize{8}
\def\hlsize{8}
\def\olsize{8}
\def\rootlrp{6}
\def\neuronsize{4mm}

\tikzset{>=latex}

\begin{figure}
\centering

\begin{tikzpicture}[shorten >=0pt, ->, draw=black!100, node distance=\layersep]

\def\percentage{40} % choose a percentage

\tikzstyle{every pin edge}=[<-,shorten <=1pt]
\tikzstyle{neuron}=[circle, draw, fill=black!100, minimum size=\neuronsize,inner sep=0pt]
\tikzstyle{input neuron}=[neuron, fill=black!0]
\tikzstyle{hidden neuron}=[neuron, fill=black!0]
\tikzstyle{output neuron}=[neuron, fill=black!0]

%%%%%%%%%%%%
% DRAW NODES
%%%%%%%%%%%%
% Draw the input layer nodes
\foreach \name / \y in {1,...,\ilsize}
\node[input neuron] (In-\name) at (0.0cm+\hsep,-\y cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H0-\name) at (1.5cm+\hsep,-\y cm) {};
% Draw the hidden layer nodes
\foreach \name / \y in {1,...,\hlsize}
\node[hidden neuron] (H1-\name) at (3.0cm+\hsep,-\y cm) {};
% Draw the output layer nodes
\foreach \name / \y in {1,...,\olsize}
\node[hidden neuron] (Out-\name) at (4.5cm+\hsep,-\y cm) {};

%%%%%%%%%%%%%%%%%%
% DRAW CONNECTIONS
%%%%%%%%%%%%%%%%%%

% there are \ilsize*\hlsize arrows from  il to hl0
% there are \hlsize*\hlsize arrows from hl0 to hl1
% there are \hlsize*\olsize arrows from hl1 to out
% total number of arrows #totalarrows = \ilsize*\hlsize + \hlsize*\hlsize + \hlsize*\olsize
% we assign to each arrow a number from 1 to #arrows
% we do this by establishing an order in which we'd draw the arrows
%
% let (1,1) be the top left node,
% with x increases denoting movement to the right,
% and with y increases denoting movement down.
% Imagine we have a 3x3 grid of arrows
% Arrow 1 = (1,1) -- (2,1)  Arrow 10 = (2,1) -- (3,1)
% Arrow 2 = (1,1) -- (2,2)  Arrow 11 = (2,1) -- (3,2)
% Arrow 3 = (1,1) -- (2,3)  Arrow 12 = (2,1) -- (3,3)
% Arrow 4 = (1,2) -- (2,1)  Arrow 13 = (2,2) -- (3,1)
% Arrow 5 = (1,2) -- (2,2)  Arrow 14 = (2,2) -- (3,2)
% Arrow 6 = (1,2) -- (2,3)  Arrow 15 = (2,2) -- (3,3)
% Arrow 7 = (1,3) -- (2,1)  Arrow 16 = (2,3) -- (3,1)
% Arrow 8 = (1,3) -- (2,2)  Arrow 17 = (2,3) -- (3,2)
% Arrow 9 = (1,3) -- (2,3)  Arrow 18 = (2,3) -- (3,3)
%
% Now, we need to know, given an arrow number, if the arrow is going to be
% one from i to h0, h0 to h1, or h1 to out. But, thankfully, this is pretty easy;
% we just need to check if the arrow number is less than \first,
% or between \first and \second, or larger than \second
%
%  #paths i to h1 = #i*#h1   #paths h1 to h2 = #h1*#h2   #paths h2 to out = #h2*#out
% ========================= =========================== =============================
%                          ^ \first                    ^ \second
%
% So, this is how we'll draw the arrows:
%
\pgfmathsetmacro{\first}{\ilsize*\hlsize+1}
\pgfmathsetmacro{\second}{\ilsize*\hlsize+\hlsize*\hlsize+1}
\pgfmathsetmacro{\totalnumberofconnections}{\ilsize*\hlsize + \hlsize*\hlsize + \hlsize*\olsize}
\pgfmathtruncatemacro{\numberofconnections}{floor(\percentage*\totalnumberofconnections/100)}
\foreach \i in {1,...,\numberofconnections}{
\drawconnection
}

\end{tikzpicture}
\end{figure}

\end{document}


• May I advertise the memberQ pgf function from here to have an easier handle of the duplicates? – user194703 Apr 23 at 22:10
• @Schrödinger's cat I'm amazed by how quickly you find these things. Though the code is somewhat too advanced for me to understand right now, I'm sure it works wonderfully, and is probably a more general alternative to my approach. Thank you for mentioning it! Feel free to add it as an edit if you want, or if not, your comment should probably suffice for future readers anyways. – steve Apr 23 at 22:37