Visualizing Classical Music Influence with networkx

0. Introduction

I came across this 1927 graphic showing the musical descendants of Czerny:

What follows is my attempt to create a similar music-ancestry graph directly from data.

1. Scraping the data

Fortunately for us, Wikipedia has an incredible comprehensive series of lists of music students by teacher that is perfect for our purposes – each teacher is given a heading and followed by a list of students.

Let's try scraping it.

In [1]:
%matplotlib notebook
import matplotlib.pyplot as plt
In [2]:
import codecs
import json
import urllib

from requests import get
from bs4 import BeautifulSoup
import networkx as nx
In [3]:
pages = [
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_A_to_B',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_C_to_F',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_G_to_J',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_K_to_M',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_N_to_Q',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_R_to_S',
    'https://en.wikipedia.org/wiki/List_of_music_students_by_teacher:_T_to_Z'
]

We'll go through each of the 7 pages and build up an adjacency list of teacher->student edges.

We'll skip over students or teachers that don't have an associated Wikipedia link to ensure that the people we're getting are all notable enough to have Wikipedia entries.

In [4]:
adjlist = ''

for url in pages:
    response = get(url)
    soup = BeautifulSoup(response.text, 'html.parser')
    
    for heading in soup.find_all('h3'):
        if not heading.find('a'):
            continue
        url = heading.find('a')['href']
        if '/wiki/' not in url:
            continue
        teacher_name = url.split('/wiki/')[1].split('_(')[0].split('#')[0]

        student_entries = heading.find_next_sibling("div", class_="columns").find_all('li')
        student_names = []
        for student in student_entries:
            if not student.find('a'):
                continue

            url = student.find('a')['href']
            if '/wiki/' in url:
                student_names.append(url.split('/wiki/')[1].split('_(')[0].split('#')[0])

        adjlist += '{} {}\n'.format(teacher_name, ' '.join(student_names))
In [5]:
adjlist[:1000]
Out[5]:
'Arkady_Abaza Nikolai_Roslavets\nChristian_Ferdinand_Abel Carl_Friedrich_Abel\nHermann_Abendroth Allard_de_Ridder Herbert_Eimert G%C3%BCnther_Herbig Wilhelm_Sch%C3%BCchter William_Steinberg\nDieter_Acker Susanne_Erding-Swiridoff\nAdolphe_Adam L%C3%A9o_Delibes Ferdinand_Poise Lo%C3%AFsa_Puget\nLouis_Adam Joseph_Daussoigne-M%C3%A9hul Ferdinand_H%C3%A9rold Friedrich_Kalkbrenner Henry_Lemoine\nJohn_Luther_Adams Corey_Dargel\nMurray_Adaskin Andrew_Dawes Boyd_McDonald Paul_Pedersen Rodney_Sharman Timothy_Williams\nGuido_Adler Karel_Navr%C3%A1til Anton_Webern Egon_Wellesz\nOskar_Adler Hans_Keller J%C3%B3zef_Koffler Dragan_Plamenac Arnold_Schoenberg\nSamuel_Adler Martin_Amlin Claude_Baker Roger_Briggs Jason_Robert_Brown David_Crumb Greg_Danner Eric_Ewazen Peng-Peng_Gong Jay_Greenberg Anthony_Iannaccone Kamran_Ince Michael_Isaacson Scott_Lindroth Marc_Mellits Carter_Pann Robert_Paterson Paul_Phillips Paul_Reller Michael_Alec_Rose Daria_Semegen Gordon_Stout Christopher_Theofanidis Michael_Sidney_Timpson Fi'

The names are URL-encoded because we took them from the link hrefs, so let's decode as UTF-8.

In [6]:
adjlist = urllib.unquote(adjlist).decode('utf-8')
In [7]:
print adjlist[:1000]
Arkady_Abaza Nikolai_Roslavets
Christian_Ferdinand_Abel Carl_Friedrich_Abel
Hermann_Abendroth Allard_de_Ridder Herbert_Eimert Günther_Herbig Wilhelm_Schüchter William_Steinberg
Dieter_Acker Susanne_Erding-Swiridoff
Adolphe_Adam Léo_Delibes Ferdinand_Poise Loïsa_Puget
Louis_Adam Joseph_Daussoigne-Méhul Ferdinand_Hérold Friedrich_Kalkbrenner Henry_Lemoine
John_Luther_Adams Corey_Dargel
Murray_Adaskin Andrew_Dawes Boyd_McDonald Paul_Pedersen Rodney_Sharman Timothy_Williams
Guido_Adler Karel_Navrátil Anton_Webern Egon_Wellesz
Oskar_Adler Hans_Keller Józef_Koffler Dragan_Plamenac Arnold_Schoenberg
Samuel_Adler Martin_Amlin Claude_Baker Roger_Briggs Jason_Robert_Brown David_Crumb Greg_Danner Eric_Ewazen Peng-Peng_Gong Jay_Greenberg Anthony_Iannaccone Kamran_Ince Michael_Isaacson Scott_Lindroth Marc_Mellits Carter_Pann Robert_Paterson Paul_Phillips Paul_Reller Michael_Alec_Rose Daria_Semegen Gordon_Stout Christopher_Theofanidis Michael_Sidney_Timpson Fisher_Tull Dan_Welcher Michael_Glenn_Will

Let's write this adjacency list to a file, and load it into networkx.

In [8]:
with codecs.open("adjlist.txt", "w", "utf-8") as temp:
    temp.write(adjlist)
In [9]:
G = nx.read_adjlist('adjlist.txt', create_using=nx.DiGraph())
In [10]:
G.number_of_nodes()
Out[10]:
4926

Let's also export our graph to a JSON format that Cytoscape.js can understand. This might be useful later on if we want to make a JavaScript visualization of the data.

In [11]:
def pretty_node(node):
    """Given a node name, format it for display purposes - e.g. 'Carl_Friedrich_Abel' => 'C. Abel'."""
    return u'{}. {}'.format(node[0], node.split('_I')[0].split('_')[-1])
In [12]:
def export_to_cytoscape_json(graph, filename, weight_fn=None):
    def exported_node(node, weight_fn=None):
        return {
            'data': {
                'id': unicode(pretty_node(node)),
                'fullName': unicode(node.replace('_', ' ')),
                'weight': weight_fn(node) if weight_fn else 1
            }
        }
    
    # Topologically sort the nodes if possible (i.e. the graph is acyclic)
    try:
        nodes = [exported_node(i, weight_fn) for i in nx.topological_sort(graph)]
    except:
        nodes = [exported_node(i, weight_fn) for i in graph.nodes()]
    links = [{'data': {'source': pretty_node(u[0]), 'target': pretty_node(u[1])}} for u in graph.edges()]
    with codecs.open(filename, "w", "utf-8") as file:
        json.dump(nodes + links, file, indent=2)
In [13]:
export_to_cytoscape_json(G, 'cytoscape.json')

2. Who are the most influential teachers?

Now that we have our graph, let's analyze it!

For starters, let's see if we can figure out who the most influential teacher is.

But how do we quantify "influence"? One natural metric to use is Katz centrality, which works similarly to Google's PageRank – nodes that are connected to influential nodes themselves gain influence, and so on.

Katz centrality is a directional measure, and it turns out we need to compute it over the reversed graph to find the most influential teachers – otherwise we just end up getting the most "influential students".

In [14]:
c = nx.katz_centrality(G.reverse())
In [16]:
for teacher in sorted(c, key=c.get)[::-1][:20]:
    print teacher, c[teacher]
Nadia_Boulanger 0.274464766203
Darius_Milhaud 0.133984907008
Karlheinz_Stockhausen 0.112095058281
Olivier_Messiaen 0.108104999101
Roger_Sessions 0.0924535118831
André_Gedalge 0.0821795297328
Vincent_d'Indy 0.0676563365134
Arnold_Schoenberg 0.067113904931
Paul_Hindemith 0.0660738864042
Charles-Marie_Widor 0.0585465871662
Gabriel_Fauré 0.0567998995
Franz_Liszt 0.0564699447567
Aaron_Copland 0.0560713201348
Antoine_François_Marmontel 0.0542184811049
Louis_Vierne 0.0539401495126
Louis_Andriessen 0.0533430371564
Charles_Villiers_Stanford 0.0533227668023
Milton_Babbitt 0.0531766068805
Ferruccio_Busoni 0.0523346432626
Salomon_Jadassohn 0.0500280441408

There's a lot of famous names there, but in the end, it's not even close. Nadia Boulanger has more than double the centrality of anybody else. Given that Boulanger taught everyone from Copland to Piazzolla to Quincy Jones, her rank is well-deserved.

And let's make a subgraph of just the 100 most-influential teachers – it'll be easier to work with than the whole graph of ~5,000 people.

In [17]:
most_important_teachers = sorted(c, key=c.get)[::-1][:100]
In [18]:
important_teachers_graph = nx.subgraph(G, most_important_teachers)

3. Let's visualize it!

I stole some code from StackOverflow to visualize hierarchical graphs in a pretty way:

In [19]:
# https://stackoverflow.com/questions/29586520/can-one-get-hierarchical-graphs-from-networkx-with-python-3/29597209
def hierarchy_pos(G, root, levels=None, width=1., height=1.):
    '''If there is a cycle that is reachable from root, then this will see infinite recursion.
       G: the graph
       root: the root node
       levels: a dictionary
               key: level number (starting from 0)
               value: number of nodes in this level
       width: horizontal space allocated for drawing
       height: vertical space allocated for drawing'''
    TOTAL = "total"
    CURRENT = "current"
    def make_levels(levels, node=root, currentLevel=0, parent=None):
        """Compute the number of nodes for each level
        """
        if not currentLevel in levels:
            levels[currentLevel] = {TOTAL : 0, CURRENT : 0}
        levels[currentLevel][TOTAL] += 1
        try:
            neighbors = G.neighbors(node)
            for neighbor in neighbors:
                if not neighbor == parent:
                    levels =  make_levels(levels, neighbor, currentLevel + 1, node)
        except:
            pass
        return levels

    def make_pos(pos, node=root, currentLevel=0, parent=None, vert_loc=0):
        dx = 1.0/levels[currentLevel][TOTAL]
        left = dx/2
        pos[node] = ((left + dx*levels[currentLevel][CURRENT])*width, vert_loc)
        levels[currentLevel][CURRENT] += 1
        try:
            neighbors = G.neighbors(node)
            for neighbor in neighbors:
                if not neighbor == parent:
                    pos = make_pos(pos, neighbor, currentLevel + 1, node, vert_loc-vert_gap)
        except:
            pass
        return pos
    
    if levels is None:
        levels = make_levels({})
    else:
        levels = {l:{TOTAL: levels[l], CURRENT:0} for l in levels}
    vert_gap = height / (max([l for l in levels])+1)
    return make_pos({})

And wrapped some helpful logic around it:

In [20]:
def draw_hierarchical_graph(graph, root=None):
    plt.figure(figsize=(10, 10))

    root = root or nx.topological_sort(graph).next()
    pos = hierarchy_pos(graph, root)
    reduced_graph = nx.subgraph(graph, pos.keys())
    
    relabeled_graph = nx.relabel_nodes(reduced_graph, pretty_node)
    relabeled_pos = {pretty_node(node): p for node, p in pos.items()}
    nx.draw(relabeled_graph, pos=relabeled_pos, with_labels=True, node_size=500, node_color='w', font_size=9, arrowsize=8)

Ok, let's try it! Note that if we don't provide a root node, the graph will be drawn from the first node in topological order (in other words, any node that doesn't have any parents):

In [21]:
draw_hierarchical_graph(important_teachers_graph)

A bit messy, but on the whole, that's a reasonable looking graph! It starts from the influential 19th-century French teacher and pianist Marmontel, and proceeds through his various students to such names as Boulanger, Messiaen, Milhaud, and Cage.

Let's expand our graph a little bit.

One thing we could do is simply add more teachers from the most-influential list – say, take the top 150 instead of the top 100. But let's try something else.

What if we find all the people who are "in between" influential teachers but not influential teachers themselves – that is, people who both taught and were taught by people on the influential-teachers list?

In [22]:
most_important_teachers_students = set([n for t in most_important_teachers for n in G.successors(t)])
In [23]:
def has_important_students(teacher):
    students = [s for s in G.successors(teacher)]
    return any([student in most_important_teachers for student in students])

most_important_teachers_and_important_students = \
    set(most_important_teachers) | set([p for p in most_important_teachers_students if has_important_students(p)])           
In [24]:
len(most_important_teachers_and_important_students) - len(most_important_teachers)
Out[24]:
32

Looks like there are 32 of these students who are "in between" influential teachers. Who are they?

In [25]:
most_important_teachers_and_important_students - set(most_important_teachers)
Out[25]:
{u'Alban_Berg',
 u'Aleksander_Micha\u0142owski',
 u'Anna_Yesipova',
 u'Charles_Koechlin',
 u'Charles_Wood',
 u'Ernst_Krenek',
 u'Eusebius_Mandyczewski',
 u'Franz_Lachner',
 u'Friedrich_Kiel',
 u'Fritz_Reiner',
 u'Gian_Carlo_Menotti',
 u'Hans_von_Koessler',
 u'Henri_Dutilleux',
 u'Henri_Vieuxtemps',
 u'Henry_Cowell',
 u'Horatio_Parker',
 u'Isidor_Seiss',
 u'Istv\xe1n_Thom\xe1n',
 u'Jacques-Nicolas_Lemmens',
 u'Johann_Nepomuk_Hummel',
 u'Louis_Niedermeyer',
 u'Ludwig_Thuille',
 u'Ludwig_van_Beethoven',
 u'Marcel_Dupr\xe9',
 u'Marion_Bauer',
 u'Maurice_Ravel',
 u'Orpha-F._Deveaux',
 u'Paul_Pisk',
 u'Peter_Mennin',
 u'Roberto_Gerhard',
 u'Romain-Octave_Pelletier_I',
 u'Stephen_Heller'}

Ah yeah, some big names here, like Beethoven (taught by Salieri and Albrechtsberger and in turn taught Czerny).

Let's make a slightly bigger subgraph consisting of both the 100 most influential teachers and their "in-between" students:

In [26]:
teacher_and_student_subgraph = nx.subgraph(G, most_important_teachers_and_important_students)

Unfortunately, by bringing these extra people into the graph, we introduce some cycles, making a topological sort (which we need to visualize the graph) no longer possible:

In [27]:
try:
    print [x for x in nx.topological_sort(teacher_and_student_subgraph)]
except Exception:
    import traceback
    traceback.print_exc()
Traceback (most recent call last):
  File "<ipython-input-27-280518520b52>", line 2, in <module>
    print [x for x in nx.topological_sort(teacher_and_student_subgraph)]
  File "/Users/alex/src/music-graph/env/lib/python2.7/site-packages/networkx/algorithms/dag.py", line 208, in topological_sort
    raise nx.NetworkXUnfeasible("Graph contains a cycle or graph changed "
NetworkXUnfeasible: Graph contains a cycle or graph changed during iteration

How many cycles are there, anyway?

In [28]:
for cycle in nx.simple_cycles(teacher_and_student_subgraph):
    print cycle
[u'Claude_Champagne', u'Orpha-F._Deveaux']
[u'Simon_Sechter', u'Henri_Vieuxtemps']

Removing individual edges seems to be a bit of a pain in networkx, so let's just take the easy way out and arbitrarily remove one node from each of the cycles:

In [29]:
most_important_teachers_and_important_students.remove(u'Orpha-F._Deveaux')
most_important_teachers_and_important_students.remove(u'Henri_Vieuxtemps')
In [30]:
teacher_and_student_subgraph = nx.subgraph(G, most_important_teachers_and_important_students)

And now we can finally visualize this augmented subgraph. Unfortunately all the extra links introduced by the "in-between" students make it a little messy:

In [31]:
draw_hierarchical_graph(teacher_and_student_subgraph)