Incorrect PageRank calculation result

Question

I referred to PageRank - Wikipedia and computed PageRank algebraically with the following equation, but I got the different result from nx.pagerank_numpy.

For instance (image from Wikipedia),

I got,

# 'A', 'B', 'C', 'D', 'E', 'F'
[[ 0.028]
 [ 0.324]
 [ 0.289]
 [ 0.033]
 [ 0.068]
 [ 0.033]]

Why are the results different?

---

Here is the source code.

import networkx as nx
import numpy as np

# Step 1: Build up a graph 
G = build_graph_wikipedia_pagerank_example()

# Step 2: PageRank calculation

# Part 1: \mathbf {1}  is the column vector of length N containing only ones.
N = len(G.nodes())      # N = 11
column_vector = np.ones((N, 1), dtype=np.int)
#print(column_vector)

# Part 2: Matrix M
# Adjacency matrix A
nodelist = ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K']  # sorted(G.nodes())
A = nx.to_numpy_matrix(G, nodelist)

# K is the diagonal matrix with the outdegrees in the diagonal.
list_outdegree = map(operator.itemgetter(1), sorted(G.out_degree().items()))
K = np.diag(list_outdegree)

K_inv = np.linalg.pinv(K)

# Matrix M
M = (K_inv * A).transpose()

# Part 3: PageRank calculation
d = 0.85
I = np.identity(N)
R = np.linalg.pinv(I - d*M) * (1-d)/N * column_vector 

To build up the graph, I use,

def build_graph_wikipedia_pagerank_example():
    """
    Build a graph for https://en.wikipedia.org/wiki/File:PageRanks-Example.svg
    """

    G = nx.DiGraph()

    # A

    # B --> 
    G.add_path(['B', 'C'])

    # C -->
    G.add_path(['C', 'B'])

    # D --> 
    G.add_path(['D', 'A'])
    G.add_path(['D', 'B'])

    # E --> 
    G.add_path(['E', 'B'])
    G.add_path(['E', 'D'])
    G.add_path(['E', 'F'])

    # F --> 
    G.add_path(['F', 'B'])
    G.add_path(['F', 'E'])

    # G --> 
    G.add_path(['G', 'B'])
    G.add_path(['G', 'E'])

    # H --> 
    G.add_path(['H', 'B'])
    G.add_path(['H', 'E'])

    # I --> 
    G.add_path(['I', 'B'])
    G.add_path(['I', 'E'])

    # J --> 
    G.add_path(['J', 'E'])

    # J --> 
    G.add_path(['K', 'E'])

    return G

Answer 1

You just need to normalize the page ranks obtained with the matrix equation, since page ranks should sum to 1.

R = R / sum(R)
print R
#[[ 0.03278149]
# [ 0.38440095]
# [ 0.34291029]
# [ 0.03908709]
# [ 0.08088569]
# [ 0.03908709]
# [ 0.01616948]
# [ 0.01616948]
# [ 0.01616948]
# [ 0.01616948]
# [ 0.01616948]]
print nx.pagerank_numpy(G, alpha=d)
#{'A': 0.032781493159344234, 'C': 0.34291028550837976, 'B': 0.3844009488135542, 'E': 0.08088569323449775, 'D': 0.03908709209996617, 'G': 0.016169479016858397, 'F': 0.03908709209996617, 'I': 0.016169479016858397, 'H': 0.016169479016858397, 'K': 0.016169479016858397, 'J': 0.016169479016858397}