(Tentative) Network Analysis with networkX : Fundamentals of network theory-2

Kyunghoon Kim
Network Analysis with networkX
Fundamentals of network theory-2
2014. 05. 28.
UNIST Mathematical Sciences
Kyunghoon Kim ( Kyunghoon@unist.ac.kr )
5/28/2014 Fundamentals of network theory-2 1

Kyunghoon Kim
Indexing
Google Glasses is a
computer with a
head-mounted
display.
He wore thick
glasses. He worked
in google corporation.
He wore glasses to
be able to read signs
at a distance.
google
glasses
is
a
computer
with
head-mounted
display
he
1 2
1 2 3
1
1 3
1
1
1
1
2 3
wore
thick
worked
in
corporation
to
be
able
read
…
2 3
2
2
2
2
3
3
3
3
1 2 3

Kyunghoon Kim
Indexing
Google Glasses is a
computer with a
head-mounted
display.
He wore thick
glasses. He worked
He wore glasses to
at a distance.
google
glasses
is
a
computer
with
head-mounted
display
he
1 2
1 2 3
1
1 3
1
1
1
1
2 3
wore
thick
worked
in
corporation
to
be
able
read
…
2 3
2
2
2
2
3
3
3
3
1 2 3

Kyunghoon Kim
Indexing with position
Google Glass is a
computer with a
head-mounted
display.
He wore thick
glasses. He worked
He wore glasses to
at a distance.
google
glasses
is
a
computer
with
head-mounted
display
he
1-1 2-8
1-2 2-4 3-3
1-3
1-4 3-11
1-5
1-6
1-7
1-8
2-1 2-5 3-1
wore
thick
worked
in
corporation
to
be
able
read
…
2-2 3-2
2-3
2-6
2-7
2-9
3-4 3-7
3-5
3-6
1 2 3

Kyunghoon Kim
Indexing with position & metatag
<title>New Google
Glass</title><body>Google
Glass is a computer with a
head-mounted
display.</body>
<title>Daily life of
David</title><body>He
wore thick glasses. He
worked in google
corporation.</body>
<title>Black
Glasses</title><body>He
wore glasses to be able to
read signs at a
distance.</body>
google
glasses
is
a
computer
with
head-mounted
display
he
1-1 2-8
1-2 2-4 3-3
1-3
1-4 3-11
1-5
1-6
1-7
1-8
2-1 2-5 3-1
wore
thick
worked
in
corporation
<title>
</title>
<body>
</body>
2-2 3-2
2-3
2-6
2-7
2-9
#
#
#
#
1 2 3

Kyunghoon Kim
Indexing with position & metatag
Altavista
“Constrained Searching of an index”, 1999

Kyunghoon Kim
“uncanny knack for returning extremely
relevant results.” – PC Magazine
The technology that launched google
Screenshot of “google.stanford.edu” 1997, http://blogoscoped.com/archive/2006-04-21-n63.html

Kyunghoon Kim
A hyperlink is a reference to data that the
reader can directly follow either by clicking or
by hovering or that is followed automatically.
– Merriam-Webster.com
Hyperlink

Kyunghoon Kim
Hyperlink Trick
Barny’s tomato pasta recipe
Cook Pasta Sheets, one at a time,
for 1 minute each. And before
serving, place pasta in the bottom
of a soup bowl.
Tony’s tomato pasta
Bring to a boil and add pasta.
Cook for 10 minutes. Drain
on paper towel.
I like barny’s recipe. I enjoyed tony’s recipe. Tony’s recipe is
amazing!
I’m in
admiration of
tony’s recipe.

Kyunghoon Kim
Authority Trick
of a soup bowl.
on paper towel.
I like barny’s recipe. I enjoyed tony’s recipe. Tony’s recipe is
amazing!
I’m in
admiration of
tony’s recipe.
100 1 1
1

Kyunghoon Kim
Authority Trick
of a soup bowl.
on paper towel.
I like barny’s recipe.
I enjoyed tony’s recipe. Tony’s recipe is
amazing!
I’m in
admiration of
tony’s recipe.
100
1 1
1
100
3
A B
100 3

Kyunghoon Kim
The iterates will not converge no matter how
long the process is run.
Cycle
1
2 3
41
2

Kyunghoon Kim
There is a person who is randomly surfing the
internet.
Random Surfer Trick

Kyunghoon Kim
internet.
Surfer starts off at a single web page selected
at random from the entire World Wide Web.
Random Surfer Trick

Kyunghoon Kim
internet.
The surfer then examines all the hyperlinks on
the page, picks one of them at random, and
clicks on it.
Random Surfer Trick

Kyunghoon Kim
internet.
clicks on it.
The new page is then examined and one of its
hyperlinks is chosen at random.
Random Surfer Trick

Kyunghoon Kim
internet.
clicks on it.
The new page is then examined and one of its
hyperlinks is chosen at random.
This process continues…
Random Surfer Trick

Kyunghoon Kim
import networkx as nx
from matplotlib import pyplot as plt
G = nx.DiGraph()
G.add_edges_from([(1,2),(2,3),(2,12),(3,4),(3,8),(3,12),(4,8),(5,1),
(6,1),(6,2),(7,2),(7,12),(8,12),(8,13),(8,14),(8,15),(8,9),(9,16),
(10,5),(10,6),(10,7),(11,10),(12,16),(13,16),(14,16),(15,16),
(16,11)])
plt.clf()
nx.draw_spring(G)
Random Surfer Trick
Example

Kyunghoon Kim
spring layout – places nodes using
Fruchterman-Reingold force-directed algorithm
circular layout – places nodes in a circle
random layout – positions nodes based on an
uniform distribution in a unit square
shell layout – places nodes in concentric circles
spectral layout – positions nodes using eigen-
vectors of the graph laplacian
Layout of NetworkX
e.g., pos = nx.spectral_layout(G)
nx.draw(G, pos)

Kyunghoon Kim
nx.spring_layout has different position at each
iteration.
To fix the position, we use nx.spectral_layout
Just be careful the following case:
networkx.spectral_layout

Kyunghoon Kim
import numpy as np
def addvalue(values,node):
values[node-1] += 1
lennode = len(G.nodes())
values = np.zeros(lennode)
Random Surfer Trick – Outline

Kyunghoon Kim
randomvector = np.random.rand(lennode)
selectednode = np.where(randomvector == max(randomvector))[0][0] + 1
print "First Node"
print selectednode
addvalue(values, selectednode)
nextnodes = G.edges(selectednode)
randomvector = np.random.rand(len(nextnodes))
selectedpos = np.where(randomvector == max(randomvector))[0][0]
selectednode = nextnodes[selectedpos][1]
print "Second Node"
print selectednode
Random Surfer Trick – Outline (Cont.)
First Node
7
Second Node
2

Kyunghoon Kim
def nextstep(selectednode):
print "choices", nextnodes
print selectednode
return selectednode
for i in range(5):
selectednode = nextstep(selectednode)
First Node
5
choices [(5, 1)]
1
choices [(1, 2)]
2
choices [(2, 3), (2, 12)]
3
choices [(3, 8), (3, 12), (3, 4)]
12
choices [(12, 16)]
16
>>> values
array([ 1., 1., 1., 0., 1., 0., 0., 0., 0., 0.,
0., 1., 0., 0., 0., 1.])

Kyunghoon Kim
for i in range(20000):
plt.clf()
nx.draw_networkx_nodes(G, pos, node_size=values)
nx.draw_networkx_edges(G, pos)
nx.draw_networkx_labels(G, pos)

Kyunghoon Kim
plt.clf()
nx.draw_networkx_nodes(G, pos, node_size=10000*values/sum(values))

Kyunghoon Kim
There is one twist : restart probability(15%)
Surfer does not click on one of the available
hyperlinks.
Instead, he restarts the procedure by picking
another page randomly from the whole web.
(boring, error of browser, etc.)
Twist

Kyunghoon Kim
def randomstart(G):
lennode = len(G.nodes())
randomvector = np.random.rand(lennode)
selectednode = np.where(randomvector == max(randomvector))[0][0] + 1
return selectednode
def nextstep(selectednode):
return selectednode
Random Surfer Trick with Twist

Kyunghoon Kim
selectednode = randomstart(G)
if np.random.rand(1)[0] > 0.15:
else:
selectednode = randomstart(G)
if i % 100000 == 0:
print i
plt.clf()
Random Surfer Trick with Twist (Cont.)

Kyunghoon Kim
plt.clf()
pos = nx.spectral_layout(G)
labels = 100*values/sum(values) % percent
labels = list(labels.astype('|S4'))
labels = dict(zip(range(1,lennode+1), labels))
nx.draw_networkx_labels(G, pos, labels)
Random Surfer Trick with Twist

Kyunghoon Kim
What is the connection between random surfer
model and the authority trick that we would
like to use for ranking web pages?
Connection

Kyunghoon Kim
The percentages calculated from random surfer
simulations turn out to be exactly what we
need to measure a page’s authority.
Connection

Kyunghoon Kim
The percentages calculated from random surfer
simulations turn out to be exactly what we
need to measure a page’s authority.
Surfer authority score
= percentage of time that a random surfer
would spend visiting that page
Connection

Kyunghoon Kim
Hyperlink Trick
: the main idea was that a page with many
incoming links should receive a high ranking.
Tricks for ranking web pages

Kyunghoon Kim
Hyperlink Trick
: the main idea was that a page with many
incoming links should receive a high ranking.
pos = nx.circular_layout(G)

Kyunghoon Kim
Authority Trick
: the main idea was that an incoming link from
a highly authoritative page should improve a
page’s ranking more than an incoming link from
a less authoritative page.

Kyunghoon Kim
Authority Trick
: the main idea was that an incoming link from
a highly authoritative page should improve a
page’s ranking more than an incoming link from
a less authoritative page.
Connection
: An incoming link from a popular page will
have more opportunities to be followed than a
link from an unpopular page.

Kyunghoon Kim
Authority Trick

Kyunghoon Kim
The random surfer model take account the
quantity(hyperlink trick) and quality(authority
trick) of incoming links at each page.
Random Surfer Model

Kyunghoon Kim
This model works perfectly well whether or not
there are cycles in the hyperlinks.
Random Surfer Model

Kyunghoon Kim
Random Surfer Model
1
2 3
4

Kyunghoon Kim
with constant term without constant term
Divide by out-degree
PageRank
x=D(D−𝜶𝑨)−1 𝟏
Degree centrality
x=𝑨D−1x
No division
Katz centrality
x=(I−𝜶𝑨)−1
𝟏
Eigenvector centrality
x=𝜅1
−1
𝑨x
Pagerank with Linear Algebra

Kyunghoon Kim
Matrix Equation
Linear Transformation
Eigenvalue
Eigenvector
Eigenvector centrality
Katz centrality
Pagerank centrality
Contents with Linear Algebra

Kyunghoon Kim
If 𝐴 is an 𝑚 × 𝑛 matrix, with columns
and if x is in ℝ 𝑛
, then the product of 𝐴 and x is
the linear combination of the columns of 𝐴
using the corresponding entries in x as weights;
that is,
The matrix equation Ax=b
 
1
1 2 1 1 2 2n n n
n
x
A x x x
x
 
      
  
x a a a a a a
1 2, , , na a a
matrix equation vector equation

Kyunghoon Kim
The matrix equation Ax=b
 
1
1 2 1 1 2 2n n n
n
x
A x x x
x
 
      
  
x a a a a a a
4
1 2 1 1 2 1 3
3 4 3 7
0 5 3 0 5 3 6
7
 
                                     

Kyunghoon Kim
import numpy as np
pointset = [[1,2],[2,2],[2,3],[3,4],[3,5],[2,5],[1,5],[1,3],[1,2]]
plt.plot(*np.transpose(pointset), marker='.')
plt.axis([0, 6, 0, 6])
Example – Plot of point set

Kyunghoon Kim
import numpy as np
pointset = [[1,2],[2,2],[2,3],[3,4],[3,5],[2,5],[1,5],[1,3],[1,2]]
plt.plot(*np.transpose(pointset), marker='.')
plt.axis([0, 6, 0, 6])
newset = []
for i in pointset:
x = np.matrix(i).transpose()
A = np.matrix([[0,1],[1,0]])
Ax = A*x
Ax = list(np.array(Ax).reshape(-1,))
newset.append(Ax)
plt.plot(*np.transpose(newset), marker='.')
Example (Cont.) – Plot of rotation

Kyunghoon Kim
newset = []
for i in pointset:
deg = pi*35/180 # deg to radian
A = np.matrix([[cos(deg),-sin(deg)],[sin(deg),cos(deg)]]) # counter clockwise
Ax = A*x
newset.append(Ax)
plt.axis([-6, 6, -6, 6])
Example (Cont.) – Plot of rotation

Kyunghoon Kim
In two dimensions,
Rotation matrix
http://librairie.immateriel.fr/fr/read_book/9780596516130/ch11s02

Kyunghoon Kim
newset = []
for i in pointset:
A = np.matrix([[1.5, 0],[0, 1]])
Ax = A*x
newset.append(Ax)
A = np.matrix([[-1.8, 0],[0, 1]])
A = np.matrix([[1, 0],[0, -1]])
Example (Cont.) – Plot of dilation

Kyunghoon Kim
3 2 2 4
1 0 1 2
     
     
     

Kyunghoon Kim
3 2 2 4 2
2
1 0 1 2 1
       
        
       

Kyunghoon Kim
3 2 2 4 2
2
1 0 1 2 1
       
        
       
A x x
The number 𝜆 is the eigenvalue. It tells
whether the special vector x is stretched or
shrunk or reversed or left unchanged – when it
is multiplied by 𝐴.

Kyunghoon Kim
The eigenvalues are a new way to see into the
heart of a matrix.
Almost all vectors change direction, when they
are multiplied by 𝐴. Certain exceptional vectors
x are in the same direction as 𝐴x. Those are the
eigenvectors.
Multiply an eigenvector by 𝐴, and the vector
𝐴x is a number 𝜆 times the original x.
Eigenvalues and Eigenvectors

Kyunghoon Kim
If (𝐴 − 𝜆𝐼)x=0 has a nonzero solution, then
(𝐴 − 𝜆𝐼) cannot have an inverse, i.e., is not
invertible. The determinant of 𝐴 − 𝜆𝐼 must be
zero.
Example – eigenvalue and eigenvector
3 2 2 4
1 0 1 2
     
     
     
( ) 0
A
A I



 
x x
x
( ) 0A I x

Kyunghoon Kim
A natural extension of the simple degree
centrality is eigenvector centrality.
All neighbors are not equivalent.
A vertex’s importance is increased by having
connections to other vertices that are
themselves important.
Eigenvector Centrality

Kyunghoon Kim
Let 𝑥𝑖 = 1 for all 𝑖.
We define to be the sum of the centralities of
𝑖’s neighbors
𝑥𝑖
′
=
𝑗
𝐴𝑖𝑗 𝑥𝑗
We can write this expression in matrix
notation as x′
= 𝐴x.
Repeating this process to make better
estimates, we have after 𝑡 steps a vector x(𝑡).

Kyunghoon Kim
x 𝑡 = 𝐴 𝑡
x(0).
Let’s write x(0) as a linear combination of the
eigenvectors v𝑖 of the adjacency matrix
x(0)=
𝑖
𝑐𝑖 𝐯𝑖
Then
x(t)=𝐴 𝑡
𝑖
𝑐𝑖 𝐯𝑖 =
𝑖
𝑐𝑖 𝑘𝑖
𝑡
𝐯𝑖 = 𝑘1
𝑡
𝑖
𝑐𝑖
𝑘𝑖
𝑘1
𝑡
𝐯𝑖
where the 𝑘𝑖 are the eigenvalues of 𝐴, 𝑘1 is largest eigenvalue.

Kyunghoon Kim
Since
𝑘 𝑖
𝑘1
< 1 for all 𝑖 ≠ 1, all terms in the sum
other than the first decay exponentially as 𝑡
becomes large.
We get x 𝑡 → 𝑐1 𝑘1
𝑡
v1 as 𝑡 → ∞.
The limiting vector of centralities is simply
proportional to the leading eigenvector of the
adjacency matrix.
Equivalently we could say that the centrality x
satisfies 𝐴x = 𝑘1x.

Kyunghoon Kim
𝐴x = 𝑘1x.
The centrality 𝑥𝑖 of vertex 𝑖 is proportional to
the sum of the centralities of 𝑖’s neighbors:
𝑥𝑖 = 𝑘1
−1
𝑗
𝐴𝑖𝑗 𝑥𝑗
It can be large
1. a vertex has many neighbors
2. a vertex has important neighbors

Kyunghoon Kim
Katz centrality.

Kyunghoon Kim
Degree centrality
x=𝑨D−1
x
D is the diagonal matrix
with elements 𝐷𝑖𝑖 = max(𝑘𝑖
𝑜𝑢𝑡
, 1).
Degree Centrality

Kyunghoon Kim
MacCormick, John. Nine Algorithms that
Changed the Future: The Ingenious Ideas that
Drive Today's Computers. Princeton University
Press, 2011.
Newman, Mark. Networks: an introduction.
Oxford University Press, 2010.
Strang, Gilbert. Introduction to linear algebra.
SIAM, 2003.
References

(Tentative) Network Analysis with networkX : Fundamentals of network theory-2

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