## Plotly plot of chord diagrams¶

Circular layout or Chord diagram is a method of visualizing data that describe relationships. It was intensively promoted through Circos, a software package in Perl that was initially designed for displaying genomic data.

This 2013 stackoverflow question, whether there is a Python library for plotting chord diagrams, was closed as being off topic. Two years later I presented in the initial version of this Jupyter Notebook a method to generate a chord diagram via Python Plotly.

This Jupyter Notebook is an updated ersion to be run using Python 3.7+, and Plotly 4.+.

We illustrate the method of generating a chord diagram from data recorded in a square matrix. The rows and columns represent the same entities.

Suppose that for a community of 5 friends on Facebook we record the number of comments posted by each member on other friends wall. The data table is given in the next cell:

In :
from IPython.display import Image
Image(filename='Data/Data-table.png')

Out: The aim of our visualization is to illustrate the total number of posts by each community member, and the flows of posts between pairs of friends.

In :
import platform, plotly
import numpy as np
from numpy import pi

In :
print(f'Python version: {platform.python_version()}')
print(f'Plotly version: {plotly.__version__}')

Python version: 3.7.3
Plotly version: 4.5.4


Define the array of data:

In :
matrix = np.array([[16,  3, 28,  0, 18],
[18,  0, 12,  5, 29],
[ 9, 11, 17, 27,  0],
[19,  0, 31, 11, 12],
[23, 17, 10,  0, 34]], dtype=int)

In :
def check_data(data_matrix):
L, M = data_matrix.shape
if L != M:
raise ValueError('Data array must have a (n,n) shape')
return L

In :
L = check_data(matrix)


A chord diagram encodes information in two graphical objects:

• ideograms, represented by distinctly colored arcs of circles;
• ribbons, that are planar shapes bounded by two quadratic Bezier curves and two arcs of circle, that can degenerate to a point;

### Ideograms¶

Summing up the entries on each matrix row, one gets a value (in our example this value is equal to the number of posts by a community member). Let us denote by total_comments the total number of posts recorded in this community.

Theoretically the interval [0, total_comments) is mapped linearly onto the unit circle, identified with the interval $[0,2\pi)$.

For a better looking plot one proceeds as follows: starting from the angular position $0$, in counter-clockwise direction, one draws succesively, around the unit circle, two parallel arcs of length equal to a mapped row sum value, minus a fixed gap. Click the image below:

In :
from IPython.display import IFrame

In :
IFrame('https://plot.ly/~empet/12234/', width=377, height=420)

Out:

Now we are defining functions that process data in order to get the ideogram ends.

As we pointed out, the unit circle is oriented counter-clockwise. In order to get an arc of circle of end angular coordinates $\theta_0<\theta_1$, we define the function, moduloAB, that resolves the case when an arc contains the point of angular coordinate $0$ (for example $\theta_0=2\pi-\pi/12$, $\theta_1=\pi/9$). The function corresponding to $a=-\pi, b=\pi$ allows to map the interval $[0,2\pi)$ onto $[-\pi, \pi)$. Via this transformation we have:

$\theta_0\mapsto \theta'_0=-\pi/12$, and

$\theta_1=\mapsto \theta'_1=\pi/9$,

and now $\theta'_0<\theta'_1$.

In :
def moduloAB(x, a, b): #maps a real number onto the unit circle identified with
#the interval [a,b), b-a=2*PI
if a>= b:
raise ValueError('Incorrect interval ends')
y = (x-a) % (b-a)
return y+b if y < 0 else y+a

In :
def test_2PI(x):
return 0 <= x < 2*pi


Compute the row sums and the lengths of corresponding ideograms:

In :
row_sum = [matrix[k,:].sum() for k in range(L)]

#set the gap between two consecutive ideograms
gap = 2*pi*0.005
ideogram_length = 2*pi * np.asarray(row_sum) / sum(row_sum) - gap*np.ones(L)


The next function returns the list of end angular coordinates for each ideogram arc:

In :
def get_ideogram_ends(ideogram_len, gap):
ideo_ends = []
left = 0
for k in range(len(ideogram_len)):
right = left + ideogram_len[k]
ideo_ends.append([left, right])
left = right + gap
return ideo_ends

In :
ideo_ends = get_ideogram_ends(ideogram_length, gap)


The function make_ideogram_arc returns equally spaced points on an ideogram arc, expressed as complex numbers in polar form:

In :
def make_ideogram_arc(R, phi, a=50):
# R is the circle radius
# phi is the list of  angle coordinates of an arc ends
# a is a parameter that controls the number of points to be evaluated on an arc
if not test_2PI(phi) or not test_2PI(phi):
phi = [moduloAB(t, 0, 2*pi) for t in phi]
length = (phi-phi) % 2*pi
nr = 5 if length <= pi/4 else int(a*length/pi)

if phi < phi:
theta = np.linspace(phi, phi, nr)
else:
phi = [moduloAB(t, -pi, pi) for t in phi]
theta = np.linspace(phi, phi, nr)
return R * np.exp(1j*theta)


The real and imaginary parts of these complex numbers will be used to define the ideogram as a Plotly shape bounded by a SVG path.

In :
make_ideogram_arc(1.3, [11*pi/6, pi/17])

Out:
array([1.12583302-0.65j      , 1.14814501-0.60972373j,
1.16901672-0.5686826j , 1.18842197-0.5269281j ,
1.20633642-0.48451259j, 1.22273759-0.44148929j,
1.23760491-0.39791217j, 1.25091973-0.3538359j ,
1.26266534-0.30931575j, 1.27282702-0.26440759j,
1.28139202-0.21916775j, 1.28834958-0.17365297j,
1.29369099-0.12792036j, 1.29740954-0.08202728j,
1.29950058-0.0360313j , 1.29996146+0.01000988j,
1.29879163+0.0560385j , 1.29599253+0.10199682j,
1.2915677 +0.1478272j , 1.28552267+0.19347214j,
1.27786503+0.23887437j])

Set ideograms labels and colors:

In :
labels=['Emma', 'Isabella', 'Ava', 'Olivia', 'Sophia']
ideo_colors=['rgba(244, 109, 67, 0.75)',
'rgba(253, 174, 97, 0.75)',
'rgba(254, 224, 139, 0.75)',
'rgba(217, 239, 139, 0.75)',
'rgba(166, 217, 106, 0.75)']#brewer colors with alpha set on 0.75


### Ribbons in a chord diagram¶

While ideograms illustrate how many comments posted each member of the Facebook community, ribbons give a comparative information on the flows of comments from one friend to another.

To illustrate this flow we map data onto the unit circle. More precisely, for each matrix row, $k$, the application:

t$\mapsto$ t*ideogram_length[k]/row_sum[k]

maps the interval [0, row_sum[k]] onto the interval [0, ideogram_length[k]]. Hence each entrymatrix[k][j]in the $k^{th}$ row is mapped tomatrix[k][j] * ideogram_length[k] / row_value[k].

The function map_data maps all matrix entries to the corresponding values in the intervals associated to ideograms:

In :
def map_data(data_matrix, row_value, ideogram_length):
mapped = np.zeros(data_matrix.shape)
for j  in range(L):
mapped[:, j] = ideogram_length * data_matrix[:,j] / row_value
return mapped

In :
mapped_data = map_data(matrix, row_sum, ideogram_length)
mapped_data

Out:
array([[0.27949818, 0.05240591, 0.48912181, 0.        , 0.31443545],
[0.31429952, 0.        , 0.20953301, 0.08730542, 0.50637144],
[0.15714976, 0.19207193, 0.29683843, 0.47144927, 0.        ],
[0.33291045, 0.        , 0.54316969, 0.19273763, 0.21025923],
[0.40429305, 0.2988253 , 0.17577959, 0.        , 0.5976506 ]])
• To each pair of values (mapped_data[k][j], mapped_data[j][k]), $k<=j$, one associates a ribbon, that is a curvilinear filled rectangle (that can be degenerate), having as opposite sides two subarcs of the $k^{th}$ ideogram, respectively $j^{th}$ ideogram, and two arcs of quadratic Bézier curves.

Here we illustrate the ribbons associated to pairs (mapped_data[j], mapped_data[j]), $j=\overline{0,4}$, that illustrate the flow of comments between Emma and all other friends, and herself:

In :
IFrame('https://plot.ly/~empet/12519/', width=420, height=420)

Out:
• For a better looking chord diagram, Circos documentation recommends to sort increasingly each row of the mapped_data.

The array idx_sort, defined below, has on each row the indices that sort the corresponding row in mapped_data:

In :
idx_sort = np.argsort(mapped_data, axis=1)
idx_sort

Out:
array([[3, 1, 0, 4, 2],
[1, 3, 2, 0, 4],
[4, 0, 1, 2, 3],
[1, 3, 4, 0, 2],
[3, 2, 1, 0, 4]], dtype=int64)

In the following we call ribbon ends, the lists l=[l, l], r=[r, r] having as elements the angular coordinates of the ends of arcs that are opposite sides in a ribbon. These arcs are sub-arcs in the internal boundaries of the ideograms, connected by the ribbon (see the image above).

• Compute the ribbon ends and store them as tuples in a list of lists ($L\times L$):
In :
def make_ribbon_ends(mapped_data, ideo_ends,  idx_sort):
L = mapped_data.shape
ribbon_boundary = np.zeros((L,L+1))
for k in range(L):
start = ideo_ends[k]
ribbon_boundary[k] = start
for j in range(1,L+1):
J = idx_sort[k][j-1]
ribbon_boundary[k][j] = start + mapped_data[k][J]
start = ribbon_boundary[k][j]
return [[(ribbon_boundary[k][j], ribbon_boundary[k][j+1] ) for j in range(L)] for k in range(L)]

In :
ribbon_ends = make_ribbon_ends(mapped_data, ideo_ends,  idx_sort)
print ('ribbon ends starting from the ideogram\n', ribbon_ends)

ribbon ends starting from the ideogram
[(2.31580258464619, 2.31580258464619), (2.31580258464619, 2.472952342161697), (2.472952342161697, 2.6650242680139837), (2.6650242680139837, 2.9618626988766086), (2.9618626988766086, 3.43331197142313)]


We note that ribbon_ends[k][j] corresponds to mapped_data[i][idx_sort[k][j]], i.e. the length of the arc of ends in ribbon_ends[k][j] is equal to mapped_data[i][idx_sort[k][j]].

Now we define a few functions that compute the control points for Bézier ribbon sides.

The function control_pts returns the cartesian coordinates of the control points, $b_0, b_1, b_2$, supposed as being initially located on the unit circle, and thus defined only by their angular coordinate. The angular coordinate of the point $b_1$ is the mean of angular coordinates of the points $b_0, b_2$.

Since for a Bézier ribbon side only $b_0, b_2$ are placed on the unit circle, one gives radius as a parameter that controls position of $b_1$. radius is the distance of $b_1$ to the circle center.

In :
def control_pts(angle, radius):
#angle is a  3-list containing angular coordinates of the control points b0, b1, b2
#radius is the distance from b1 to the  origin O(0,0)

if len(angle) != 3:
raise InvalidInputError('angle must have len =3')
b_cplx = np.array([np.exp(1j*angle[k]) for k in range(3)])
return list(zip(b_cplx.real, b_cplx.imag))

In :
def ctrl_rib_chords(l, r, radius):
# this function returns a 2-list containing control poligons of the two quadratic Bezier
#curves that are opposite sides in a ribbon
#l (r) the list of angular variables of the ribbon arc ends defining
#the ribbon starting (ending) arc
# radius is a common parameter for both control polygons
if len(l) != 2 or len(r) != 2:
raise ValueError('the arc ends must be elements in a list of len 2')
return [control_pts([l[j], (l[j]+r[j])/2, r[j]], radius) for j in range(2)]


Each ribbon is colored with the color of one of the two ideograms it connects. We define an L-list of L-lists of colors for ribbons. Denote it by ribbon_color.

ribbon_color[k][j] is the Plotly color string for the ribbon associated to mapped_data[k][j] and mapped_data[j][k], i.e. the ribbon connecting two subarcs in the $k^{th}$, respectively, $j^{th}$ ideogram. Hence this structure is symmetric.

Initially we define:

In :
ribbon_color = [L * [ideo_colors[k]] for k in range(L)]


and then eventually we are changing the color in a few positions.

For our example we are perfotming the following color change:

In :
ribbon_color=ideo_colors
ribbon_color=ideo_colors
ribbon_color=ideo_colors
ribbon_color=ideo_colors


The symmetric locations are not modified, because we do not access ribbon_color[k][j], $k>j$, when drawing the ribbons.

Functions that return the Plotly SVG paths that are ribbon boundaries:

In :
def make_q_bezier(b):# defines the Plotly SVG path for a quadratic Bezier curve defined by the
#list of its control points
if len(b) != 3:
raise valueError('control poligon must have 3 points')
A, B, C = b
return f'M {A}, {A} Q {B}, {B} {C}, {C}'

In :
b=[(1,4), (-0.5, 2.35), (3.745, 1.47)]

In :
make_q_bezier(b)

Out:
'M 1, 4 Q -0.5, 2.35 3.745, 1.47'

make_ribbon_arc returns the Plotly SVG path corresponding to an arc represented by its end angular coordinates, theta0, theta1.

In :
def make_ribbon_arc(theta0, theta1):

if test_2PI(theta0) and test_2PI(theta1):
if theta0 < theta1:
theta0 = moduloAB(theta0, -pi, pi)
theta1 = moduloAB(theta1, -pi, pi)
if theta0  *theta1 > 0:
raise ValueError('incorrect angle coordinates for ribbon')

nr = int(40 * (theta0 - theta1) / pi)
if nr <= 2: nr = 3
theta = np.linspace(theta0, theta1, nr)
pts=np.exp(1j*theta)# points in polar complex form, on the given arc

string_arc = ''
for k in range(len(theta)):
string_arc += f'L {pts.real[k]}, {pts.imag[k]} '
return   string_arc
else:
raise ValueError('the angle coordinates for an arc side of a ribbon must be in [0, 2*pi]')

In :
make_ribbon_arc(np.pi/3, np.pi/6)

Out:
'L 0.5000000000000001, 0.8660254037844386 L 0.5877852522924732, 0.8090169943749473 L 0.6691306063588582, 0.7431448254773941 L 0.7431448254773944, 0.6691306063588581 L 0.8090169943749475, 0.5877852522924731 L 0.8660254037844387, 0.49999999999999994 '

Finally we are ready to define data and layout for the Plotly plot of the chord diagram.

In :
import plotly.graph_objects as go

In :
def plot_layout(title, plot_size):

return dict(title=title,
xaxis=dict(visible=False),
yaxis=dict(visible=False),
showlegend=False,
width=plot_size,
height=plot_size,
margin=dict(t=25, b=25, l=25, r=25),
hovermode='closest',
)


Function that returns the Plotly shape of an ideogram:

In :
def make_ideo_shape(path, line_color, fill_color):
#line_color is the color of the shape boundary
#fill_collor is the color assigned to an ideogram

return  dict(line=dict(color=line_color,
width=0.45),
path=path,
layer='below',
type='path',
fillcolor=fill_color)


We generate two types of ribbons: a ribbon connecting subarcs in two distinct ideograms, respectively a ribbon from one ideogram to itself (it corresponds to mapped_data[k][k], i.e. it gives the flow of comments from a community member to herself).

In :
def make_ribbon(l, r, line_color, fill_color, radius=0.2):
#l=[l, l], r=[r, r]  represent the opposite arcs in the ribbon
#line_color is the color of the shape boundary
#fill_color is the fill color for the ribbon shape

b, c = poligon

return  dict(line=dict(color=line_color,
width=0.5),
path=make_q_bezier(b) + make_ribbon_arc(r, r)+
make_q_bezier(c[::-1]) + make_ribbon_arc(l, l),
type='path',
layer='below',
fillcolor = fill_color,
)

b = control_pts([l, (l+l)/2, l], radius)

return  dict(line = dict(color=line_color,
width=0.5),
path =  make_q_bezier(b)+make_ribbon_arc(l, l),
type = 'path',
layer = 'below',
fillcolor = fill_color
)

In :
def invPerm(perm):
# function that returns the inverse of a permutation, perm
inv =  * len(perm)
for i, s in enumerate(perm):
inv[s] = i
return inv

In :
layout=plot_layout('Chord diagram', 400)


Now let us explain the key point of associating ribbons to right data:

From the definition of ribbon_ends we notice that ribbon_ends[k][j] corresponds to data stored in matrix[k][sigma[j]], where sigma is the permutation of indices $0, 1, \ldots L-1$, that sort the row k in mapped_data. If sigma_inv is the inverse permutation of sigma, we get that to matrix[k][j] corresponds the ribbon_ends[k][sigma_inv[j]].

ribbon_info is a list of dicts setting the information that is displayed when hovering the mouse over the ribbon ends.

Set the radius of Bézier control point, $b_1$, for each ribbon associated to a diagonal data entry:

In :
radii_sribb = [0.4, 0.30, 0.35, 0.39, 0.12]# these value are set after a few trials

In :
ribbon_info = []
shapes = []
for k in range(L):

sigma = idx_sort[k]
sigma_inv = invPerm(sigma)
for j in range(k, L):
if matrix[k][j] == 0 and matrix[j][k]==0: continue
eta = idx_sort[j]
eta_inv = invPerm(eta)
l = ribbon_ends[k][sigma_inv[j]]

if j == k:
shapes.append(make_self_rel(l, 'rgb(175,175,175)' ,
z = 0.9*np.exp(1j*(l+l)/2)

#the text below will be displayed when hovering the mouse over the ribbon
text = f'{labels[k]} commented on {int(matrix[k][k])} of herself Fb posts'

ribbon_info.append(go.Scatter(x=[z.real],
y=[z.imag],
mode='markers',
marker=dict(size=0.5, color=ideo_colors[k]),
text=text,
hoverinfo='text'
)
)
else:
r = ribbon_ends[j][eta_inv[k]]
zi = 0.9 * np.exp(1j*(l+l)/2)
zf = 0.9 * np.exp(1j*(r+r)/2)
#texti and textf are the strings that will be displayed when hovering the mouse
#over the two ribbon ends
texti = f'{labels[k]} commented on {int(matrix[k][j])} of {labels[j]} Fb posts'
textf = f'{labels[j]} commented on {int(matrix[j][k])} of {labels[k]} Fb posts'

ribbon_info.append(go.Scatter(x=[zi.real],
y=[zi.imag],
mode='markers',
marker=dict(size=0.5, color=ribbon_color[k][j]),
text=texti,
hoverinfo='text'
)
),
ribbon_info.append(go.Scatter(x=[zf.real],
y=[zf.imag],
mode='markers',
marker=dict(size=0.5, color=ribbon_color[k][j]),
text=textf,
hoverinfo='text'
)
)
r = (r, r) # IMPORTANT!!!  Reverse these arc ends because otherwise you get
# a twisted ribbon
#append the ribbon shape
shapes.append(make_ribbon(l, r, 'rgb(175,175,175)' , ribbon_color[k][j]))



ideograms is a list of dicts that set the position, and color of ideograms, as well as the information associated to each ideogram.

In :
ideograms = []
for k in range(len(ideo_ends)):
z =  make_ideogram_arc(1.1, ideo_ends[k])
zi = make_ideogram_arc(1.0, ideo_ends[k])
m = len(z)
n = len(zi)
ideograms.append(go.Scatter(x=z.real,
y=z.imag,
mode='lines',
line=dict(color=ideo_colors[k], shape='spline', width=0.25),
hoverinfo='text'
)
)

path = 'M '
for s in range(m):
path += f'{z.real[s]}, {z.imag[s]} L '

Zi = np.array(zi.tolist()[::-1])

for s in range(m):
path += f'{Zi.real[s]}, {Zi.imag[s]} L '
path += f'{z.real} ,{z.imag}'

shapes.append(make_ideo_shape(path,'rgb(150,150,150)' , ideo_colors[k]))


In :
data = ideograms + ribbon_info
layout['shapes'] = shapes
fig = go.Figure(data=data, layout=layout)
init_notebook_mode(connected=True)
iplot(fig)


Here is a chord diagram associated to a community of 8 Facebook friends:

In :
IFrame('https://plot.ly/~empet/12148/chord-diagram-of-facebook-comments-in-a-community/',
width=500, height=500)

Out:
In :
from IPython.core.display import HTML
def  css_styling():


`