In [170]:
nx.draw_networkx(G)
Python offers the library NetworkX for manipulating graphs. You can learn more here:
https://networkx.github.io/documentation/stable/tutorial.html
import networkx as nx
import matplotlib.pyplot as plt
%matplotlib inline
G = nx.Graph()
Add nodes to the graph
G.add_node(1)
G.add_nodes_from([2,3])
G.add_node('Alice')
G.add_node('Bob')
print(G.nodes())
[1, 2, 3, 'Alice', 'Bob', 'Charlie']
Draw the graph
nx.draw_networkx(G)
Add edges to the graph
G.add_edge(1,2)
G.add_edges_from([(1,3),('Alice','Bob')])
e = (1,'Alice')
G.add_edge(*e)
nx.draw_networkx(G)
Adding an edge with a new node will create the node Charlie in the graph
G.add_edge('Alice','Charlie')
print(G.edges())
print(G.nodes())
nx.draw_networkx(G)
[(1, 2), (1, 3), (1, 'Alice'), ('Alice', 'Bob'), ('Alice', 'Charlie')]
[1, 2, 3, 'Alice', 'Bob', 'Charlie']
Creating a graph from edges
This is the most common way to create a graph
G2 = nx.Graph()
G2.add_edges_from([(1,2),(1,3),('Alice','Bob'),(1,'Alice')])
print(G2.nodes())
print(G2.edges())
nx.draw_networkx(G2)
[1, 2, 3, 'Alice', 'Bob']
[(1, 2), (1, 3), (1, 'Alice'), ('Alice', 'Bob')]
Reading a graph from a file with list of edges
https://networkx.org/documentation/stable/reference/readwrite/index.html
#Read a graph from a list of edges
G3 = nx.read_edgelist('graph_edges.txt')
print(G3.nodes())
print(G3.edges())
nx.draw_networkx(G3)
['1', '2', '3', 'Alice', 'Bob', 'Charlie']
[('1', '2'), ('1', '3'), ('1', 'Alice'), ('2', '3'), ('Alice', 'Bob'), ('Bob', 'Charlie')]
Removing edges and nodes
G2.remove_edge(1,3)
G2.remove_node(3)
G2.remove_node(1)
print(G2.nodes())
print(G2.edges())
nx.draw_networkx(G2)
[2, 'Alice', 'Bob']
[('Alice', 'Bob')]
A graph can be thought of as a dictionary with nodes as the keys
Each node is a dictionary with the neighbors as the keys, and the edge properties as values
type(G)
networkx.classes.graph.Graph
for n in G: print(n,':',G[n])
1 : {2: {}, 3: {}, 'Alice': {}}
2 : {1: {}}
3 : {1: {}}
Alice : {'Bob': {}, 1: {}, 'Charlie': {}}
Bob : {'Alice': {}}
Charlie : {'Alice': {}}
G[1]
AtlasView({2: {}, 3: {}, 'Alice': {}})
print(G.nodes)
print(G.edges)
[1, 2, 3, 'Alice', 'Bob', 'Charlie']
[(1, 2), (1, 3), (1, 'Alice'), ('Alice', 'Bob'), ('Alice', 'Charlie')]
#For every node we can obtain the properties of the node (in this example this is empty)
print(G.nodes['Alice'])
#For every edge we can obtain the properties of the edge (in this example this is empty)
print(G.edges[(1,2)])
{}
{}
You can assign attributes and values to the nodes of the graph
G3.nodes['Alice']['gender'] = 'female'
G3.nodes['Bob']['gender'] = 'male'
G3.nodes['Charlie']['gender'] = 'male'
G3.nodes['1']['value'] = 1
G3.nodes['2']['value'] = -1
G3.nodes['3']['value'] = 0
for n in G3.nodes():
print(n,':',G3.nodes[n])
1 : {'value': 1}
2 : {'value': -1}
3 : {'value': 0}
Alice : {'gender': 'female'}
Bob : {'gender': 'male'}
Charlie : {'gender': 'male'}
for n in G3.nodes():
print(n,':',G3[n])
1 : {'2': {}, '3': {}, 'Alice': {}}
2 : {'1': {}, '3': {}}
3 : {'1': {}, '2': {}}
Alice : {'Bob': {}, '1': {}}
Bob : {'Alice': {}, 'Charlie': {}}
Charlie : {'Bob': {}}
G3.nodes['Alice']['value'] = 1
G3.nodes['Bob']['value'] = -1
G3.nodes['Charlie']['value'] = 1
for n in G3.nodes():
print(n+ ":" + str(G3.nodes[n]['value']))
for n in G3.nodes():
print(n,':',G3.nodes[n])
1:1
2:-1
3:0
Alice:1
Bob:-1
Charlie:1
1 : {'value': 1}
2 : {'value': -1}
3 : {'value': 0}
Alice : {'gender': 'female', 'value': 1}
Bob : {'gender': 'male', 'value': -1}
Charlie : {'gender': 'male', 'value': 1}
You can also add attributes and values to the edges in the graph
G3['Alice']['Bob']['label'] = 'strong'
print(G3['Bob']['Alice'])
print(G3['Alice'])
print(G3['Bob'])
{'label': 'strong'}
{'Bob': {'label': 'strong'}, '1': {}}
{'Alice': {'label': 'strong'}, 'Charlie': {}}
for e in G3.edges:
print(e,':',G3.edges[e])
('1', '2') : {}
('1', '3') : {}
('1', 'Alice') : {}
('2', '3') : {}
('Alice', 'Bob') : {'label': 'strong'}
('Bob', 'Charlie') : {}
print(G3.edges[('Alice','Bob')])
print(G3.edges[('Alice','Bob')]['label'])
{'label': 'strong'}
strong
# The method get_edge_attributes returns a dictionary with keys the edges
#and values the values of the attribute we requested (for the edges that have it)
nx.get_edge_attributes(G3,'label')
{('Alice', 'Bob'): 'strong'}
pos=nx.spring_layout(G3)
nx.draw_networkx(G3, pos)
my_labels = nx.get_edge_attributes(G3,'label')
print(my_labels)
nx.draw_networkx_edge_labels(G3, pos, edge_labels=my_labels)
#plt.show()
{('Alice', 'Bob'): 'strong'}
{('Alice', 'Bob'): Text(-0.14520541461629888, 0.3747206779133405, 'strong')}
A special attribute of a an edge is the "weight". When adding weighted edges, you enter triples consisting of the two edge endpoints and the weight of the edge. This weight is stored in an attribute "weight" by default.
G4 = nx.Graph()
G4.add_weighted_edges_from([(1,2,0.5),(2,3,0.1),(3,4,0.7)])
for (a,b) in G4.edges():
print (a,b,':',G4[a][b])
for (a,b,w) in G4.edges(data =True): #data=True returns weight as well
print (str(a)+" "+ str(b) + " " + str(w['weight']))
for n in G4:
print(n,':',G4[n])
for e in G4.edges:
print(e,':',G4.edges[e])
1 2 : {'weight': 0.5}
2 3 : {'weight': 0.1}
3 4 : {'weight': 0.7}
1 2 0.5
2 3 0.1
3 4 0.7
1 : {2: {'weight': 0.5}}
2 : {1: {'weight': 0.5}, 3: {'weight': 0.1}}
3 : {2: {'weight': 0.1}, 4: {'weight': 0.7}}
4 : {3: {'weight': 0.7}}
(1, 2) : {'weight': 0.5}
(2, 3) : {'weight': 0.1}
(3, 4) : {'weight': 0.7}
weights = nx.get_edge_attributes(G4,'weight')
print(weights)
{(1, 2): 0.5, (2, 3): 0.1, (3, 4): 0.7}
pos=nx.spring_layout(G4)
nx.draw_networkx(G4, pos)
nx.draw_networkx_edge_labels(G4, pos, edge_labels=weights)
{(1, 2): Text(-0.1568874759358344, -0.7526589763309715, '0.5'),
(2, 3): Text(0.0028747654380344365, 0.013983266304825603, '0.1'),
(3, 4): Text(0.1568874759358344, 0.7526589763309715, '0.7')}
print(weights)
{(1, 2): 0.5, (2, 3): 0.1, (3, 4): 0.7}
DG1=nx.DiGraph()
DG1.add_edges_from([(1,2), (2,1), (3,1), (1,4), (2,4), (4,3)])
pos=nx.spring_layout(DG1)
nx.draw_networkx(DG1,pos)
DG2=nx.DiGraph()
DG2.add_weighted_edges_from([(1,2,0.5), (3,1,0.75), (1,4,0.1)])
print(DG2.edges())
for n in DG2:
print(n, DG2[n])
pos=nx.spring_layout(DG2)
nx.draw_networkx(DG2,pos)
weights = nx.get_edge_attributes(DG2,'weight')
nx.draw_networkx_edge_labels(DG2, pos, edge_labels=weights)
[(1, 2), (1, 4), (3, 1)]
1 {2: {'weight': 0.5}, 4: {'weight': 0.1}}
2 {}
3 {1: {'weight': 0.75}}
4 {}
{(1, 2): Text(-0.41071094043126943, -0.03983116068898572, '0.5'),
(1, 4): Text(0.2914365096122925, 0.435178557883492, '0.1'),
(3, 1): Text(0.2596752269996655, -0.5249902814275222, '0.75')}
You can transform a directed graph into an undirected on using to_undirected()
UG1 = DG1.to_undirected()
print(UG1.edges())
for n in UG1:
print(n, UG1[n])
nx.draw_networkx(UG1)
[(1, 2), (1, 4), (1, 3), (2, 4), (3, 4)]
1 {2: {}, 4: {}, 3: {}}
2 {1: {}, 4: {}}
3 {1: {}, 4: {}}
4 {1: {}, 2: {}, 3: {}}
UG2 = DG2.to_undirected()
print(UG2.edges())
for n in UG2:
print(n, UG2[n])
nx.draw_networkx(UG2)
[(1, 2), (1, 4), (1, 3)]
1 {2: {'weight': 0.5}, 4: {'weight': 0.1}, 3: {'weight': 0.75}}
2 {1: {'weight': 0.5}}
3 {1: {'weight': 0.75}}
4 {1: {'weight': 0.1}}
Some common graph functions and algorithms are already implemented in networkx library.
Neighbors, degrees and adjancency matrix
nx.draw_networkx(G)
A = nx.adjacency_matrix(G)
print(A)
#the adjacency matrix is stored as a sparse matrix
print(type(A))
(0, 1) 1 (0, 2) 1 (0, 3) 1 (1, 0) 1 (2, 0) 1 (3, 0) 1 (3, 4) 1 (3, 5) 1 (4, 3) 1 (5, 3) 1 <class 'scipy.sparse._csr.csr_matrix'>
print(G.neighbors(1)) # returns the neighbors of a node
print(list(G.neighbors(1)))
for x in G.neighbors(1):
print(x)
print('degree of 1:',G.degree(1)) # returns the degree of a node
<dict_keyiterator object at 0x000002035D61B950> [2, 3, 'Alice'] 2 3 Alice degree of 1: 3
print(G4[3])
print(G4.degree(3, weight='weight'))
print(G4.degree(3))
{2: {'weight': 0.1}, 4: {'weight': 0.7}}
0.7999999999999999
2
Neighbors and degrees for directed or weighted graphs
pos=nx.spring_layout(DG2)
nx.draw_networkx(DG2,pos)
weights = nx.get_edge_attributes(DG2,'weight')
nx.draw_networkx_edge_labels(DG2, pos, edge_labels=weights)
{(1, 2): Text(-0.5049838391744116, 0.07440616425642085, '0.5'),
(1, 4): Text(0.4089437064379942, -0.226453271208447, '0.1'),
(3, 1): Text(-0.08607245438759417, 0.17343192430832471, '0.75')}
print(list(DG2.successors(1)))
print(list(DG2.neighbors(1)))
print(list(DG2.predecessors(1)))
print(DG2.out_degree(1))
print(DG2.in_degree(1))
print(DG2.out_degree(1,weight='weight'))
[2, 4] [2, 4] [3] 2 1 0.6
G5 = nx.Graph()
G5.add_edges_from([(1,2),(1,3),(1,4),(2,3),('Alice','Bob'),('Bob','Charlie'),('Alice','Charlie')])
nx.draw_networkx(G5)
C = nx.connected_components(G5)
print(type(C))
for c in C:
print(c)
<class 'generator'>
{1, 2, 3, 4}
{'Charlie', 'Bob', 'Alice'}
Get the connected component subgraphs
connected_subgraphs = [nx.subgraph(G5,c) for c in nx.connected_components(G5)]
for GC in connected_subgraphs:
print('Connected Component')
print(GC.nodes())
print(GC.edges())
print(len(GC))
Connected Component
[1, 2, 3, 4]
[(1, 2), (1, 3), (1, 4), (2, 3)]
4
Connected Component
['Charlie', 'Bob', 'Alice']
[('Charlie', 'Bob'), ('Charlie', 'Alice'), ('Bob', 'Alice')]
3
Get the largest connected component
# Get the nodes
largest_cc = max(nx.connected_components(G5), key=len)
print(largest_cc)
{1, 2, 3, 4}
#Get the subgraph
largest_cc = max(nx.connected_components(G5), key=len)
print(largest_cc)
CC_max = nx.subgraph(G5,largest_cc)
nx.draw_networkx(CC_max)
{1, 2, 3, 4}
DG = nx.DiGraph()
DG.add_edges_from([(1,2),(2,3),(3,4),(4,2),(4,3),(4,5)])
nx.draw_networkx(DG)
largest_SCC = max(nx.strongly_connected_components(DG), key=len)
#print(largest_SCC)
SCC_max = nx.subgraph(DG,largest_SCC)
nx.draw_networkx(SCC_max)
G6 = nx.Graph()
G6.add_edges_from([(1,2),(1,3),(1,4),(2,3),(1,'Alice'),('Alice','Bob'),('Alice','Charlie'),('Bob','Charlie')])
nx.draw_networkx(G6)
sp = nx.shortest_path(G6,3,'Bob')
print(sp)
print(len(sp)-1)
print(nx.shortest_path_length(G6,3,'Bob'))
[3, 1, 'Alice', 'Bob'] 3 3
SP1 = nx.single_source_shortest_path(G6,1)
print(SP1)
#print(nx.single_source_shortest_path_length(G3,'1'))
{1: [1], 2: [1, 2], 3: [1, 3], 4: [1, 4], 'Alice': [1, 'Alice'], 'Bob': [1, 'Alice', 'Bob'], 'Charlie': [1, 'Alice', 'Charlie']}
SP = nx.all_pairs_shortest_path(G6)
print(type(SP))
SP = dict(SP)
print(dict(SP))
<class 'generator'>
{1: {1: [1], 2: [1, 2], 3: [1, 3], 4: [1, 4], 'Alice': [1, 'Alice'], 'Bob': [1, 'Alice', 'Bob'], 'Charlie': [1, 'Alice', 'Charlie']}, 2: {2: [2], 1: [2, 1], 3: [2, 3], 4: [2, 1, 4], 'Alice': [2, 1, 'Alice'], 'Bob': [2, 1, 'Alice', 'Bob'], 'Charlie': [2, 1, 'Alice', 'Charlie']}, 3: {3: [3], 1: [3, 1], 2: [3, 2], 4: [3, 1, 4], 'Alice': [3, 1, 'Alice'], 'Bob': [3, 1, 'Alice', 'Bob'], 'Charlie': [3, 1, 'Alice', 'Charlie']}, 4: {4: [4], 1: [4, 1], 2: [4, 1, 2], 3: [4, 1, 3], 'Alice': [4, 1, 'Alice'], 'Bob': [4, 1, 'Alice', 'Bob'], 'Charlie': [4, 1, 'Alice', 'Charlie']}, 'Alice': {'Alice': ['Alice'], 1: ['Alice', 1], 'Bob': ['Alice', 'Bob'], 'Charlie': ['Alice', 'Charlie'], 2: ['Alice', 1, 2], 3: ['Alice', 1, 3], 4: ['Alice', 1, 4]}, 'Bob': {'Bob': ['Bob'], 'Alice': ['Bob', 'Alice'], 'Charlie': ['Bob', 'Charlie'], 1: ['Bob', 'Alice', 1], 2: ['Bob', 'Alice', 1, 2], 3: ['Bob', 'Alice', 1, 3], 4: ['Bob', 'Alice', 1, 4]}, 'Charlie': {'Charlie': ['Charlie'], 'Alice': ['Charlie', 'Alice'], 'Bob': ['Charlie', 'Bob'], 1: ['Charlie', 'Alice', 1], 2: ['Charlie', 'Alice', 1, 2], 3: ['Charlie', 'Alice', 1, 3], 4: ['Charlie', 'Alice', 1, 4]}}
print(SP1['Bob'])
print(SP[1]['Bob'])
[1, 'Alice', 'Bob'] [1, 'Alice', 'Bob']
DG2 = nx.DiGraph()
DG2.add_edges_from([(1,2),(1,3),(3,2),(2,5),(4,1),(4,2),(4,3),(5,1),(5,4)])
nx.draw_networkx(DG2)
The result of the Pagerank algorithm is a dictionary with the node ids as keys, and the pagerank values as values.
pr = nx.pagerank(DG2)
print(pr)
print('\n')
pr = nx.pagerank(G6)
print(pr)
{1: 0.18064505060873784, 2: 0.27131643087724033, 3: 0.14665711544131713, 5: 0.26061906832422166, 4: 0.140762334748483}
{1: 0.2420308196543696, 2: 0.12671382905463274, 3: 0.12671382905463274, 4: 0.07286059991932929, 'Alice': 0.17987731223897474, 'Bob': 0.12590180503903042, 'Charlie': 0.12590180503903042}
[h,a] = nx.hits(DG2)
print(h)
print(a)
print(a[2])
{1: 0.3028419093958839, 2: -1.1622198651869745e-17, 3: 0.16745199268671332, 5: 0.12544122612673908, 4: 0.4042648717906636}
{1: 0.23681287910395024, 2: 0.39098432508292885, 3: 0.3161224561036187, 5: -2.7136717951052335e-17, 4: 0.056080339709502096}
0.39098432508292885
We will see the example we presented in class
G7 = nx.read_edgelist('graph-example.txt')
nx.draw_networkx(G7)
To define the personalized Pagerank, we will use the "personalization" parameter, were we define the jump vector as a dictionary, with node ids as the keys, and the jump probability to each node as the values
print(nx.pagerank(G7))
print(nx.pagerank(G7, personalization = {'1':1}))
print(nx.pagerank(G7, personalization = {'6':1}))
{'1': 0.12755099866537586, '2': 0.18232887836414127, '3': 0.2394858218322733, '4': 0.18433868256410274, '5': 0.1326787034099837, '6': 0.13361691516412283}
{'1': 0.2583391334031776, '2': 0.20094608623757965, '3': 0.24190126489812175, '4': 0.14028792201015408, '5': 0.0833530743914921, '6': 0.07517251905947459}
{'1': 0.07517273333798744, '2': 0.12532699696556127, '3': 0.1866533325726192, '4': 0.18958031704255388, '5': 0.15407113246217508, '6': 0.26919548761910295}
We will now change the jump probability. To change the jump probability we will set the parameter alpha.
Attention: The parameter alpha is not the jump probability as we have described in class, but the probability of following an out-link. The jump probability is 1-alpha.
print(nx.pagerank(G7, alpha = 0.5))
print(nx.pagerank(G7, alpha = 0.5, personalization = {'1':1}))
print(nx.pagerank(G7, alpha = 0.5, personalization = {'6':1}))
{'1': 0.1390335398088414, '2': 0.1741686298166486, '3': 0.2133752383994728, '4': 0.17643097727782997, '5': 0.14740281022602297, '6': 0.14958880447118414}
{'1': 0.5510064790190851, '2': 0.16964493982349704, '3': 0.18185949066208873, '4': 0.054964764048023335, '5': 0.026690643658381363, '6': 0.015833682788924434}
{'1': 0.015833407896886878, '2': 0.039357701756408944, '3': 0.07419162031220411, '4': 0.15675161523961573, '5': 0.150192049257395, '6': 0.5636736055374894}
BC = nx.edge_betweenness_centrality(G6)
print(BC)
{('1', '2'): 0.23809523809523808, ('1', '3'): 0.23809523809523808, ('1', '4'): 0.2857142857142857, ('1', 'Alice'): 0.5714285714285714, ('2', '3'): 0.047619047619047616, ('Alice', 'Bob'): 0.23809523809523808, ('Alice', 'Charlie'): 0.23809523809523808, ('Bob', 'Charlie'): 0.047619047619047616}
nx.draw_networkx(G6)
nx.draw(G6, with_labels = True)
nx.draw_circular(G6)
nx.draw_spectral(G6, with_labels=True)
nx.draw_spring(G6)
nx.draw_spring(DG2)
karate=nx.read_gml("karate.gml",label='id')
nx.draw_networkx(karate)
nx.draw_spring(karate,with_labels=True)
Change the size of the nodes depending on their pagerank value
pr = nx.pagerank(karate)
nx.draw_networkx(karate,node_size=[10000*pr[x] for x in pr])
We will now do our own implementation of Pagerank. Pagerank values for node $i$ are computed by iterating the following formula:
$$p_i = 0.85\sum_{j\rightarrow i}\frac{p_j}{d_{out}(j)} +0.15\frac 1n$$We will associate each node with two values: the old pagerank in the previous step and the new one that is currently computed. We initialize the old pagerank to $1/n$
for x in karate.nodes:
karate.nodes[x]['old_pr'] = 1/len(karate.nodes)
karate.nodes[x]['pr'] = 0
The algorithm goes over the edges in the graph, and for each edge (x,y) transfers a fraction of the Pagerank of x to y (and vice versa since the graph is undirected).
For convergece check we want the maximum difference between old and new Pagerank values to be less than eps.
eps = 0.0000001
while (True):
for (x,y) in karate.edges:
karate.nodes[y]['pr'] += karate.nodes[x]['old_pr']/karate.degree(x)
karate.nodes[x]['pr'] += karate.nodes[y]['old_pr']/karate.degree(y)
diff = 0
for x in karate.nodes:
karate.nodes[x]['pr'] = karate.nodes[x]['pr']*0.85 + 0.15/len(karate.nodes)
diff = max(diff, abs(karate.nodes[x]['pr'] - karate.nodes[x]['old_pr']))
if diff < eps: break
for x in karate.nodes:
karate.nodes[x]['old_pr'] = karate.nodes[x]['pr']
karate.nodes[x]['pr'] = 0
print({x:karate.nodes[x]['pr'] for x in karate.nodes})
{1: 0.09699746639416912, 2: 0.05287697716512755, 3: 0.05707850530246944, 4: 0.03585989886187514, 5: 0.02197802085797975, 6: 0.02911125727906845, 7: 0.02911125727906845, 8: 0.024490521601363044, 9: 0.029766036239543145, 10: 0.014309384124315726, 11: 0.02197802085797975, 12: 0.009564757473502347, 13: 0.014644911694305714, 14: 0.029536468425771824, 15: 0.014535968166646367, 16: 0.014535968166646367, 17: 0.016784065326985863, 18: 0.014558694786535084, 19: 0.014535968166646367, 20: 0.019604641610943857, 21: 0.014535968166646367, 22: 0.014558694786535084, 23: 0.014535968166646367, 24: 0.03152244833588808, 25: 0.021075999887935973, 26: 0.02100616170212189, 27: 0.01504401086799329, 28: 0.0256397267700412, 29: 0.019573438474829792, 30: 0.02628848291686834, 31: 0.024590131585182876, 32: 0.037158037426708354, 33: 0.07169310822672904, 34: 0.10091903290492975}
We got essentially the same values as for the Pagerank vector.
pr
{1: 0.09700181758983706,
2: 0.052878391037427,
3: 0.05707842304763673,
4: 0.035860643223064786,
5: 0.021979406974834498,
6: 0.02911334166344221,
7: 0.029113341663442205,
8: 0.02449075803950918,
9: 0.029765339186167028,
10: 0.014308950284462798,
11: 0.021979406974834494,
12: 0.009564916863537146,
13: 0.014645186487916188,
14: 0.02953631497720298,
15: 0.014535161524273824,
16: 0.014535161524273824,
17: 0.016785378110253487,
18: 0.01455885977424349,
19: 0.014535161524273824,
20: 0.01960441671193729,
21: 0.014535161524273824,
22: 0.01455885977424349,
23: 0.014535161524273824,
24: 0.03152091531163227,
25: 0.021075455001162945,
26: 0.02100562817474579,
27: 0.015043395360629753,
28: 0.025638803528350497,
29: 0.019572960509438537,
30: 0.026287262837112076,
31: 0.024589336534292478,
32: 0.037156635922679405,
33: 0.07169213006588289,
34: 0.1009179167487121}
Pagerank implementation using matrix operations
import numpy as np
A = nx.adjacency_matrix(karate)
d = np.array([karate.degree(n) for n in karate.nodes])
D = np.diag(1/d)
P = D@A
N = len(karate.nodes)
u = np.ones(N)/N
p = p_old = u
while(True):
p = (0.85*p_old.T@P+0.15*u.T).T
if np.linalg.norm(p-p_old) < eps: break
p_old = p
p
array([0.09699741, 0.05287696, 0.05707851, 0.03585989, 0.021978 ,
0.02911123, 0.02911123, 0.02449052, 0.02976604, 0.01430939,
0.021978 , 0.00956476, 0.01464491, 0.02953647, 0.01453598,
0.01453598, 0.01678405, 0.01455869, 0.01453598, 0.01960464,
0.01453598, 0.01455869, 0.01453598, 0.03152247, 0.02107601,
0.02100617, 0.01504402, 0.02563974, 0.01957344, 0.0262885 ,
0.02459014, 0.03715806, 0.07169312, 0.10091905])