(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 ( [email protected] )

5/28/2014 Fundamentals of network theory-2 1

mailto:[email protected]

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

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“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

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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


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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!


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!


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

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There is a person who is randomly surfing the internet.

Random Surfer Trick


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Surfer starts off at a single web page selected at random from the entire World Wide Web.

Random Surfer Trick


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The surfer then examines all the hyperlinks on the page, picks one of them at random, and clicks on it.

Random Surfer Trick


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The new page is then examined and one of its hyperlinks is chosen at random.

Random Surfer Trick


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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

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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)

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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


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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)



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plt.clf()

nx.draw_networkx_nodes(G, pos, node_size=10000*values/sum(values))





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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


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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


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

Connection


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

Connection


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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


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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)

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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


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This model works perfectly well whether or not there are cycles in the hyperlinks.

Random Surfer Model


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Random Surfer Model


1

2 3

4

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Random Surfer Model


1

2 3

4

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with constant term without constant term

Divide by out-degreePageRank

x=D(D−𝜶𝑨)−1𝟏

Degree centrality

x=𝑨D−1x

No divisionKatz centrality

x=(I−𝜶𝑨)−1𝟏

Eigenvector centrality

x=𝜅1−1𝑨x

Pagerank with Linear Algebra


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Matrix Equation

Linear Transformation

Eigenvalue

Eigenvector

Eigenvector centrality

Katz centrality

Pagerank centrality

Contents with Linear Algebra


Kyunghoon Kim

If 𝐴 is an 𝑚 × 𝑛matrix, with columnsand if x is in ℝ𝑛, then the product of 𝐴 and x is the linear combination of the columns of 𝐴using 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

41 2 1 1 2 1 3

3 4 3 70 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


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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


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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


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3 2 2 4

1 0 1 2

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3 2 2 4 22

1 0 1 2 1

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3 2 2 4 22

1 0 1 2 1

A x xThe 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


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Let 𝑥𝑖 = 1 for all 𝑖.

We define to be the sum of the centralities of 𝑖’s neighbors

𝑥𝑖′ =

𝑗

𝐴𝑖𝑗𝑥𝑗

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 = 𝐴𝑡x(0).

Let’s write x(0) as a linear combination of the eigenvectors v𝑖 of the adjacency matrix

x(0)=

𝑖

𝑐𝑖𝐯𝑖

Then

x(t)=𝐴𝑡

𝑖

𝑐𝑖𝐯𝑖 =

𝑖

𝑐𝑖𝑘𝑖𝑡𝐯𝑖 = 𝑘1

𝑡

𝑖

𝑐𝑖𝑘𝑖𝑘1

𝑡

𝐯𝑖

where the 𝑘𝑖 are the eigenvalues of 𝐴, 𝑘1 is largest eigenvalue.



Kyunghoon Kim

Since 𝑘𝑖

𝑘1< 1 for all 𝑖 ≠ 1, all terms in the sum

other than the first decay exponentially as 𝑡becomes large.

We get x 𝑡 → 𝑐1𝑘1𝑡v1 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 xsatisfies 𝐴x = 𝑘1x.



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𝐴x = 𝑘1x.

The centrality 𝑥𝑖 of vertex 𝑖 is proportional to the sum of the centralities of 𝑖’s neighbors:

𝑥𝑖 = 𝑘1−1

𝑗

𝐴𝑖𝑗𝑥𝑗

It can be large1. a vertex has many neighbors2. a vertex has important neighbors



Kyunghoon Kim

Katz centrality.


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Degree centrality

x=𝑨D−1x

D is the diagonal matrix

with elements 𝐷𝑖𝑖 = max(𝑘𝑖𝑜𝑢𝑡, 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

Education

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