Random Walk in Python

Learn how to use Python to make a Random Walk


New to Plotly?

Plotly is a free and open-source graphing library for Python. We recommend you read our Getting Started guide for the latest installation or upgrade instructions, then move on to our Plotly Fundamentals tutorials or dive straight in to some Basic Charts tutorials.

A random walk can be thought of as a random process in which a token or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\mathbb{Z}$), two dimensions ($\mathbb{Z}^2$) or three dimensions ($\mathbb{Z}^3$) in Cartesian space, where $\mathbb{Z}$ represents the set of integers. In the visualizations below, we will be using scatter plots as well as a colorscale to denote the time sequence of the walk.

Random Walk in 1D

The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendancy of the random walk is.

In [1]:
import plotly.graph_objects as go
import numpy as np
np.random.seed(1)

l = 100
steps = np.random.choice([-1, 1], size=l) + 0.05 * np.random.randn(l) # l steps
position = np.cumsum(steps) # integrate the position by summing steps values
y = 0.05 * np.random.randn(l)

fig = go.Figure(data=go.Scatter(
    x=position,
    y=y,
    mode='markers',
    name='Random Walk in 1D',
    marker=dict(
        color=np.arange(l),
        size=7,
        colorscale='Reds',
        showscale=True,
    )
))

fig.update_layout(yaxis_range=[-1, 1])
fig.show()

Random Walk in 2D

In [2]:
import plotly.graph_objects as go
import numpy as np

l = 1000
x_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
y_steps = np.random.choice([-1, 1], size=l) + 0.2 * np.random.randn(l) # l steps
x_position = np.cumsum(x_steps) # integrate the position by summing steps values
y_position = np.cumsum(y_steps) # integrate the position by summing steps values

fig = go.Figure(data=go.Scatter(
    x=x_position,
    y=y_position,
    mode='markers',
    name='Random Walk',
    marker=dict(
        color=np.arange(l),
        size=8,
        colorscale='Greens',
        showscale=True
    )
))

fig.show()

Random walk and diffusion

In the two following charts we show the link between random walks and diffusion. We compute a large number N of random walks representing for examples molecules in a small drop of chemical. While all trajectories start at 0, after some time the spatial distribution of points is a Gaussian distribution. Also, the average distance to the origin grows as $\sqrt(t)$.

In [3]:
import plotly.graph_objects as go
import numpy as np

l = 1000
N = 10000
steps = np.random.choice([-1, 1], size=(N, l)) + 0.05 * np.random.standard_normal((N, l)) # l steps
position = np.cumsum(steps, axis=1) # integrate all positions by summing steps values along time axis

fig = go.Figure(data=go.Histogram(x=position[:, -1])) # positions at final time step
fig.show()
In [4]:
import plotly.graph_objects as go
from plotly.subplots import make_subplots
import numpy as np

l = 1000
N = 10000
t = np.arange(l)
steps = np.random.choice([-1, 1], size=(N, l)) + 0.05 * np.random.standard_normal((N, l)) # l steps
position = np.cumsum(steps, axis=1) # integrate the position by summing steps values
average_distance = np.std(position, axis=0) # average distance

fig = make_subplots(1, 2)
fig.add_trace(go.Scatter(x=t, y=average_distance, name='mean distance'), 1, 1)
fig.add_trace(go.Scatter(x=t, y=average_distance**2, name='mean squared distance'), 1, 2)
fig.update_xaxes(title_text='$t$')
fig.update_yaxes(title_text='$l$', col=1)
fig.update_yaxes(title_text='$l^2$', col=2)
fig.update_layout(showlegend=False)
fig.show()

Advanced Tip

We can formally think of a 1D random walk as a point jumping along the integer number line. Let $Z_i$ be a random variable that takes on the values +1 and -1. Let this random variable represent the steps we take in the random walk in 1D (where +1 means right and -1 means left). Also, as with the above visualizations, let us assume that the probability of moving left and right is just $\frac{1}{2}$. Then, consider the sum

$$ \begin{align*} S_n = \sum_{i=0}^{n}{Z_i} \end{align*} $$

where S_n represents the point that the random walk ends up on after n steps have been taken.

To find the expected value of $S_n$, we can compute it directly. Since each $Z_i$ is independent, we have

$$ \begin{align*} \mathbb{E}(S_n) = \sum_{i=0}^{n}{\mathbb{E}(Z_i)} \end{align*} $$

but since $Z_i$ takes on the values +1 and -1 then

$$ \begin{align*} \mathbb{E}(Z_i) = 1 \cdot P(Z_i=1) + -1 \cdot P(Z_i=-1) = \frac{1}{2} - \frac{1}{2} = 0 \end{align*} $$

Therefore, we expect our random walk to hover around $0$ regardless of how many steps we take in our walk.

What About Dash?

Dash is an open-source framework for building analytical applications, with no Javascript required, and it is tightly integrated with the Plotly graphing library.

Learn about how to install Dash at https://dash.plot.ly/installation.

Everywhere in this page that you see fig.show(), you can display the same figure in a Dash application by passing it to the figure argument of the Graph component from the built-in dash_core_components package like this:

import plotly.graph_objects as go # or plotly.express as px
fig = go.Figure() # or any Plotly Express function e.g. px.bar(...)
# fig.add_trace( ... )
# fig.update_layout( ... )

import dash
import dash_core_components as dcc
import dash_html_components as html

app = dash.Dash()
app.layout = html.Div([
    dcc.Graph(figure=fig)
])

app.run_server(debug=True, use_reloader=False)  # Turn off reloader if inside Jupyter