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Filled Chord Diagram in Python

How to make an interactive filled-chord diagram in Python with Plotly and iGraph.

Filled-Chord Diagrams with Plotly

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.

M Bostock developed reusable charts for chord diagrams in d3.js. Two years ago on stackoverflow, the exsistence of a Python package for plotting chord diagrams was adressed, but the question was closed due to being off topic.
Here we show that a chord diagram can be generated in Python with Plotly. We illustrate the method of generating a chord diagram from data recorded in a square matrix. The rows and columns represent the same entities.

This example considers a community of 5 friends on Facebook. We record the number of comments posted by each member on the other friends' walls. The data table is given in the next cell:

In [1]:
import plotly.plotly as py
import plotly.figure_factory as ff
import plotly.graph_objs as go

data = [['', 'Emma', 'Isabella', 'Ava', 'Olivia', 'Sophia', 'row-sum'],
        ['Emma', 16, 3, 28, 0, 18, 65],
        ['Isabella', 18, 0, 12, 5, 29, 64],
        ['Ava', 9, 11, 17, 27, 0, 64],
        ['Olivia', 19, 0, 31, 11, 12, 73],
        ['Sophia', 23, 17, 10, 0, 34, 84]]

table = ff.create_table(data, index=True)
py.iplot(table, filename='Data-Table')
Out[1]:

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 [2]:
import numpy as np

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)

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

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:

Ideograms

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

As we stressed 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 a 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 [3]:
PI=np.pi

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

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

Compute the row sums and the lengths of corresponding ideograms:

In [4]:
row_sum=[np.sum(matrix[k,:]) 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 [5]:
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

ideo_ends=get_ideogram_ends(ideogram_length, gap)
ideo_ends
Out[5]:
[[0, 1.1354613447974538],
 [1.1668772713333517, 2.284386658110292],
 [2.31580258464619, 3.43331197142313],
 [3.464727897959028, 4.7438049069205865],
 [4.775220833456484, 6.251769380643687]]

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

In [6]:
def make_ideogram_arc(R, phi, a=50):
    # R is the circle radius
    # phi is the list of ends angle coordinates of an arc
    # a is a parameter that controls the number of points to be evaluated on an arc
    if not test_2PI(phi[0]) or not test_2PI(phi[1]):
        phi=[moduloAB(t, 0, 2*PI) for t in phi]
    length=(phi[1]-phi[0])% 2*PI
    nr=5 if length<=PI/4 else int(a*length/PI)

    if phi[0] < phi[1]:
        theta=np.linspace(phi[0], phi[1], nr)
    else:
        phi=[moduloAB(t, -PI, PI) for t in phi]
        theta=np.linspace(phi[0], phi[1], 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 [7]:
z=make_ideogram_arc(1.3, [11*PI/6, PI/17])
print z
[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 [8]:
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 entry matrix[k][j] of the $k^{th}$ row is mapped to matrix[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 [9]:
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

mapped_data=map_data(matrix, row_sum, ideogram_length)
mapped_data
Out[9]:
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[0][j], mapped_data[j][0]), $j=\overline{0,4}$, that illustrate the flow of comments between Emma and all other friends, and herself:

Ribbons
  • 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 [10]:
idx_sort=np.argsort(mapped_data, axis=1)
idx_sort
Out[10]:
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]])

In the following we call ribbon ends, the lists l=[l[0], l[1]], r=[r[0], r[1]] 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 [11]:
def make_ribbon_ends(mapped_data, ideo_ends,  idx_sort):
    L=mapped_data.shape[0]
    ribbon_boundary=np.zeros((L,L+1))
    for k in range(L):
        start=ideo_ends[k][0]
        ribbon_boundary[k][0]=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)]

ribbon_ends=make_ribbon_ends(mapped_data, ideo_ends,  idx_sort)
print 'ribbon ends starting from the ideogram[2]\n', ribbon_ends[2]
ribbon ends starting from the ideogram[2]
[(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] correspond 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 [12]:
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)])
    b_cplx[1]=radius*b_cplx[1]
    return zip(b_cplx.real, b_cplx.imag)
In [13]:
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 [14]:
ribbon_color=[L*[ideo_colors[k]] for k in range(L)]

and then eventually we change the color in a few positions.

For our example we change:

In [15]:
ribbon_color[0][4]=ideo_colors[4]
ribbon_color[1][2]=ideo_colors[2]
ribbon_color[2][3]=ideo_colors[3]
ribbon_color[2][4]=ideo_colors[4]

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 [16]:
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 'M '+str(A[0])+',' +str(A[1])+' '+'Q '+\
                str(B[0])+', '+str(B[1])+ ' '+\
                str(C[0])+', '+str(C[1])

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

make_q_bezier(b)
Out[16]:
'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 [17]:
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 on arc in polar complex form

        string_arc=''
        for k in range(len(theta)):
            string_arc+='L '+str(pts.real[k])+', '+str(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]')

make_ribbon_arc(np.pi/3, np.pi/6)
Out[17]:
'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 [21]:
def make_layout(title, plot_size):
    axis=dict(showline=False, # hide axis line, grid, ticklabels and  title
          zeroline=False,
          showgrid=False,
          showticklabels=False,
          title=''
          )

    return go.Layout(title=title,
                  xaxis=dict(axis),
                  yaxis=dict(axis),
                  showlegend=False,
                  width=plot_size,
                  height=plot_size,
                  margin=dict(t=25, b=25, l=25, r=25),
                  hovermode='closest',
                  shapes=[]# to this list one appends below the dicts defining the ribbon,
                           #respectively the ideogram shapes
                 )

Function that returns the Plotly shape of an ideogram:

In [22]:
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,
            type='path',
            fillcolor=fill_color,
            layer='below'
        )

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 [23]:
def make_ribbon(l, r, line_color, fill_color, radius=0.2):
    #l=[l[0], l[1]], r=[r[0], r[1]]  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
    poligon=ctrl_rib_chords(l,r, radius)
    b,c =poligon

    return  dict(
                line=dict(
                color=line_color, width=0.5
            ),
            path=  make_q_bezier(b)+make_ribbon_arc(r[0], r[1])+
                   make_q_bezier(c[::-1])+make_ribbon_arc(l[1], l[0]),
            type='path',
            fillcolor=fill_color,
            layer='below'
        )

def make_self_rel(l, line_color, fill_color, radius):
    #radius is the radius of Bezier control point b_1
    b=control_pts([l[0], (l[0]+l[1])/2, l[1]], radius)
    return  dict(
                line=dict(
                color=line_color, width=0.5
            ),
            path=  make_q_bezier(b)+make_ribbon_arc(l[1], l[0]),
            type='path',
            fillcolor=fill_color,
            layer='below'
        )

def invPerm(perm):
    # function that returns the inverse of a permutation, perm
    inv = [0] * len(perm)
    for i, s in enumerate(perm):
        inv[s] = i
    return inv

layout=make_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 [24]:
radii_sribb=[0.4, 0.30, 0.35, 0.39, 0.12]# these value are set after a few trials 
In [25]:
ribbon_info=[]
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:
            layout['shapes'].append(make_self_rel(l, 'rgb(175,175,175)' ,
                                    ideo_colors[k], radius=radii_sribb[k]))
            z=0.9*np.exp(1j*(l[0]+l[1])/2)
            #the text below will be displayed when hovering the mouse over the ribbon
            text=labels[k]+' commented on '+ '{:d}'.format(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[0]+l[1])/2)
            zf=0.9*np.exp(1j*(r[0]+r[1])/2)
            #texti and textf are the strings that will be displayed when hovering the mouse 
            #over the two ribbon ends
            texti=labels[k]+' commented on '+ '{:d}'.format(matrix[k][j])+' of '+\
                  labels[j]+ ' Fb posts',

            textf=labels[j]+' commented on '+ '{:d}'.format(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[1], r[0])#IMPORTANT!!!  Reverse these arc ends because otherwise you get
                          # a twisted ribbon
            #append the ribbon shape
            layout['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 [28]:
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),
                             text=labels[k]+'<br>'+'{:d}'.format(row_sum[k]),
                             hoverinfo='text'
                             )
                     )


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

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

    for s in range(m):
        path+=str(Zi.real[s])+', '+str(Zi.imag[s])+' L '
    path+=str(z.real[0])+' ,'+str(z.imag[0])

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

data = go.Data(ideograms+ribbon_info)
fig = go.Figure(data=data, layout=layout)

import plotly.offline as off
off.init_notebook_mode()

off.iplot(fig, filename='chord-diagram-Fb')