###Si `nx.erdos_renyi_graph` arroja el error random_state_index is incorrect, intentar:
# import decoratorNetwork Science - UDD
Centralidad y Mundo Pequeño
Cristian Candia-Castro Vallejos, Ph.D.\(^{1,2,3,4}\)
- [1] Data Science Institute (IDS), Universidad del Desarrollo, Chile
- [2] Northwestern Institute on Complex Systems, Kellogg School of Management, Northwestern Unviersity, USA
- [3] Centro de Investigación en Complejidad Social, Universidad del Desarrollo, Chile
- [4] Computational Research in Social Science Laboratory, Facultad de Ingeniería y Facultad de Gobierno, Universidad del Desarrollo, Chile
Referencias:
- https://github.com/MridulS/pydata-networkx
Redes? Grafos?
Una estructura matemática utilizada para modelar relaciones por pares entre objetos, donde los objetos generalmente se denominan nodos y la relación entre ellos enlaces.
\(G = (V, E)\)
\(V\) = conjunto de nodos/vértices
\(E\) = conjunto de \((x, y)\) enlaces
# print(decorator.__version__)# ! pip install decorator==5.0.9 #, 5.1.0####Luego reiniciar Jupyter y correr:
# ! pip install networkx# Module 1: Comenzando con NetworkX
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import collections
import networkx as nx
%matplotlib inline
import urllib.request as urllib
import io
# path = 'https://saref.github.io/teaching/PyNetworkshop/'G_rand=nx.erdos_renyi_graph(14, .01)type(G_rand)networkx.classes.graph.Graphnx.draw_circular(nx.erdos_renyi_graph(14, 0.2), with_labels=True)
¿Ejemplos de redes?
¿Puedes pensar en algunas redes del mundo real?
Usando NetworkX
# Creating a graph/network object
G = nx.Graph()# accessing nodes
G.nodes()NodeView(())# accessing edges
G.edges()EdgeView([])# Creemos una red de recetas
# G.add_node('Tomato')
# G.add_node('Eggs')
# G.add_node('Lamb')
# G.add_node('Chicken')
G.add_nodes_from(['Tomato', 'Eggs', 'Lamb', 'Chicken'])# G.add_edge('Tomato', 'Eggs')
# G.add_edge('Lamb', 'Tomato')
# G.add_edge('Chicken', 'Tomato')
G.add_edges_from([('Tomato', 'Eggs'), ('Tomato', 'Lamb'), ('Tomato', 'Chicken')])G.nodes()
#list(G.nodes())NodeView(('Tomato', 'Eggs', 'Lamb', 'Chicken'))G.edges()EdgeView([('Tomato', 'Eggs'), ('Tomato', 'Lamb'), ('Tomato', 'Chicken')])nx.draw(G, with_labels=True)
# cualquier objeto "hasheable" puede ser un nodo en la red
G.add_node([1, 2])--------------------------------------------------------------------------- TypeError Traceback (most recent call last) Cell In[28], line 2 1 # cualquier objeto "hasheable" puede ser un nodo en la red ----> 2 G.add_node([1, 2]) File ~/anaconda3/envs/candialab2/lib/python3.10/site-packages/networkx/classes/graph.py:563, in Graph.add_node(self, node_for_adding, **attr) 524 def add_node(self, node_for_adding, **attr): 525 """Add a single node `node_for_adding` and update node attributes. 526 527 Parameters (...) 561 doesn't change on mutables. 562 """ --> 563 if node_for_adding not in self._node: 564 if node_for_adding is None: 565 raise ValueError("None cannot be a node") TypeError: unhashable type: 'list'
En Python, cualquier objeto inmutable (como un número entero, booleano, cadena, tupla) es hash, lo que significa que su valor no cambia durante su vida útil.
Creemos un grafo que se parezca al siguiente:
# Los datos de una red se pueden representar como un edgelist, una matriz o un grafo
G = nx.Graph()
G.add_nodes_from([1, 2, 3, 4, 5])
G.add_edges_from([(1, 2), (2,1), (2, 3), (1,3), (1,4), (1,5), (3,5)])# 7 tuplasnx.draw(G, with_labels=True)
G.number_of_nodes()5G.number_of_edges() # observe que 1,2 y 2,1 cuentan como 1 enlace. Ya que este es un grafo no-dirigido6list(G.nodes)[1, 2, 3, 4, 5]list(G.edges)[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (3, 5)]#remueve nodos
G.remove_node(1) # take notice that once you remove node 1, all edges linked with 1 will be removed as well.list(G.nodes)[2, 3, 4, 5]list(G.edges)[(2, 3), (3, 5)]nx.draw(G, with_labels=True)
#remueve enlaces
G.remove_edge(3,5) # remove edge from 3 to 5 list(G.edges)[(2, 3)]nx.draw(G, with_labels=True)
# agrega atributos a nodos y enlaces
G.add_node(1, occupation="Estudiante")
G.add_edge(1,2, weight="5") # creating a weighted network#Atajo para crear enlaces con pesos
G.add_weighted_edges_from([(1, 2, 10), (1, 3, 1.30)])nx.draw(G, with_labels=True)
# Estábamos trabajando con una red no dirigida. Intentemos crear una versión dirigida.
DiG= nx.DiGraph()#Base class for directed graphs.
DiG.add_nodes_from([1, 2, 3, 4, 5])
DiG.add_edges_from([(1, 2), (2, 1), (2, 3), (1,3), (1,4), (1,5), (3,5)])nx.draw(DiG, with_labels=True)
DiG.number_of_nodes()5DiG.number_of_edges() # ahora observe que 1,2 y 2,1 se cuentan dos veces, ya que este es un grafo dirigido.7# otro atajo entre redes dirigidas y no-dirigidas
DirectedG = G.to_directed()
UndirectedG = DiG.to_undirected()nx.draw(UndirectedG, with_labels=True)
nx.draw(DirectedG, with_labels=True)
Estadísticas descriptivas para una red en múltiples niveles (ego, díada, tríada, subgrupo, red)
# Let's load a well-known dataset, Krackkite graph (use nx. for existing datasets)
G = nx.krackhardt_kite_graph()Nota: En teoría de grafos, el grafo de cometas de Krackhardt es un grafo simple con diez nodos. El grafo lleva el nombre de David Krackhardt, un investigador de la teoría de redes sociales.
Krackhardt introdujo el grafo en 1990 para distinguir diferentes conceptos de centralidad. Tiene la propiedad de que el vértice con grado máximo (etiquetado 3 en la figura, con grado 6), el vértice con centralidad máxima de intermediación (etiquetado 7) y los dos vértices con centralidad máxima de proximidad (etiquetado 5 y 6) son todos diferentes unos de otros.
plt.figure(figsize =(10, 10))
nx.draw_networkx(G)
Análisis a nivel de nodo
Centralidad
Los índices de centralidad son respuestas a la pregunta “¿Qué caracteriza a un nodo importante?” La palabra “importancia” tiene un gran número de significados, lo que lleva a muchas definiciones diferentes de centralidad.
Calcularemos (1) centralidad de grado, (2) centralidad de intermediación, (3) centralidad de proximidad, (4) centralidad de vector propio y las relaciones entre ellos.
# degree.items()# list(reversed(sorted(degree.items(), key=lambda x: x[1])))Centralidad de grado
Si importancia significa:
Que tan popular eres? Cuantas personas cononces?
también puede calcularse sin normalizar, dependiendo de la pregunta que se quiera responder.
nx.degree_centrality(G)
#degree centrality (cuántas conocen / cuántas personas pueden conocer)
degree = nx.degree_centrality(G)
print(degree)
[print('El nodo ',nodo," tiene centralidad: ",centralidad) for (nodo, centralidad) in reversed(sorted(degree.items(), key=lambda x: x[1]))]{0: 0.4444444444444444, 1: 0.4444444444444444, 2: 0.3333333333333333, 3: 0.6666666666666666, 4: 0.3333333333333333, 5: 0.5555555555555556, 6: 0.5555555555555556, 7: 0.3333333333333333, 8: 0.2222222222222222, 9: 0.1111111111111111}
El nodo 3 tiene centralidad: 0.6666666666666666
El nodo 6 tiene centralidad: 0.5555555555555556
El nodo 5 tiene centralidad: 0.5555555555555556
El nodo 1 tiene centralidad: 0.4444444444444444
El nodo 0 tiene centralidad: 0.4444444444444444
El nodo 7 tiene centralidad: 0.3333333333333333
El nodo 4 tiene centralidad: 0.3333333333333333
El nodo 2 tiene centralidad: 0.3333333333333333
El nodo 8 tiene centralidad: 0.2222222222222222
El nodo 9 tiene centralidad: 0.1111111111111111[None, None, None, None, None, None, None, None, None, None]Nota: Una lambda es una función anónima a menudo se usa en funciones como sorted() que toman un invocable como parámetro (a menudo el parámetro de palabra clave key). También puede proporcionar una función existente en lugar de una lambda, siempre que sea un objeto invocable.
Nota: Un objeto invocable es un objeto que le permite usar paréntesis redondos () y eventualmente pasar algunos parámetros, al igual que las funciones.
Betweenness Centrality
Si importancia significa:
Capacidad de intermediación entre grupos.
Probabilidad de que la información que se origina en cualquier lugar de la red te alcance
- \(d_{jk}\) el número de caminos mas cortos que pasan entre los nodos \(j\) y \(k\).
- \(d_{jk}(i)\) número de caminos mas cortos entre j y k que pasan por el nodo \(i\). ######
nx.betweenness_centrality(G, k=None, normalized=True, weight=None, endpoints=False, seed=None)
## Betweenness centrality: ¿Con qué frecuencia un nodo está en el camino más corto entre otros 2 nodos?
# (con qué frecuencia nos encontramos con Alice en nuestro camino para llegar a otros)
between = nx.betweenness_centrality(G, normalized = True, endpoints = False) # normalize betweenness centrality when values get large
[print(key,":",value) for (key, value) in reversed(sorted(between.items(), key=lambda x: x[1]))]7 : 0.38888888888888884
6 : 0.23148148148148148
5 : 0.23148148148148148
8 : 0.2222222222222222
3 : 0.10185185185185183
1 : 0.023148148148148143
0 : 0.023148148148148143
9 : 0.0
4 : 0.0
2 : 0.0[None, None, None, None, None, None, None, None, None, None]Closeness centrality
Si importancia significa:
Estar cerca de todos los nodos
- \(d_{ij}\) es la distancia entre los nodos \(i\) y \(j\).
nx.closeness_centrality(G, u=None, distance=None, normalized=True)
## Closeness centrality: qué tan accesible es alguien para el resto de la red (networkx usa la distancia *inward* a un nodo, no hacia outward. Para usar la distancia outward, use `G.reverse())
closeness = nx.closeness_centrality(G)
[print(key," : ",value) for (key, value) in reversed(sorted(closeness.items(), key=lambda x: x[1]))]#ordena en funcion del element en la segunda posicion (1)6 : 0.6428571428571429
5 : 0.6428571428571429
7 : 0.6
3 : 0.6
1 : 0.5294117647058824
0 : 0.5294117647058824
4 : 0.5
2 : 0.5
8 : 0.42857142857142855
9 : 0.3103448275862069[None, None, None, None, None, None, None, None, None, None]Eigenvector centrality
Si importancia significa:
Estar conectado a nodos importantes
\(\lambda\) es el valor propio.
La solución (cuando existe) da la centralidad del nodo. Tomamos el mas alto \(\lambda\).
Este concepto es el núcleo del algoritmo de clasificación de Google.
nx.eigenvector_centrality(G, max_iter=100, tol=1e-06, nstart=None, weight='weight')
## Eigenvector centrality: alguien es tan importante en la medida en que el vecino de ese alguien es importante
eigenvector = nx.eigenvector_centrality(G) # eigenvector centrality
[print(key," : ",value) for (key, value) in reversed(sorted(eigenvector.items(), key=lambda x: x[1]))]3 : 0.4810204881221006
6 : 0.3976910106255469
5 : 0.3976910106255469
1 : 0.35220898139203594
0 : 0.35220898139203594
4 : 0.2858347353163241
2 : 0.2858347353163241
7 : 0.19586185175360382
8 : 0.04807477501420294
9 : 0.011164058575824238[None, None, None, None, None, None, None, None, None, None]Resumen
Ejemplo del mundo real: la red de familias de élite florentinas del siglo XIV
Ejemplo
- Prestigio de las familias florentinas durante el Renacimiento.
G2=nx.florentine_families_graph()plt.figure(figsize =(8, 8))
nx.draw_networkx(G2)
TAREA 2.1
P: ¿Quiénes son las 3 familias principales en términos de: 1. ¿Centralidad de grado? 2. ¿Centralidad de intermediación? 3. ¿Centralidad de cercanía? 4. ¿Centralidad del vector propio?
Una matriz que muestra las relaciones entre las medidas de centralidad: 
PageRank
Redes dirigidas.
El algoritmo del billón de dólares, PageRank, funciona contando el número y la importancia de los enlaces a una página para determinar una estimación aproximada de la importancia del sitio web. El algoritmo asigna un valor de importancia a cada página web en función del número y la calidad de los enlaces a la página. La suposición subyacente es que es probable que los sitios web más importantes reciban más enlaces de otros sitios web. Fue desarrollado por los fundadores de Google, Larry Page y Sergey Brin.
# Crear un grafo dirigido
G = nx.DiGraph()
# Añadir bordes (estos también añaden nodos)
G.add_edge('A', 'B')
G.add_edge('B', 'C')
G.add_edge('C', 'A')
G.add_edge('A', 'D')
G.add_edge('D', 'E')
G.add_edge('E', 'F')
G.add_edge('B', 'F')
G.add_edge('C', 'G')
G.add_edge('G', 'A')
# Aplicar el algoritmo PageRank
pagerank = nx.pagerank(G, alpha=0.85)
# Ordenar los valores de PageRank de mayor a menor
pagerank_sorted = sorted(pagerank.items(), key=lambda item: item[1], reverse=True)
# Imprimir los valores de PageRank ordenados
for node, pagerank_value in pagerank_sorted:
print(f"Node: {node}, PageRank: {pagerank_value}")
# Crear una lista de tamaños para los nodos basada en los valores de PageRank
sizes = [pagerank[node] * 20000 for node in G.nodes]
# Dibujar la red
nx.draw(G, with_labels=True, node_size=sizes)
# Mostrar la red
plt.show()Node: F, PageRank: 0.2329773721880849
Node: A, PageRank: 0.1721303892141963
Node: E, PageRank: 0.15416106912747013
Node: B, PageRank: 0.12287387024080189
Node: D, PageRank: 0.12287387024080189
Node: C, PageRank: 0.10193997532982814
Node: G, PageRank: 0.09304345365881689
Análisis a nivel de díadas: recuentos de díadas, caminos, geodésicas
## El número de diadas (same as calculating number of edges)
# For undirected networks: N(N-1)/2
# For directed networks: N(N-1)
G2.number_of_edges()20G2.nodes # lookup of the nodenamesNodeView(('Acciaiuoli', 'Medici', 'Castellani', 'Peruzzi', 'Strozzi', 'Barbadori', 'Ridolfi', 'Tornabuoni', 'Albizzi', 'Salviati', 'Pazzi', 'Bischeri', 'Guadagni', 'Ginori', 'Lamberteschi'))G2.number_of_edges('Medici',"Peruzzi") #'Barbadori' comprueba si hay un enlace entre dos nodos0nx.has_path(G2,'Medici','Peruzzi') # comprobar si existe ruta entre los nodos Medici y PeruzziTrue# Lista de todos los caminos posibles entre Medici y Peruzzi
paths = nx.all_simple_paths(G2, source='Medici', target='Peruzzi', cutoff=3)# cutoff fija el corte para la longitud de el manimo menor o igual que el valor dado (3)
print(list(paths))[['Medici', 'Barbadori', 'Castellani', 'Peruzzi'], ['Medici', 'Ridolfi', 'Strozzi', 'Peruzzi']]nx.shortest_path(G2,'Medici','Peruzzi') # encuentra una ruta geodésica
#Visualiza el camino mas corto
geodesic = nx.shortest_path(G2,'Medici','Peruzzi')
layout = nx.spring_layout(G2)
edges = [(geodesic[i], geodesic[i+1]) for i in range(len(geodesic) - 1)]
nx.draw_networkx_edges(G2, layout, edgelist = edges, edge_color="r", width=10)
nx.draw(G2, layout, with_labels=True)
# Que tan similares son los vecinos de los 2 nodos
list(nx.common_neighbors(G2,'Medici', 'Ginori'))['Albizzi']Trabajemos en una red real
La red de colaboración Arxiv GR-QC (Relatividad General y Cosmología Cuántica) es de e-print arXiv y cubre las colaboraciones científicas entre los artículos de los autores enviados a la categoría de Relatividad General y Cosmología Cuántica. Si un autor \(i\) es coautor de un artículo con el autor \(j\), el grafo contiene un enlace no dirigido entre \(i\) y \(j\). Si el artículo es co-autoreado por \(k\) autores, esto genera un (sub) grafo completamente conectado con \(k\) nodos.
fuente: http://snap.stanford.edu/data/index.html#canets
# Si seis autores escribieron un artículo juntos, tendrán un grafo completo
nx.draw(nx.complete_graph(5),with_labels=True)
# crear un grafo de autor a partir del conjunto de datos
import csv
authors_graph = nx.Graph()
with open('./Data/CA-GrQc.txt', 'r') as f:
reader = csv.reader(f, delimiter='\t')
for row in reader:
authors_graph.add_edge(row[0], row[1])type(authors_graph)networkx.classes.graph.Graphprint(authors_graph.number_of_edges())
print(authors_graph.number_of_nodes())14496
5242authors_graph.nodes()NodeView(('3466', '937', '5233', '8579', '10310', '15931', '17038', '18720', '19607', '1854', '4583', '9572', '10841', '13056', '14982', '16310', '19640', '23855', '24372', '24814', '5052', '899', '1796', '2287', '3096', '3386', '4472', '5346', '5740', '6094', '6376', '9124', '10235', '10427', '10597', '15159', '16148', '16741', '18235', '18549', '19297', '20511', '20595', '20613', '24371', '24559', '24731', '25102', '25271', '25396', '1658', '4822', '6864', '7689', '7926', '10268', '12971', '18600', '20421', '20886', '21048', '22393', '23186', '23214', '23298', '23945', '24939', '339', '624', '3731', '4743', '5407', '6610', '6700', '8045', '9099', '9639', '9785', '12141', '15184', '15784', '18719', '19870', '20532', '22527', '23576', '23577', '23649', '24199', '24293', '25201', '10243', '6774', '8049', '8053', '8517', '11964', '15538', '16694', '18648', '19423', '21012', '22457', '22691', '23452', '16174', '16470', '17822', '14265', '392', '2485', '2949', '3173', '3441', '3593', '3853', '3927', '3937', '3939', '5107', '5218', '5230', '6030', '7350', '7504', '7601', '8718', '9522', '11621', '12498', '12691', '15251', '16020', '16261', '17156', '17626', '18622', '19059', '19525', '19738', '20122', '20432', '21866', '22074', '23721', '8916', '13556', '14485', '8612', '615', '743', '2076', '4515', '5773', '9482', '10822', '11175', '11604', '14004', '15003', '15552', '15814', '16083', '17932', '20001', '20100', '23481', '16258', '1356', '1727', '2752', '4125', '6667', '6825', '10039', '10351', '11082', '14123', '16676', '21194', '10912', '14534', '17268', '19783', '21705', '22836', '2710', '62', '106', '260', '2959', '3677', '4708', '5172', '5541', '5794', '5807', '6575', '8458', '10601', '11401', '13026', '13205', '13659', '13989', '14007', '14009', '14599', '15301', '18757', '20934', '21543', '22184', '23647', '23708', '25916', '26023', '26051', '26100', '214', '5435', '6512', '10590', '23559', '1765', '3032', '5302', '7383', '7442', '7768', '13276', '17266', '22415', '10794', '7050', '25850', '10113', '10657', '12130', '17172', '4846', '676', '824', '2133', '2654', '4748', '5672', '10549', '12928', '13220', '14419', '17330', '17439', '18487', '20850', '22779', '23382', '24029', '11785', '45', '46', '570', '773', '1653', '2212', '2741', '2952', '3372', '4046', '4164', '4511', '4513', '5262', '6179', '6830', '7956', '8879', '11241', '11472', '12365', '12496', '12678', '12781', '12851', '14540', '14807', '15659', '16159', '17655', '17692', '18894', '19961', '20108', '20562', '20635', '21281', '21508', '21847', '22798', '22887', '23293', '24955', '25346', '25758', '934', '5579', '9755', '10550', '16032', '17331', '17603', '20644', '22497', '23387', '23907', '24924', '25080', '12422', '1339', '3164', '15580', '16393', '20478', '20956', '3890', '5621', '8824', '11613', '12306', '12860', '14547', '18182', '21707', '24696', '2661', '7899', '8067', '8208', '11132', '11402', '12980', '13364', '14969', '16389', '18109', '18365', '23038', '24845', '25379', '13740', '4550', '4702', '7264', '13096', '14128', '19489', '19527', '19784', '22476', '25006', '25486', '26', '1407', '1488', '8219', '10762', '11801', '12665', '12688', '13142', '15108', '15321', '20647', '20827', '20879', '23614', '3909', '17979', '3872', '5109', '7533', '12409', '20101', '23096', '8862', '78', '4877', '7459', '8254', '12155', '22598', '24932', '888', '1520', '6468', '6627', '7007', '7712', '10711', '13614', '14102', '18517', '18676', '23351', '23689', '24114', '2465', '2592', '3977', '5055', '5993', '9265', '12334', '19890', '20341', '21560', '17309', '24833', '543', '1958', '2193', '3917', '6858', '8148', '9092', '12478', '15366', '18125', '18398', '19675', '21806', '23693', '26196', '10115', '10134', '23916', '7893', '593', '5510', '9360', '12627', '16778', '18037', '18051', '13385', '19578', '12386', '13333', '23896', '8978', '9017', '15170', '15455', '16589', '2255', '3056', '6158', '7307', '7324', '8365', '9023', '11444', '12324', '12472', '13831', '14746', '16128', '17075', '18875', '19900', '20000', '20806', '21944', '21968', '23302', '23665', '23758', '24722', '12045', '12287', '14181', '20257', '21613', '7510', '197', '8851', '1343', '2991', '8299', '15416', '18088', '25286', '1254', '3420', '10130', '2250', '3243', '7717', '7985', '11015', '12085', '13714', '14767', '16056', '16994', '17414', '18971', '19216', '20534', '21776', '21860', '25205', '178', '1248', '1403', '2368', '2420', '16210', '18681', '20641', '24762', '2307', '6934', '22423', '231', '345', '1186', '1234', '1841', '1997', '2404', '2450', '2980', '3409', '5134', '5578', '8503', '9341', '9889', '12503', '13060', '13597', '16611', '18208', '18543', '18866', '22421', '22937', '23363', '23628', '25053', '25251', '2982', '4036', '4115', '12938', '13032', '19215', '21432', '22726', '22834', '22966', '23511', '25528', '25836', '14376', '8710', '22483', '1375', '2846', '5555', '5564', '5787', '9721', '10158', '10942', '13600', '13929', 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'5261', '6282', '18908', '25531', '12050', '5364', '14', '14171', '14845', '5232', '19521'))(authors_graph['3466'])AtlasView({'937': {}, '5233': {}, '8579': {}, '10310': {}, '15931': {}, '17038': {}, '18720': {}, '19607': {}})nx.draw(authors_graph,with_labels=True)
¿Podemos encontrar al investigador más influyente/importante en esta red?
¿Cómo evaluamos la importancia de algunas personas en una red?
Dentro de una red social, habrá determinadas personas que desempeñen determinadas funciones importantes. Por ejemplo, puede haber personas hiperconectadas que estén conectadas con muchas, muchas más personas. Ellas serán claves para la difusión de información. Alternativamente, si se tratara de una red de contactos de enfermedades, identificarlos sería útil para detener la propagación de enfermedades.
¿Cómo se identificaría a estas personas?
Tarea 2.2
Cree una lista de tuplas (nodo, grado de nodo) y busque el nodo con el grado máximo.
grado de nodo = número de vecinos
Repita el ejercicio para todas mas métricas de centralidad vistas en este notebook (grado, betweenness, closseness, eigenvector y pagerank). Interprete sus resultados en función de la métrica.
El grado de un nodo se traduce en grado de centralidad (que es una versión normalizada de grado)
nx.degree_centrality(authors_graph){'3466': 0.0015264262545315779,
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...}Tarea 2.3
Trace un histograma de centralidad de grado (versión normalizada) y grado (número de enlaces) de author_graph.
Sugerencia: plt.hist(list_of_values) trazará un histograma
(comaprar count vs centralidad grado)
Repita el ejercicio para todas las métricas de centralidad vistas en este notebook.
Interprte sus resultados.
Echemos un vistazo a los componentes conectados de un grafo.
En la teoría de grafos, un componente conectado (o simplemente un componente) de un grafo no dirigido es un subgrafo en el que dos vértices cualesquiera están conectados entre sí por caminos, y que no está conectado a ningún vértice adicional en el supergrafo.
G = nx.erdos_renyi_graph(n=10, p=0.15, seed=1)
nx.draw(G, with_labels=True)
sorted(nx.connected_components(authors_graph),key=len, reverse=True)[{'14376',
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{'10897', '5548'},
{'23881', '5577'},
{'18098', '3100'},
{'1695', '19380'},
{'23416', '3074'},
{'19911', '25487'},
{'10148', '10847'},
{'25853', '8404'},
{'22891', '25'},
{'10559', '18245'},
{'10002', '7468'},
{'12041', '14384'},
{'17951', '833'},
{'6161', '7055'},
{'13022', '8077'},
{'2117', '6300'},
{'15609', '21809'},
{'21018', '5479'},
{'15218', '17585'},
{'18895', '7957'},
{'15181', '2331'},
{'21579', '373'},
{'10356', '19062'},
{'11828', '7807'},
{'24202', '2624'},
{'15572', '7356'},
{'22021', '8811'},
{'7534', '7536'},
{'1344', '18126'},
{'1376', '4267'},
{'15712', '19596'},
{'17471', '3989'},
{'12504', '9053'},
{'1589', '7603'},
{'15208', '23759'},
{'10517', '5119'},
{'6446', '8036'},
{'5137', '6884'},
{'24464', '8336'},
{'25510', '3279'},
{'19906', '26102'},
{'19941', '6079'},
{'16338', '21390'},
{'3844', '82'},
{'15688', '22692'},
{'4954', '4955'},
{'5409', '8623'},
{'2557', '4272'},
{'2981', '4427'},
{'6669', '9865'},
{'7588', '7590'},
{'14130', '3599'},
{'18581', '2938'},
{'21593', '22325'},
{'1187', '6290'},
{'15423', '18144'},
{'254', '7572'},
{'11623', '23621'},
{'11971', '25211'},
{'12554', '7636'},
{'18183', '4278'},
{'10522', '10561'},
{'12863', '18390'},
{'1836', '8400'},
{'24504', '8054'},
{'21623', '22605'},
{'10382', '25482'},
{'16643', '20035'},
{'14131', '372'},
{'25062', '8735'},
{'11911', '9039'},
{'21303', '734'},
{'17470', '19221'},
{'18194', '21949'},
{'14358', '5697'},
{'13717', '8558'},
{'8534', '8676'},
{'16622', '18454'},
{'16312', '16314'},
{'11991', '3186'},
{'14182', '17537'},
{'18156', '5387'},
{'19145', '23760'},
{'22110', '5231'},
{'20084', '22311'},
{'12751', '6307'},
{'11280', '831'},
{'14132', '15154'},
{'11641', '24462'},
{'14338', '22729'},
{'11411', '24242'},
{'5492', '5853'},
{'12473', '9831'},
{'18388', '20654'},
{'15401', '4196'},
{'15962', '25447'},
{'18417', '19892'},
{'20320', '4643'},
{'14990', '25975'},
{'14763', '4432'},
{'11718', '20766'},
{'15374', '15553'},
{'5490', '8474'},
{'1976', '23672'},
{'12248', '98'},
{'17992', '4181'},
{'2248', '8409'},
{'11579', '15073'},
{'16922', '20925'},
{'23772', '735'},
{'10638', '8717'},
{'23094', '2561'},
{'24733', '6776'},
{'11215', '23156'},
{'17503', '3748'},
{'10602', '20805'},
{'11467', '19575'},
{'25095', '9130'},
{'1010', '2749'},
{'11642', '25230'},
{'16015', '22938'},
{'20333', '4262'},
{'10407', '6701'},
{'22435', '5406'},
{'410', '4366'},
{'13413', '9211'},
{'19473', '23812'},
{'24214', '4116'},
{'20707', '24820'},
{'5486', '8366'},
{'5261', '6282'},
{'18908', '25531'},
{'12050', '5364'},
{'14', '14171'},
{'12295'}]print([len(c) for c in sorted(nx.connected_components(authors_graph),key=len, reverse=True)])[4158, 14, 12, 10, 9, 9, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1]Nota: la función sorted () tiene un parámetro opcional llamado key que toma una función como su valor. Esta función key transforma cada elemento antes de ordenar, toma el valor y se usa dentro de sorted en lugar del valor original.
# Guardamos subgrafos en una lista
graphs = [authors_graph.subgraph(c).copy() for c in sorted(nx.connected_components(authors_graph), key=len, reverse=True)]graphs[<networkx.classes.graph.Graph at 0x131703dc0>,
<networkx.classes.graph.Graph at 0x130caca60>,
<networkx.classes.graph.Graph at 0x130caca30>,
<networkx.classes.graph.Graph at 0x13103b6d0>,
<networkx.classes.graph.Graph at 0x130caf400>,
<networkx.classes.graph.Graph at 0x130c79c30>,
<networkx.classes.graph.Graph at 0x130c7a500>,
<networkx.classes.graph.Graph at 0x130c79e70>,
<networkx.classes.graph.Graph at 0x12f7aad10>,
<networkx.classes.graph.Graph at 0x130ec4400>,
<networkx.classes.graph.Graph at 0x130ec48b0>,
<networkx.classes.graph.Graph at 0x130ec4430>,
<networkx.classes.graph.Graph at 0x130ec4610>,
<networkx.classes.graph.Graph at 0x130ec4880>,
<networkx.classes.graph.Graph at 0x130ec42b0>,
<networkx.classes.graph.Graph at 0x13103acb0>,
<networkx.classes.graph.Graph at 0x131701b70>,
<networkx.classes.graph.Graph at 0x131700550>,
<networkx.classes.graph.Graph at 0x1317038e0>,
<networkx.classes.graph.Graph at 0x131703970>,
<networkx.classes.graph.Graph at 0x131701930>,
<networkx.classes.graph.Graph at 0x1317039d0>,
<networkx.classes.graph.Graph at 0x131701e10>,
<networkx.classes.graph.Graph at 0x131703940>,
<networkx.classes.graph.Graph at 0x131703910>,
<networkx.classes.graph.Graph at 0x131700700>,
<networkx.classes.graph.Graph at 0x131703790>,
<networkx.classes.graph.Graph at 0x131703370>,
<networkx.classes.graph.Graph at 0x131703cd0>,
<networkx.classes.graph.Graph at 0x1317031c0>,
<networkx.classes.graph.Graph at 0x131701ea0>,
<networkx.classes.graph.Graph at 0x131700340>,
<networkx.classes.graph.Graph at 0x131703ee0>,
<networkx.classes.graph.Graph at 0x131703c10>,
<networkx.classes.graph.Graph at 0x131703b80>,
<networkx.classes.graph.Graph at 0x131701e40>,
<networkx.classes.graph.Graph at 0x131702c50>,
<networkx.classes.graph.Graph at 0x131700c40>,
<networkx.classes.graph.Graph at 0x131702620>,
<networkx.classes.graph.Graph at 0x1317025f0>,
<networkx.classes.graph.Graph at 0x131703760>,
<networkx.classes.graph.Graph at 0x131701ae0>,
<networkx.classes.graph.Graph at 0x131701360>,
<networkx.classes.graph.Graph at 0x131700430>,
<networkx.classes.graph.Graph at 0x131703e80>,
<networkx.classes.graph.Graph at 0x131703010>,
<networkx.classes.graph.Graph at 0x131703af0>,
<networkx.classes.graph.Graph at 0x131701600>,
<networkx.classes.graph.Graph at 0x131703310>,
<networkx.classes.graph.Graph at 0x1317019c0>,
<networkx.classes.graph.Graph at 0x131703340>,
<networkx.classes.graph.Graph at 0x131701960>,
<networkx.classes.graph.Graph at 0x1317011b0>,
<networkx.classes.graph.Graph at 0x131702c80>,
<networkx.classes.graph.Graph at 0x131701ab0>,
<networkx.classes.graph.Graph at 0x1317000d0>,
<networkx.classes.graph.Graph at 0x131701ed0>,
<networkx.classes.graph.Graph at 0x131703d00>,
<networkx.classes.graph.Graph at 0x1317009a0>,
<networkx.classes.graph.Graph at 0x1317017e0>,
<networkx.classes.graph.Graph at 0x131701b10>,
<networkx.classes.graph.Graph at 0x131702a40>,
<networkx.classes.graph.Graph at 0x131703400>,
<networkx.classes.graph.Graph at 0x1317036a0>,
<networkx.classes.graph.Graph at 0x1317007f0>,
<networkx.classes.graph.Graph at 0x131701a50>,
<networkx.classes.graph.Graph at 0x131701a80>,
<networkx.classes.graph.Graph at 0x131701990>,
<networkx.classes.graph.Graph at 0x131702a70>,
<networkx.classes.graph.Graph at 0x131702860>,
<networkx.classes.graph.Graph at 0x131702c20>,
<networkx.classes.graph.Graph at 0x131700c70>,
<networkx.classes.graph.Graph at 0x1317026e0>,
<networkx.classes.graph.Graph at 0x131702110>,
<networkx.classes.graph.Graph at 0x1317020e0>,
<networkx.classes.graph.Graph at 0x1317000a0>,
<networkx.classes.graph.Graph at 0x131702590>,
<networkx.classes.graph.Graph at 0x131701330>,
<networkx.classes.graph.Graph at 0x131701fc0>,
<networkx.classes.graph.Graph at 0x131700040>,
<networkx.classes.graph.Graph at 0x131701f30>,
<networkx.classes.graph.Graph at 0x131701f90>,
<networkx.classes.graph.Graph at 0x131701f60>,
<networkx.classes.graph.Graph at 0x1317010f0>,
<networkx.classes.graph.Graph at 0x1317011e0>,
<networkx.classes.graph.Graph at 0x131700b20>,
<networkx.classes.graph.Graph at 0x131700af0>,
<networkx.classes.graph.Graph at 0x131700ac0>,
<networkx.classes.graph.Graph at 0x131700a30>,
<networkx.classes.graph.Graph at 0x131701840>,
<networkx.classes.graph.Graph at 0x131700b50>,
<networkx.classes.graph.Graph at 0x131700a90>,
<networkx.classes.graph.Graph at 0x131700bb0>,
<networkx.classes.graph.Graph at 0x1317008b0>,
<networkx.classes.graph.Graph at 0x131700880>,
<networkx.classes.graph.Graph at 0x1317008e0>,
<networkx.classes.graph.Graph at 0x131700910>,
<networkx.classes.graph.Graph at 0x1317039a0>,
<networkx.classes.graph.Graph at 0x131703b20>,
<networkx.classes.graph.Graph at 0x131703df0>,
<networkx.classes.graph.Graph at 0x131700c10>,
<networkx.classes.graph.Graph at 0x131700250>,
<networkx.classes.graph.Graph at 0x131700790>,
<networkx.classes.graph.Graph at 0x131700be0>,
<networkx.classes.graph.Graph at 0x131702980>,
<networkx.classes.graph.Graph at 0x1317027a0>,
<networkx.classes.graph.Graph at 0x131703a00>,
<networkx.classes.graph.Graph at 0x1317025c0>,
<networkx.classes.graph.Graph at 0x131700ca0>,
<networkx.classes.graph.Graph at 0x131700cd0>,
<networkx.classes.graph.Graph at 0x131700d30>,
<networkx.classes.graph.Graph at 0x131700d00>,
<networkx.classes.graph.Graph at 0x131700d60>,
<networkx.classes.graph.Graph at 0x131700d90>,
<networkx.classes.graph.Graph at 0x131700dc0>,
<networkx.classes.graph.Graph at 0x131700190>,
<networkx.classes.graph.Graph at 0x1317001f0>,
<networkx.classes.graph.Graph at 0x1317001c0>,
<networkx.classes.graph.Graph at 0x1317002e0>,
<networkx.classes.graph.Graph at 0x1317002b0>,
<networkx.classes.graph.Graph at 0x131700df0>,
<networkx.classes.graph.Graph at 0x131700e20>,
<networkx.classes.graph.Graph at 0x131700e50>,
<networkx.classes.graph.Graph at 0x131700eb0>,
<networkx.classes.graph.Graph at 0x131700e80>,
<networkx.classes.graph.Graph at 0x131700ee0>,
<networkx.classes.graph.Graph at 0x131700f10>,
<networkx.classes.graph.Graph at 0x131700f40>,
<networkx.classes.graph.Graph at 0x1317003d0>,
<networkx.classes.graph.Graph at 0x1317004c0>,
<networkx.classes.graph.Graph at 0x1317003a0>,
<networkx.classes.graph.Graph at 0x131700160>,
<networkx.classes.graph.Graph at 0x1317005e0>,
<networkx.classes.graph.Graph at 0x131700520>,
<networkx.classes.graph.Graph at 0x1317005b0>,
<networkx.classes.graph.Graph at 0x131700490>,
<networkx.classes.graph.Graph at 0x1317006d0>,
<networkx.classes.graph.Graph at 0x131700760>,
<networkx.classes.graph.Graph at 0x131700580>,
<networkx.classes.graph.Graph at 0x131700640>,
<networkx.classes.graph.Graph at 0x1317004f0>,
<networkx.classes.graph.Graph at 0x131700100>,
<networkx.classes.graph.Graph at 0x131700130>,
<networkx.classes.graph.Graph at 0x131700f70>,
<networkx.classes.graph.Graph at 0x131700fa0>,
<networkx.classes.graph.Graph at 0x131701000>,
<networkx.classes.graph.Graph at 0x131700fd0>,
<networkx.classes.graph.Graph at 0x131701030>,
<networkx.classes.graph.Graph at 0x131701060>,
<networkx.classes.graph.Graph at 0x131701090>,
<networkx.classes.graph.Graph at 0x131702050>,
<networkx.classes.graph.Graph at 0x131703280>,
<networkx.classes.graph.Graph at 0x131702800>,
<networkx.classes.graph.Graph at 0x1317021d0>,
<networkx.classes.graph.Graph at 0x131703190>,
<networkx.classes.graph.Graph at 0x131703040>,
<networkx.classes.graph.Graph at 0x131702fe0>,
<networkx.classes.graph.Graph at 0x131702890>,
<networkx.classes.graph.Graph at 0x1317035b0>,
<networkx.classes.graph.Graph at 0x1317029e0>,
<networkx.classes.graph.Graph at 0x131702350>,
<networkx.classes.graph.Graph at 0x1317022c0>,
<networkx.classes.graph.Graph at 0x131702ec0>,
<networkx.classes.graph.Graph at 0x131702e90>,
<networkx.classes.graph.Graph at 0x131702ce0>,
<networkx.classes.graph.Graph at 0x131702ef0>,
<networkx.classes.graph.Graph at 0x131702e30>,
<networkx.classes.graph.Graph at 0x131703100>,
<networkx.classes.graph.Graph at 0x1317034c0>,
<networkx.classes.graph.Graph at 0x131702f50>,
<networkx.classes.graph.Graph at 0x131703130>,
<networkx.classes.graph.Graph at 0x131702e60>,
<networkx.classes.graph.Graph at 0x131702da0>,
<networkx.classes.graph.Graph at 0x131703ac0>,
<networkx.classes.graph.Graph at 0x131702f20>,
<networkx.classes.graph.Graph at 0x131703430>,
<networkx.classes.graph.Graph at 0x131703580>,
<networkx.classes.graph.Graph at 0x131702cb0>,
<networkx.classes.graph.Graph at 0x1317024d0>,
<networkx.classes.graph.Graph at 0x131703520>,
<networkx.classes.graph.Graph at 0x131703550>,
<networkx.classes.graph.Graph at 0x1317034f0>,
<networkx.classes.graph.Graph at 0x131703250>,
<networkx.classes.graph.Graph at 0x1317030a0>,
<networkx.classes.graph.Graph at 0x131702aa0>,
<networkx.classes.graph.Graph at 0x1317009d0>,
<networkx.classes.graph.Graph at 0x131703610>,
<networkx.classes.graph.Graph at 0x1317031f0>,
<networkx.classes.graph.Graph at 0x131702230>,
<networkx.classes.graph.Graph at 0x131702fb0>,
<networkx.classes.graph.Graph at 0x1317033a0>,
<networkx.classes.graph.Graph at 0x131703160>,
<networkx.classes.graph.Graph at 0x131703670>,
<networkx.classes.graph.Graph at 0x1317033d0>,
<networkx.classes.graph.Graph at 0x131702500>,
<networkx.classes.graph.Graph at 0x1317023e0>,
<networkx.classes.graph.Graph at 0x131700400>,
<networkx.classes.graph.Graph at 0x1317021a0>,
<networkx.classes.graph.Graph at 0x131702530>,
<networkx.classes.graph.Graph at 0x131701810>,
<networkx.classes.graph.Graph at 0x131701c60>,
<networkx.classes.graph.Graph at 0x131701150>,
<networkx.classes.graph.Graph at 0x1317022f0>,
<networkx.classes.graph.Graph at 0x131701c90>,
<networkx.classes.graph.Graph at 0x131701750>,
<networkx.classes.graph.Graph at 0x131701f00>,
<networkx.classes.graph.Graph at 0x131702680>,
<networkx.classes.graph.Graph at 0x131701c30>,
<networkx.classes.graph.Graph at 0x131701d50>,
<networkx.classes.graph.Graph at 0x131700610>,
<networkx.classes.graph.Graph at 0x1317018d0>,
<networkx.classes.graph.Graph at 0x131701cc0>,
<networkx.classes.graph.Graph at 0x131701d80>,
<networkx.classes.graph.Graph at 0x131701900>,
<networkx.classes.graph.Graph at 0x131701d20>,
<networkx.classes.graph.Graph at 0x1317012a0>,
<networkx.classes.graph.Graph at 0x131703a30>,
<networkx.classes.graph.Graph at 0x131703e50>,
<networkx.classes.graph.Graph at 0x131703be0>,
<networkx.classes.graph.Graph at 0x1317010c0>,
<networkx.classes.graph.Graph at 0x131701cf0>,
<networkx.classes.graph.Graph at 0x1317035e0>,
<networkx.classes.graph.Graph at 0x131700220>,
<networkx.classes.graph.Graph at 0x131701de0>,
<networkx.classes.graph.Graph at 0x131703c40>,
<networkx.classes.graph.Graph at 0x131703eb0>,
<networkx.classes.graph.Graph at 0x131703f70>,
<networkx.classes.graph.Graph at 0x131701570>,
<networkx.classes.graph.Graph at 0x131701120>,
<networkx.classes.graph.Graph at 0x131703f10>,
<networkx.classes.graph.Graph at 0x131703700>,
<networkx.classes.graph.Graph at 0x131703fd0>,
<networkx.classes.graph.Graph at 0x131703ca0>,
<networkx.classes.graph.Graph at 0x131703d60>,
<networkx.classes.graph.Graph at 0x131703f40>,
<networkx.classes.graph.Graph at 0x131703d30>,
<networkx.classes.graph.Graph at 0x131703fa0>,
<networkx.classes.graph.Graph at 0x1317030d0>,
<networkx.classes.graph.Graph at 0x131703880>,
<networkx.classes.graph.Graph at 0x131703460>,
<networkx.classes.graph.Graph at 0x131702740>,
<networkx.classes.graph.Graph at 0x131703c70>,
<networkx.classes.graph.Graph at 0x131700310>,
<networkx.classes.graph.Graph at 0x1317014e0>,
<networkx.classes.graph.Graph at 0x1317037f0>,
<networkx.classes.graph.Graph at 0x131702200>,
<networkx.classes.graph.Graph at 0x131702bc0>,
<networkx.classes.graph.Graph at 0x131702bf0>,
<networkx.classes.graph.Graph at 0x131702dd0>,
<networkx.classes.graph.Graph at 0x131702e00>,
<networkx.classes.graph.Graph at 0x131702d10>,
<networkx.classes.graph.Graph at 0x131701720>,
<networkx.classes.graph.Graph at 0x1317016c0>,
<networkx.classes.graph.Graph at 0x1317016f0>,
<networkx.classes.graph.Graph at 0x131701690>,
<networkx.classes.graph.Graph at 0x131701660>,
<networkx.classes.graph.Graph at 0x131701630>,
<networkx.classes.graph.Graph at 0x131701c00>,
<networkx.classes.graph.Graph at 0x131701bd0>,
<networkx.classes.graph.Graph at 0x131700850>,
<networkx.classes.graph.Graph at 0x131700820>,
<networkx.classes.graph.Graph at 0x131700b80>,
<networkx.classes.graph.Graph at 0x1317029b0>,
<networkx.classes.graph.Graph at 0x131702d70>,
<networkx.classes.graph.Graph at 0x131702d40>,
<networkx.classes.graph.Graph at 0x131702b90>,
<networkx.classes.graph.Graph at 0x131702b60>,
<networkx.classes.graph.Graph at 0x1317023b0>,
<networkx.classes.graph.Graph at 0x131702380>,
<networkx.classes.graph.Graph at 0x131700460>,
<networkx.classes.graph.Graph at 0x131701270>,
<networkx.classes.graph.Graph at 0x131701db0>,
<networkx.classes.graph.Graph at 0x1317032b0>,
<networkx.classes.graph.Graph at 0x131703bb0>,
<networkx.classes.graph.Graph at 0x131701180>,
<networkx.classes.graph.Graph at 0x131703070>,
<networkx.classes.graph.Graph at 0x131700a00>,
<networkx.classes.graph.Graph at 0x131703e20>,
<networkx.classes.graph.Graph at 0x131700a60>,
<networkx.classes.graph.Graph at 0x1317028c0>,
<networkx.classes.graph.Graph at 0x131702650>,
<networkx.classes.graph.Graph at 0x1317024a0>,
<networkx.classes.graph.Graph at 0x130c780a0>,
<networkx.classes.graph.Graph at 0x1312d3460>,
<networkx.classes.graph.Graph at 0x1312d2e30>,
<networkx.classes.graph.Graph at 0x1312d28c0>,
<networkx.classes.graph.Graph at 0x1312d29b0>,
<networkx.classes.graph.Graph at 0x1312d07f0>,
<networkx.classes.graph.Graph at 0x1312d0af0>,
<networkx.classes.graph.Graph at 0x1312d0b20>,
<networkx.classes.graph.Graph at 0x1312d3490>,
<networkx.classes.graph.Graph at 0x1312d2a70>,
<networkx.classes.graph.Graph at 0x1312d2a40>,
<networkx.classes.graph.Graph at 0x1312d34c0>,
<networkx.classes.graph.Graph at 0x1312d2cb0>,
<networkx.classes.graph.Graph at 0x1312d2b00>,
<networkx.classes.graph.Graph at 0x1312d2bf0>,
<networkx.classes.graph.Graph at 0x1312d2c20>,
<networkx.classes.graph.Graph at 0x1312d3bb0>,
<networkx.classes.graph.Graph at 0x1312d3be0>,
<networkx.classes.graph.Graph at 0x1312d3550>,
<networkx.classes.graph.Graph at 0x1312d3ac0>,
<networkx.classes.graph.Graph at 0x1312d3af0>,
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<networkx.classes.graph.Graph at 0x1312d2980>,
<networkx.classes.graph.Graph at 0x1312d3a00>,
<networkx.classes.graph.Graph at 0x1312d3580>,
<networkx.classes.graph.Graph at 0x1312d2f20>,
<networkx.classes.graph.Graph at 0x1312d35b0>,
<networkx.classes.graph.Graph at 0x1312d3b50>,
<networkx.classes.graph.Graph at 0x1312d3b80>,
<networkx.classes.graph.Graph at 0x1312d3a60>,
<networkx.classes.graph.Graph at 0x1312d3a90>,
<networkx.classes.graph.Graph at 0x1312d3a30>,
<networkx.classes.graph.Graph at 0x1312d2d10>,
<networkx.classes.graph.Graph at 0x1312d3730>,
<networkx.classes.graph.Graph at 0x1312d3880>,
<networkx.classes.graph.Graph at 0x1312d37c0>,
<networkx.classes.graph.Graph at 0x1312d39a0>,
<networkx.classes.graph.Graph at 0x1312d3910>,
<networkx.classes.graph.Graph at 0x1312d36d0>,
<networkx.classes.graph.Graph at 0x1312d37f0>,
<networkx.classes.graph.Graph at 0x1312d3760>]len(graphs[0])4158Camino más corto de la red
print(nx.shortest_path(graphs[0], '22504', '23991'))#camino entre a y b.
print(len(nx.shortest_path(graphs[0], '22504', '23991')))#distancia en nodos (incluye ppio y fin)
print(nx.shortest_path_length(graphs[0], '22504', '23991'))#distancia en enlaces (#nodos-1)['22504', '6512', '18757', '2710', '3677', '23991']
6
5# help(nx.shortest_path)Plot redes con nodos de distinto tamaño
g = graphs[1]# ejemplo
pos=pos = nx.spring_layout(g) # Layout para la red (coordenadas de los nodos y enlaces)nx.draw(g,pos)
#Codigo para el tamaño del nodo
# g = graphs[0]# ejemplo
# pos=pos = nx.spring_layout(g) # Layout para la red (coordenadas de los nodos y enlaces)
degree_g = dict(nx.degree(g))
clustering_g=dict(nx.clustering(g))
nx.draw(g,pos, nodelist=degree_g.keys(), node_size=[v*100 for v in degree_g.values()])
plt.show()
nx.draw(g, pos,nodelist=clustering_g.keys(), node_size=[(v+.01) *1000 for v in clustering_g.values()])
plt.show()
# nx.draw(g,pos, nodelist=list(degree_g.keys()), node_size=[v**(1/2) for v in degree_g.values()])
# plt.show()
Grado de nodo vs coeficiente de clustering
degree_g = dict(nx.degree(graphs[0]))
clustering_g=dict(nx.clustering(graphs[0]))
x = degree_g.values()
y = clustering_g.values()
# colors = np.random.rand(N)
# area = (30 * np.random.rand(N))**2 # 0 to 15 point radii
plt.scatter(x, y, alpha=0.5)
plt.show()
MUNDO PEQUEÑO
Seis grados de separación, número de Erdos, número de Bacon !!
Tarea 2.4
Encuentre el “número” del autor ‘22504’ del grafo author_graph, si no hay conexión entre los nodos, asignele el número ‘-1’. También trace un histograma del “número” autor ‘22504’.
Encuentre la longitud de ruta más corta promedio en el primer componente, es decir, graphs[0]
SUGERENCIA: nx.shortest_path_length
Interprete sus resultados
d = {}
for node in authors_graph.nodes():#para cada nodo
try:
#calcula la longitud del camino mas corto entre node y `22504`
d[node] = nx.shortest_path_length(graphs[0], '22504', node) #21012;22504
except:
#COMPLETAR FUNCION: si arroja error (no hay camino) asinga un -1.
next# dplt.hist(list(d.values()))
plt.show()
# Calcule el promedio de la longitud de los caminos más cortos de todo el grafo aca (filtren out los -1):
np.mean(list(d.values()))5.61014911014911Cambiemos de tema y hablemos de Juego de tronos (Network of Thrones)
¿Es sorprendente verdad? ¿Cuál es la relación entre un programa de televisión / novela de fatansia y la ciencia de redes o Python?
Si no has oído hablar de Game of Thrones, entonces debes ser muy bueno para aislarte. Game of Thrones es la popular serie de televisión de HBO basada en la (también) popular serie de libros “A Song of Ice and Fire” de George R.R. Martin.
Ahora, analizaremos la red de co-ocurrencia de los personajes en los libros de Juego de Tronos. Aquí, se considera que dos personajes coexisten si sus nombres aparecen en hasta a 15 palabras de distancia entre sí en los libros.
# !conda install --yes --prefix {sys.prefix} community #
# # !conda install --yes --prefix {sys.prefix} python-louvain #
# !pip install python-louvain
Collecting python-louvain
Downloading python-louvain-0.16.tar.gz (204 kB)
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Preparing metadata (setup.py) ... done
Requirement already satisfied: networkx in /Users/crcandia/anaconda3/envs/candialab2/lib/python3.10/site-packages (from python-louvain) (3.4.2)
Requirement already satisfied: numpy in /Users/crcandia/anaconda3/envs/candialab2/lib/python3.10/site-packages (from python-louvain) (1.26.4)
Building wheels for collected packages: python-louvain
Building wheel for python-louvain (setup.py) ... done
Created wheel for python-louvain: filename=python_louvain-0.16-py3-none-any.whl size=9389 sha256=cdc525604c7339e87ad69e9d240f4594802c17d0318889e12123746dfdc8a436
Stored in directory: /Users/crcandia/Library/Caches/pip/wheels/d0/b0/d7/6dd26c3817810fa379088eaeb755a01d9a2a411c37632079d1
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Installing collected packages: python-louvain
Successfully installed python-louvain-0.16
import pandas as pd
import networkx as nx
import matplotlib.pyplot as plt
from matplotlib import cm
# import community
from community import community_louvain
import numpy as np
%matplotlib inlineCargamos los datasets
book1 = pd.read_csv('data/asoiaf-book1-edges.csv')
book2 = pd.read_csv('data/asoiaf-book2-edges.csv')
book3 = pd.read_csv('data/asoiaf-book3-edges.csv')
book4 = pd.read_csv('data/asoiaf-book4-edges.csv')
book5 = pd.read_csv('data/asoiaf-book5-edges.csv')El DataFrame resultante book1 tiene 5 columnas: origen, destino, tipo, peso y libro. El origen y el destino son los dos nodos que están vinculados por un enlace. Una red puede tener enlaces dirigidos o no dirigidos y en esta red todos los enlaces no están dirigidos. El atributo de peso de cada enlace nos dice el número de interacciones que los personajes han tenido sobre el libro, y la columna del libro nos dice el número del libro.
# book1.head()
book1.sort_values('weight',ascending=False).head(10)| Source | Target | Type | weight | book | |
|---|---|---|---|---|---|
| 329 | Eddard-Stark | Robert-Baratheon | Undirected | 291 | 1 |
| 134 | Bran-Stark | Robb-Stark | Undirected | 112 | 1 |
| 62 | Arya-Stark | Sansa-Stark | Undirected | 104 | 1 |
| 249 | Daenerys-Targaryen | Drogo | Undirected | 101 | 1 |
| 479 | Joffrey-Baratheon | Sansa-Stark | Undirected | 87 | 1 |
| 504 | Jon-Snow | Samwell-Tarly | Undirected | 81 | 1 |
| 454 | Jeor-Mormont | Jon-Snow | Undirected | 81 | 1 |
| 320 | Eddard-Stark | Petyr-Baelish | Undirected | 81 | 1 |
| 257 | Daenerys-Targaryen | Jorah-Mormont | Undirected | 75 | 1 |
| 225 | Cersei-Lannister | Robert-Baratheon | Undirected | 72 | 1 |
Una vez que tenemos los datos cargados como un DataFrame de pandas, es hora de crear una red. Creamos un grafo para cada libro. Es posible crear un MultiGraph en lugar de 5 grafos, pero es más fácil jugar con diferentes grafos.
G_book1 = nx.Graph()
G_book2 = nx.Graph()
G_book3 = nx.Graph()
G_book4 = nx.Graph()
G_book5 = nx.Graph()Completemos el grafo con los enlaces del DataFrame de pandas.
for row in book1.iterrows(): #Itera las filas de DataFrame como pares (index, series).
G_book1.add_edge(row[1]['Source'], row[1]['Target'],weight=row[1]['weight'], book=row[1]['book'])for row in book2.iterrows():
G_book2.add_edge(row[1]['Source'], row[1]['Target'], weight=row[1]['weight'], book=row[1]['book'])
for row in book3.iterrows():
G_book3.add_edge(row[1]['Source'], row[1]['Target'], weight=row[1]['weight'], book=row[1]['book'])
for row in book4.iterrows():
G_book4.add_edge(row[1]['Source'], row[1]['Target'], weight=row[1]['weight'], book=row[1]['book'])
for row in book5.iterrows():
G_book5.add_edge(row[1]['Source'], row[1]['Target'], weight=row[1]['weight'], book=row[1]['book'])books = [G_book1, G_book2, G_book3, G_book4, G_book5]Echemos un vistazo a estos enlaces
list(G_book1.edges(data=True))[16]#data=True muestra los atributos del enlace: peso y libro en este caso('Jaime-Lannister', 'Loras-Tyrell', {'weight': 3, 'book': 1})list(G_book1.edges(data=True))[400]('Benjen-Stark', 'Theon-Greyjoy', {'weight': 4, 'book': 1})len(list(G_book1.edges()))684print(G_book1.number_of_edges())
print(G_book1.number_of_nodes())684
187Encontrar el nodo/personaje más importante en estas redes.
¿Es Jon Snow, Tyrion, Daenerys o alguien más? ¡Vamos a ver! Network Science nos ofrece muchas métricas diferentes para medir la importancia de un nodo en una red como vimos en la primera parte del taller.
Tenga en cuenta que no existe una forma “correcta” de calcular el nodo más importante de una red, cada métrica tiene un significado diferente.
Primero, midamos la importancia de un nodo en una red observando la cantidad de vecinos que tiene, es decir, la cantidad de nodos a los que está conectado. Por ejemplo, una cuenta influyente en Twitter, donde la relación seguidor-seguidor forma la red, es una cuenta que tiene un alto número de seguidores. Esta medida de importancia se llama centralidad de grado.
Usando esta medida, extraigamos los diez personajes más importantes del primer libro (libro [0]) y el quinto libro (libro [4]).
deg_cen_book1 = nx.degree_centrality(books[0])deg_cen_book5 = nx.degree_centrality(books[4])sorted(deg_cen_book1.items(), key=lambda x:x[1], reverse=True)[0:10][('Eddard-Stark', 0.3548387096774194),
('Robert-Baratheon', 0.2688172043010753),
('Tyrion-Lannister', 0.24731182795698928),
('Catelyn-Stark', 0.23118279569892475),
('Jon-Snow', 0.19892473118279572),
('Robb-Stark', 0.18817204301075272),
('Sansa-Stark', 0.18817204301075272),
('Bran-Stark', 0.17204301075268819),
('Cersei-Lannister', 0.16129032258064518),
('Joffrey-Baratheon', 0.16129032258064518)]sorted(deg_cen_book5.items(), key=lambda x:x[1], reverse=True)[0:10]
#key recive la funcion para ordenar, lambda es la funcion "al vuelo" para ordenar[('Jon-Snow', 0.1962025316455696),
('Daenerys-Targaryen', 0.18354430379746836),
('Stannis-Baratheon', 0.14873417721518986),
('Tyrion-Lannister', 0.10443037974683544),
('Theon-Greyjoy', 0.10443037974683544),
('Cersei-Lannister', 0.08860759493670886),
('Barristan-Selmy', 0.07911392405063292),
('Hizdahr-zo-Loraq', 0.06962025316455696),
('Asha-Greyjoy', 0.056962025316455694),
('Melisandre', 0.05379746835443038)]# Plot a histogram of degree centrality
plt.hist(list(dict(nx.degree(G_book1)).values()))
plt.show()
d= {}
for i, j in dict(nx.degree(G_book1)).items():
if j in d:
d[j] += 1
else:
d[j] = 1
x = list((d.keys()))
y = list(d.values())
plt.scatter(x, y, alpha=0.9)
plt.show()
d = {}
for i, j in dict(nx.degree(G_book1)).items():
if j in d:
d[j] += 1
else:
d[j] = 1
x = np.log10(list((d.keys())))
y = np.log10(list(d.values()))
plt.scatter(x, y, alpha=0.9)
plt.show()
Tarea 2.5
Cree una nueva medida de centralidad, weighted_degree(Graph, weight) que toma Graph y el atributo de peso y devuelve un diccionario de grados ponderados. El grado ponderado se calcula sumando el peso de todos los enlaces de un nodo. Luego, encuentre los cinco personajes más importantes de acuerdo con esta medida.
Interprete sus resultados con respecto a lo obtenido usando degree (sin peso)
#Escriba su código acá (siempre pueden agregar más líneas de ser necesario):
weighted_degree= lindo código!
# print(list(nx.neighbors(G_book1,'Eddard-Stark')))# plt.hist(list(weighted_degree(G_book1, 'weight').values()))
# plt.show()d = {}
for i, j in dict(nx.degree(G_book1)).items():
# print(i,j)
if j in d:
d[j] += 1 #cuenta la cantidad de veces que el mismo grado "ocurre"
else:
d[j] = 1
x = np.log2(list((d.keys())))
y = np.log2(list(d.values()))
plt.scatter(x, y, alpha=0.9)
plt.show()
# sorted(weighted_degree(G_book1, 'weight').items(), key=lambda x:x[1], reverse=True)[0:5]Hagamos esto para la centralidad de Betweenness y verifiquemos si esto hace alguna diferencia
G = nx.barbell_graph(5, 1) #Recordemos: El nodo del centro es el de mayor betweennes
nx.draw(G)
# Primero verifique el grafo no ponderado, solo la estructura
sorted(nx.betweenness_centrality(G_book1).items(), key=lambda x:x[1], reverse=True)[0:10][('Eddard-Stark', 0.2696038913836117),
('Robert-Baratheon', 0.21403028397371796),
('Tyrion-Lannister', 0.1902124972697492),
('Jon-Snow', 0.17158135899829566),
('Catelyn-Stark', 0.1513952715347627),
('Daenerys-Targaryen', 0.08627015537511595),
('Robb-Stark', 0.07298399629664767),
('Drogo', 0.06481224290874964),
('Bran-Stark', 0.05579958811784442),
('Sansa-Stark', 0.03714483664326785)]# Consideremos el peso ahora
sorted(nx.betweenness_centrality(G_book1, weight='weight').items(), key=lambda x:x[1], reverse=True)[0:10][('Robert-Baratheon', 0.23341885664466297),
('Eddard-Stark', 0.18703429235687297),
('Tyrion-Lannister', 0.15311225972516293),
('Robb-Stark', 0.1024018949825402),
('Catelyn-Stark', 0.10169012330302643),
('Jon-Snow', 0.09027684366394043),
('Jaime-Lannister', 0.07745109164464009),
('Rodrik-Cassel', 0.07667992877670296),
('Drogo', 0.06894355184677767),
('Jorah-Mormont', 0.0627085149665795)]# por defecto, el atributo de peso en el pagerank es `weight`,
# por lo que usamos `peso = None` para encontrar los resultados no ponderados
sorted(nx.pagerank(G_book1, weight=None).items(), key=lambda x:x[1], reverse=True)[0:10][('Eddard-Stark', 0.04550213151218522),
('Tyrion-Lannister', 0.03301449867019286),
('Catelyn-Stark', 0.030189729639356143),
('Robert-Baratheon', 0.02982527420823678),
('Jon-Snow', 0.026840579868198603),
('Robb-Stark', 0.02155918676623074),
('Sansa-Stark', 0.019998050283349653),
('Bran-Stark', 0.019940543257472354),
('Jaime-Lannister', 0.017500555351025744),
('Cersei-Lannister', 0.017074603349338583)]sorted(nx.pagerank(G_book1, weight='weight').items(), key=lambda x:x[1], reverse=True)[0:10][('Eddard-Stark', 0.07236162026570049),
('Robert-Baratheon', 0.04849367196106829),
('Jon-Snow', 0.04770801150205558),
('Tyrion-Lannister', 0.04367631315626212),
('Catelyn-Stark', 0.034666613211363564),
('Bran-Stark', 0.02977004993932226),
('Robb-Stark', 0.029214217154195955),
('Daenerys-Targaryen', 0.027098612952214246),
('Sansa-Stark', 0.02694466267974004),
('Cersei-Lannister', 0.02162037092438613)]Tarea 2.6
¿Existe una correlación entre estas métricas para todos los libros?
Encuentre la correlación entre estas cuatro métricas de centralidad.
- Eigenvector Centrality (PageRank, no apto para redes no-dirigidas).
- betweenness_centrality
- weighted_degree
- centralidad de grado
Inteprete sus resultados en función de como se relacionan las métricas, ej. Alto Eigenvector y bajo Betweenness Centrality, etc.
cor = pd.DataFrame.from_records(
[nx.pagerank(G_book1, weight='weight'),
nx.betweenness_centrality(G_book1, weight='weight'),
# weighted_degree(G_book1, 'weight'),
nx.degree_centrality(G_book1)])cor.T.corr()| 0 | 1 | 2 | |
|---|---|---|---|
| 0 | 1.000000 | 0.870210 | 0.949258 |
| 1 | 0.870210 | 1.000000 | 0.871385 |
| 2 | 0.949258 | 0.871385 | 1.000000 |
Evolución de la importancia de los personajes sobre los libros
Según la centralidad de grado, el personaje más importante del primer libro es Eddard Stark, pero ni siquiera está en el top 10 del quinto libro. La importancia cambia en el transcurso de cinco libros (hay muchas muertes :B)
Veamos la evolución de la centralidad de grado de un par de personajes como Eddard Stark, Jon Snow, Tyrion, que aparecieron en el top 10 de centralidad de grado en el primer libro.
Creamos un DataFrame con columnas de personajes y los libros como índices donde cada entrada es la centralidad de grado del personaje en ese libro en particular y trazamos la evolución de la centralidad de grado para Eddard Stark, Jon Snow y Tyrion.
Podemos ver que la importancia de Eddard Stark en la red diminuye y con Jon Snow hay una caída en el cuarto libro pero un aumento repentino en el quinto libro.
evol = [nx.degree_centrality(book) for book in books]#para cada red calcula el grado de nodo
evol_df = pd.DataFrame.from_records(evol).fillna(0)#Crea un objeto DataFrame a partir de un ndarray estructurado, una secuencia de tuplas o diccionarios, o un DataFrame.
evol_df| Addam-Marbrand | Jaime-Lannister | Tywin-Lannister | Aegon-I-Targaryen | Daenerys-Targaryen | Eddard-Stark | Aemon-Targaryen-(Maester-Aemon) | Alliser-Thorne | Bowen-Marsh | Chett | ... | Yellow-Dick | Walda-Frey-(daughter-of-Merrett) | Roose-Ryswell | Scar | Shrouded-Lord | Theomore | William-Foxglove | Willow-Witch-eye | Thistle | Wulfe | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.010753 | 0.155914 | 0.118280 | 0.010753 | 0.112903 | 0.354839 | 0.037634 | 0.053763 | 0.026882 | 0.016129 | ... | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 1 | 0.000000 | 0.081395 | 0.065891 | 0.019380 | 0.069767 | 0.085271 | 0.023256 | 0.015504 | 0.000000 | 0.003876 | ... | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 2 | 0.026490 | 0.149007 | 0.096026 | 0.009934 | 0.079470 | 0.052980 | 0.059603 | 0.023179 | 0.033113 | 0.016556 | ... | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 3 | 0.014652 | 0.234432 | 0.058608 | 0.000000 | 0.000000 | 0.029304 | 0.025641 | 0.000000 | 0.000000 | 0.000000 | ... | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| 4 | 0.000000 | 0.031646 | 0.028481 | 0.003165 | 0.183544 | 0.044304 | 0.015823 | 0.003165 | 0.028481 | 0.000000 | ... | 0.009494 | 0.006329 | 0.003165 | 0.006329 | 0.003165 | 0.006329 | 0.003165 | 0.003165 | 0.003165 | 0.003165 |
5 rows × 796 columns
evol_df[['Eddard-Stark', 'Tyrion-Lannister', 'Jon-Snow','Daenerys-Targaryen']].plot()
#Nota: Los corchetes interiores son para la lista y los corchetes externos son operadores de indexación, es decir,
#debe usar corchetes dobles si seleccionas dos o más columnas. Con un nombre de columna, un solo par de corchetes
#devuelve una Series, mientras que los corchetes dobles devuelven un dataframe.
evol_df['Eddard-Stark']#, 'Tyrion-Lannister', 'Jon-Snow']0 0.354839
1 0.085271
2 0.052980
3 0.029304
4 0.044304
Name: Eddard-Stark, dtype: float64# Buscamos a los 3 personajes mas importantes de cada libro y creamos un arreglo con sus nombres.
set_of_char = set()
for i in range(5):
set_of_char |= set(list(evol_df.T[i].sort_values(ascending=False)[0:3].index))# `|=` hace una unión de conjuntos y reasigna (junta los 5 libros)
# print(set_of_char)
set_of_char
# VER: https://stackoverflow.com/questions/3929278/what-does-ior-do-in-python{'Brienne-of-Tarth',
'Cersei-Lannister',
'Daenerys-Targaryen',
'Eddard-Stark',
'Jaime-Lannister',
'Joffrey-Baratheon',
'Jon-Snow',
'Robert-Baratheon',
'Stannis-Baratheon',
'Tyrion-Lannister'}Ejercicio resuelto
Trace la evolución de la centralidad de grado ponderado de los personajes antes mencionados sobre los 5 libros y repita el mismo ejercicio para la centralidad de intermediación.
evol_df[list(set_of_char)].plot(figsize=(19,10))
plt.xlabel('Capitulo de libro')
plt.ylabel('Centralidad de grado')Text(0, 0.5, 'Centralidad de grado')
evol = [nx.betweenness_centrality(graph, weight='weight') for graph in [G_book1, G_book2, G_book3, G_book4, G_book5]]
evol_df = pd.DataFrame.from_records(evol).fillna(0)
set_of_char = set()
for i in range(5):
set_of_char |= set(list(evol_df.T[i].sort_values(ascending=False)[0:3].index))
evol_df[list(set_of_char)].plot(figsize=(19,10))
plt.ylabel('Betweenness Centrality')
plt.xlabel('Capitulo de libro')Text(0.5, 0, 'Capitulo de libro')
Entonces, ¿qué pasa con Stannis Baratheon?
nx.draw(nx.barbell_graph(5, 1), with_labels=True)
sorted(nx.degree_centrality(G_book5).items(), key=lambda x:x[1], reverse=True)[:5]#ordena por centralidad (con 0 ordena por el nombre)[('Jon-Snow', 0.1962025316455696),
('Daenerys-Targaryen', 0.18354430379746836),
('Stannis-Baratheon', 0.14873417721518986),
('Tyrion-Lannister', 0.10443037974683544),
('Theon-Greyjoy', 0.10443037974683544)]sorted(nx.betweenness_centrality(G_book5).items(), key=lambda x:x[1], reverse=True)[:5][('Stannis-Baratheon', 0.45283060689247934),
('Daenerys-Targaryen', 0.2959459062106149),
('Jon-Snow', 0.24484873673158666),
('Tyrion-Lannister', 0.20961613179551256),
('Robert-Baratheon', 0.17716906651536968)]Detección de comunidad en redes (Spoiler)
Se dice que una red tiene una estructura de comunidades si los nodos de la red se pueden agrupar “fácilmente” en conjuntos de nodos (potencialmente superpuestos) de modo que cada conjunto de nodos esté densamente conectado internamente (dentro de cada comunidad).
Usaremos el algoritmo de detección de la comunidad de Louvain para encontrar los módulos en nuestro grafo.
def draw_communities(G, partition, pos, n_labels=10):
"""Dibuja el grafo G con una coloración de los nodos de acuerdo a la partición de la comunidad proporcionada.
Parámetros:
- G: un objeto networkx.Graph
- partition: un diccionario mapeando nombres de nodos a etiquetas de comunidad
- pos: un diccionario mapeando nombres de nodos a posiciones en el espacio 2D
- n_labels: la cantidad de nodos con el grado más alto para los que se mostrarán las etiquetas
"""
# Creamos un mapa de colores
cmap = cm.get_cmap('viridis', max(partition.values()) + 1)
# Dibujamos los nodos, coloreándolos por su comunidad y ajustando su tamaño por su grado
nx.draw_networkx_nodes(G, pos, node_color=[cmap(partition[n]) for n in G.nodes()],
node_size=[G.degree(n) * 100 for n in G.nodes()])
# Dibujamos las aristas
nx.draw_networkx_edges(G, pos, alpha=0.2)
# Seleccionamos los n_labels nodos de mayor grado para etiquetar
degree_dict = dict(G.degree(G.nodes()))
highest_degree_nodes = sorted(degree_dict.items(), key=lambda item: item[1], reverse=True)[:n_labels]
labels = {node[0]:node[0] for node in highest_degree_nodes}
# Ajustamos las posiciones de las etiquetas
label_pos = {node: (pos[node][0], pos[node][1] + 0.04) for node in labels} # Movemos las etiquetas un poco hacia arriba
nx.draw_networkx_labels(G, label_pos, labels=labels)
# nx.draw_networkx_labels(G, pos, labels=labels)
plt.show()
def community_layout(g, partition):
"""
Calcule el diseño para una red modular. Esto mejora la separación de las comunidades.
"""
# Crear un nuevo gráfico que esté ponderado en función de la comunidad (partición)
inter_community_edges = [(u, v) for u, v in g.edges() if partition[u] != partition[v]]
# Ponderamos las aristas inter-comunidad a 0.01 y las intra-comunidad a 1
weights = {edge: 0.01 if edge in inter_community_edges else 1 for edge in g.edges()}
nx.set_edge_attributes(g, weights, "weight")
# Recalculamos el diseño con los pesos ajustados
pos_communities = nx.spring_layout(g, k=0.2, weight='weight', iterations=50)
return pos_communities
plt.figure(figsize=(15, 10))
# Uso de la función:
partition = community_louvain.best_partition(G_book1)
pos = community_layout(G_book1, partition)
draw_communities(G_book1, partition, pos, n_labels=15) # Se etiquetan los 15 nodos de mayor grado/var/folders/q8/5g879lls62l1z8smyjkh4h700000gn/T/ipykernel_63607/3088408559.py:12: MatplotlibDeprecationWarning: The get_cmap function was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use ``matplotlib.colormaps[name]`` or ``matplotlib.colormaps.get_cmap(obj)`` instead.
cmap = cm.get_cmap('viridis', max(partition.values()) + 1)
partition{'Addam-Marbrand': 7,
'Jaime-Lannister': 7,
'Tywin-Lannister': 7,
'Aegon-I-Targaryen': 1,
'Daenerys-Targaryen': 1,
'Eddard-Stark': 2,
'Aemon-Targaryen-(Maester-Aemon)': 3,
'Alliser-Thorne': 3,
'Bowen-Marsh': 3,
'Chett': 3,
'Clydas': 3,
'Jeor-Mormont': 3,
'Jon-Snow': 3,
'Samwell-Tarly': 3,
'Aerys-II-Targaryen': 2,
'Brandon-Stark': 2,
'Gerold-Hightower': 2,
'Jon-Arryn': 2,
'Robert-Baratheon': 2,
'Aggo': 1,
'Drogo': 1,
'Jhogo': 1,
'Jorah-Mormont': 1,
'Quaro': 1,
'Rakharo': 1,
'Albett': 3,
'Halder': 3,
'Rast': 3,
'Grenn': 3,
'Pypar': 3,
'Tyrion-Lannister': 7,
'Alyn': 2,
'Harwin': 2,
'Jory-Cassel': 2,
'Tomard': 2,
'Arthur-Dayne': 2,
'Arya-Stark': 4,
'Benjen-Stark': 3,
'Bran-Stark': 5,
'Catelyn-Stark': 7,
'Cersei-Lannister': 2,
'Desmond': 4,
'Ilyn-Payne': 4,
'Jeyne-Poole': 4,
'Joffrey-Baratheon': 4,
'Meryn-Trant': 4,
'Mordane': 4,
'Mycah': 4,
'Myrcella-Baratheon': 4,
'Petyr-Baelish': 2,
'Rickon-Stark': 5,
'Robb-Stark': 5,
'Rodrik-Cassel': 7,
'Sandor-Clegane': 4,
'Sansa-Stark': 4,
'Syrio-Forel': 4,
'Tommen-Baratheon': 4,
'Vayon-Poole': 2,
'Yoren': 3,
'Arys-Oakheart': 2,
'Balon-Greyjoy': 2,
'Balon-Swann': 4,
'Renly-Baratheon': 2,
'Barristan-Selmy': 2,
'Boros-Blount': 4,
'Pycelle': 2,
'Varys': 2,
'Jaremy-Rykker': 3,
'Luwin': 5,
'Mance-Rayder': 3,
'Theon-Greyjoy': 5,
'Waymar-Royce': 6,
'Beric-Dondarrion': 4,
'Gregor-Clegane': 4,
'Loras-Tyrell': 4,
'Thoros-of-Myr': 4,
'Hali': 5,
'Hallis-Mollen': 5,
'Hodor': 5,
'Hullen': 5,
'Joseth': 5,
'Nan': 5,
'Osha': 5,
'Rickard-Karstark': 5,
'Rickard-Stark': 5,
'Stiv': 5,
'Lyanna-Stark': 2,
'Bronn': 7,
'Chiggen': 7,
'Marillion': 7,
'Shae': 7,
'Shagga': 7,
'Vardis-Egen': 7,
'Willis-Wode': 7,
'Brynden-Tully': 7,
'Edmure-Tully': 7,
'Hoster-Tully': 7,
'Lysa-Arryn': 7,
'Nestor-Royce': 7,
'Walder-Frey': 7,
'Colemon': 7,
'Donnel-Waynwood': 7,
'Eon-Hunter': 7,
'Jon-Umber-(Greatjon)': 5,
'Masha-Heddle': 7,
'Moreo-Tumitis': 7,
'Mya-Stone': 7,
'Mychel-Redfort': 7,
'Robert-Arryn': 7,
'Stevron-Frey': 7,
'Tytos-Blackwood': 7,
'Wendel-Manderly': 7,
'Cayn': 2,
'Janos-Slynt': 2,
'Stannis-Baratheon': 2,
'Chella': 7,
'Clement-Piper': 7,
'Karyl-Vance': 7,
'Cohollo': 1,
'Haggo': 1,
'Qotho': 1,
'Conn': 7,
'Coratt': 7,
'Doreah': 1,
'Eroeh': 1,
'Illyrio-Mopatis': 1,
'Irri': 1,
'Jhiqui': 1,
'Mirri-Maz-Duur': 1,
'Rhaegar-Targaryen': 2,
'Viserys-Targaryen': 1,
'Danwell-Frey': 0,
'Hosteen-Frey': 0,
'Jared-Frey': 0,
'Dareon': 3,
'Daryn-Hornwood': 2,
'Torrhen-Karstark': 2,
'Dolf': 7,
'Donal-Noye': 3,
'Jommo': 1,
'Ogo': 1,
'Rhaego': 1,
'Dywen': 3,
'Galbart-Glover': 5,
'Gendry': 2,
'High-Septon-(fat_one)': 4,
'Howland-Reed': 2,
'Jacks': 2,
'Joss': 2,
'Marq-Piper': 4,
'Porther': 2,
'Raymun-Darry': 2,
'Tobho-Mott': 2,
'Tregar': 2,
'Varly': 2,
'Wyl-(guard)': 2,
'Wylla': 2,
'Fogo': 1,
'Roose-Bolton': 5,
'Gared': 6,
'Will-(prologue)': 6,
'Oswell-Whent': 2,
'Todder': 3,
'Gunthor-son-of-Gurn': 7,
'Harys-Swyft': 7,
'Heward': 2,
'Hobb': 3,
'Hugh': 2,
'Jafer-Flowers': 3,
'Kevan-Lannister': 7,
'Matthar': 3,
'Othor': 3,
'Maege-Mormont': 5,
'Jonos-Bracken': 7,
'Jyck': 7,
'Morrec': 7,
'Kurleket': 7,
'Lancel-Lannister': 2,
'Leo-Lefford': 7,
'Mace-Tyrell': 4,
'Lyn-Corbray': 7,
'Paxter-Redwyne': 4,
'Maegor-I-Targaryen': 4,
'Mord': 7,
'Randyll-Tarly': 3,
'Timett': 7,
'Ulf-son-of-Umar': 7}# Creamos un diccionario de personajes y sus respectivas comunidades
d = {}
for character, par in partition.items():
if par in d:
d[par].append(character)
else:
d[par] = [character]
d{7: ['Addam-Marbrand',
'Jaime-Lannister',
'Tywin-Lannister',
'Tyrion-Lannister',
'Catelyn-Stark',
'Rodrik-Cassel',
'Bronn',
'Chiggen',
'Marillion',
'Shae',
'Shagga',
'Vardis-Egen',
'Willis-Wode',
'Brynden-Tully',
'Edmure-Tully',
'Hoster-Tully',
'Lysa-Arryn',
'Nestor-Royce',
'Walder-Frey',
'Colemon',
'Donnel-Waynwood',
'Eon-Hunter',
'Masha-Heddle',
'Moreo-Tumitis',
'Mya-Stone',
'Mychel-Redfort',
'Robert-Arryn',
'Stevron-Frey',
'Tytos-Blackwood',
'Wendel-Manderly',
'Chella',
'Clement-Piper',
'Karyl-Vance',
'Conn',
'Coratt',
'Dolf',
'Gunthor-son-of-Gurn',
'Harys-Swyft',
'Kevan-Lannister',
'Jonos-Bracken',
'Jyck',
'Morrec',
'Kurleket',
'Leo-Lefford',
'Lyn-Corbray',
'Mord',
'Timett',
'Ulf-son-of-Umar'],
1: ['Aegon-I-Targaryen',
'Daenerys-Targaryen',
'Aggo',
'Drogo',
'Jhogo',
'Jorah-Mormont',
'Quaro',
'Rakharo',
'Cohollo',
'Haggo',
'Qotho',
'Doreah',
'Eroeh',
'Illyrio-Mopatis',
'Irri',
'Jhiqui',
'Mirri-Maz-Duur',
'Viserys-Targaryen',
'Jommo',
'Ogo',
'Rhaego',
'Fogo'],
2: ['Eddard-Stark',
'Aerys-II-Targaryen',
'Brandon-Stark',
'Gerold-Hightower',
'Jon-Arryn',
'Robert-Baratheon',
'Alyn',
'Harwin',
'Jory-Cassel',
'Tomard',
'Arthur-Dayne',
'Cersei-Lannister',
'Petyr-Baelish',
'Vayon-Poole',
'Arys-Oakheart',
'Balon-Greyjoy',
'Renly-Baratheon',
'Barristan-Selmy',
'Pycelle',
'Varys',
'Lyanna-Stark',
'Cayn',
'Janos-Slynt',
'Stannis-Baratheon',
'Rhaegar-Targaryen',
'Daryn-Hornwood',
'Torrhen-Karstark',
'Gendry',
'Howland-Reed',
'Jacks',
'Joss',
'Porther',
'Raymun-Darry',
'Tobho-Mott',
'Tregar',
'Varly',
'Wyl-(guard)',
'Wylla',
'Oswell-Whent',
'Heward',
'Hugh',
'Lancel-Lannister'],
3: ['Aemon-Targaryen-(Maester-Aemon)',
'Alliser-Thorne',
'Bowen-Marsh',
'Chett',
'Clydas',
'Jeor-Mormont',
'Jon-Snow',
'Samwell-Tarly',
'Albett',
'Halder',
'Rast',
'Grenn',
'Pypar',
'Benjen-Stark',
'Yoren',
'Jaremy-Rykker',
'Mance-Rayder',
'Dareon',
'Donal-Noye',
'Dywen',
'Todder',
'Hobb',
'Jafer-Flowers',
'Matthar',
'Othor',
'Randyll-Tarly'],
4: ['Arya-Stark',
'Desmond',
'Ilyn-Payne',
'Jeyne-Poole',
'Joffrey-Baratheon',
'Meryn-Trant',
'Mordane',
'Mycah',
'Myrcella-Baratheon',
'Sandor-Clegane',
'Sansa-Stark',
'Syrio-Forel',
'Tommen-Baratheon',
'Balon-Swann',
'Boros-Blount',
'Beric-Dondarrion',
'Gregor-Clegane',
'Loras-Tyrell',
'Thoros-of-Myr',
'High-Septon-(fat_one)',
'Marq-Piper',
'Mace-Tyrell',
'Paxter-Redwyne',
'Maegor-I-Targaryen'],
5: ['Bran-Stark',
'Rickon-Stark',
'Robb-Stark',
'Luwin',
'Theon-Greyjoy',
'Hali',
'Hallis-Mollen',
'Hodor',
'Hullen',
'Joseth',
'Nan',
'Osha',
'Rickard-Karstark',
'Rickard-Stark',
'Stiv',
'Jon-Umber-(Greatjon)',
'Galbart-Glover',
'Roose-Bolton',
'Maege-Mormont'],
6: ['Waymar-Royce', 'Gared', 'Will-(prologue)'],
0: ['Danwell-Frey', 'Hosteen-Frey', 'Jared-Frey']}# la densidad de un grafo es una propiedad que determina la proporción de aristas que posee.
nx.density(G_book1)0.03933068828704502nx.density(nx.subgraph(G_book1, d[4]))0.19927536231884058nx.density(nx.subgraph(G_book1, d[6]))1.0nx.density(nx.subgraph(G_book1, d[1]))#/nx.density(G_book1)0.2813852813852814Tarea 2.7
Encuentre el nodo más importante en cada partición/comunidad según su centralidad de grado (HINT: use la centralidad normalizada, por qué?).
from sinfo import sinfo
sinfo()-----
community 0.15
matplotlib 3.5.0
networkx 2.6.3
numpy 1.22.4
pandas 1.3.4
sinfo 0.3.1
-----
IPython 7.29.0
jupyter_client 7.1.0
jupyter_core 4.9.1
jupyterlab 3.2.4
notebook 6.4.6
-----
Python 3.10.0 | packaged by conda-forge | (default, Nov 20 2021, 02:25:38) [Clang 11.1.0 ]
macOS-10.15.7-x86_64-i386-64bit
16 logical CPU cores, i386
-----
Session information updated at 2023-06-16 01:06