Lab 4: Working with NetDraw

Opening a dataset File>Open

Editing node attributes Transform>Node attribute editor

Editing link properties Properties>Lines

Configuring networks Layout>

Highlighting parts of the network Analysis>

Storing the dataset or the diagram File>Save…

Using an external editor (Notepad) to edit a NetDraw dataset (a .vna file)

2

The Development of Social Network Analysis—with an Emphasis on Recent Events (Freeman, 2011)

Group Level (Macro Level) Cohesive Subgroups or Communities

Individual Level (Micro Level) Position

Centralities

In-between (Meso Level) Blockmodeling

3

Cohesive Subgroups

Clique

Cohesive subgroups are subgraphs that are more tightly interconnected embedded within a graph. Clique is a subgraph that contains at least three nodes

and all the shortest path lengths between nodes are one.

1 3 6

5

4

2

7

Clique: {1,2,3}、{1,3,4}

、 {3,4,6,7}

Loosely-Connected Subgroups

A clique is a fully connected subgraph and the most tightly-connected cohesive subgroup.

We can release the constrain of the clique: Based on path length

n-clique n-clan

Based on node degree k-plex k-core

N-Clique

n-clique d(i, j) ≦ n for all i, j∈V

1

2

6

4 5

3 2-cliques ：

and{1,2,3,4,5}

{2,3,4,5,6 }

N-clan

n-clan considers only the shortest paths that pass through the nodes within the

subgraph of n-clan and requires all the shortest path lengths be not greater than n.

An n-clan is also an n-clique, but an n-clique may not be an n-clan.

1

2

6

4 5

32-clan ：

{2,3,4,5,6 }

K-plex

k-plex is a subgraph that has s nodes, and each node connects to at least other s-k nodes of the subgraph:

d(i) (≧ s-k) for all i ∈V

3

14

2

2-plex ：{1,2,3,4}

K-core

k-core is a subgraph that each node connects to at least other k nodes of the subgraph:

d(i)≧k for all i ∈V

3

14

2

2-core ：{1,2,3,4}

10

Network Data Collection

3

12

3

5

12

3

54

12

3

54

2

3

54

3

54

12

3

54

12

… …

… …

Whole Network Partial Network Ego Networks

Data Structures: adjacency matrix or adjacency list

11

Collecting Network from Blogosphere

Homepage ( 首頁 ) Blogroll

Post ( 文章 ) Citation (hyperlink in content) Trackback Comment

Select Top 100 blogs in 2008, 2009, 2010, 2011 from 「部落格觀察」 http://look.urs.tw (Closed Now…)

3

54

12

12

A Blogroll Network (2008)

Top 100 + Random 100 Blogroll Network

Analysis of Cliques

Number of Cliques

3-node cliques 4-node cliques 4-node cliques

67 41 22 4

Component of Cliques

The Largest Cliques

Clique member1 1、 8、 10、 33、 402 17、 48、 55、 75、

973 48、 55、 75、 92、

974 48、 71、 75、 92、

97

Lab 5: Finding all the cliques out

Network>Subgroups>Cliques Input dataset:

2008.vna

Output dataset: …

Interpret the output dataset Hierarchical clustering

http://www.analytictech.com/networks/hiclus.htm

Visualize the result in NetDaw

Analysis of 2-clique and 2-clan

The largest 2-clique and 2-clan

2-plex member1 17、 48、 55、 75、 92、 9

72 48、 55、 71、 75、 92、 9

7

Analysis of 2-plex

The largest 2-plex

Lab 6: Finding n-cliques, n-clans and k-plex

Network>Subgroups>… Input dataset:

2008.vna

Output dataset: …

Interpret the output dataset

Visualize the result in NetDaw

19

Analysis of k-cores

20

Number of nodes

Number of edges

Reciprocal edges

Average degree

Path lengthClustering coefficient

clique (1) 5 10 3 4 1 1clique (2) 5 10 6 4 1 1clique (3) 5 10 6 4 1 1clique (4) 5 10 10 4 1 12-clique 22 52 9 4.727 1.932 0.5192-clan 22 52 9 4.727 1.932 0.5192-plex (1) 6 14 9 4.667 1.067 0.9332-plex (2) 6 14 10 4.667 1.067 0.9335-core 14 50 24 7.143 1.567 0.424

Comparison of Cohesive Subgroups

21

Inclusion Property of K-core

k-core 的定義具有包含性（ inclusion ）： 若一個節點屬於 c-core ，則必然也屬於 (c-1)-core 。

1-core

2-core

4-core

0-core

22

2008 Blogroll Network

23

2009 Blogroll Network

24

K-core Analysis for Exploring Network Changes

If a node belongs to c-core, but not (c+1)-core, then the coreness of the nodes is c.

Coreness change between 2008 and 2009:

coreness

coreness

2009/9/11

0 1 2 3 4 5 6 7

2008/11/1

0 10 8 9 8 0 5 0 3

1 1 6 1 4 1 0 0 0

2 0 4 6 2 1 0 0 1

3 0 2 1 7 4 0 0 0

4 1 1 1 1 14 3 0 7

5 0 0 1 0 2 9 0 0

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0

Lab 7: Analyzing k-core

Input dataset: 2008.vna

Analyzing: In UCINET In NetDraw

Interpret the output dataset

26

Reciprocity, Transitivity, and Closure

27

On the Second Thought: Open vs. Closed

28

Structural Hole

29

Structural Holes

30

Triad types andBalance-theoreticModels

Lab 8: Conducting Triad Census

Network>Triad Census Input dataset:

2008.vna

Output dataset: …

Interpret the output dataset

Suggested Readings (ftp://163.25.117.117/nplu)

In the directory: / 資訊網絡分析 /Readings 17 The Development of Social Network Analysis.pdf 18 Analysing Social Networks Via the Internet.pdf 19 Graph Theoretical Approaches to Social Network

Analysis.pdf

33

Centralities

34

Background

At the individual level, one dimension of position in the network can be captured through centrality.

Conceptually, centrality is fairly straight forward: we want to identify which nodes are in the ‘center’ of the network.

In practice, identifying exactly what we mean by ‘center’ is somewhat complicated.

Approaches: Degree Closeness Betweenness Power

The graph level measures: Centralization

Degree Centrality

An index that measures the degree of a node A local measure

A B CD

E

F G

H J

I

KML

36

Formula of Degree Centrality

in-degree centrality:

out-degree centrality:

normalized in-degree centrality:

normalized out-degree centrality:

ini

inD kic

outi

outD kic

1' vkic ini

inD

1' vkic outi

outD

3

54

12

37

Degree Centrality in the Examples

38

Another Example

Degree centrality, however, can be deceiving, because it is a purely local measure.

Closeness Centrality

An index that measures the distance from a node to the other nodes A global measure

A B CD

E

F G

H J

I

KML

40

Formula of Closeness Centrality

節點 i 的連入接近中心性(in-closeness centrality) ：

節點 i 的連出接近中心性(out-closeness centrality) ：

節點 i 的正規連入接近中心性：

節點 i 的正規連出接近中心性：

3

54

12

Betweenness Centrality

An index that measures the intermediate importance of a node A global measure

A B CD

E

F G

H J

I

KML

42

Formula of Betweenness Centrality

betweenness centrality:

normalized betweenness centrality:

3

54

12

43

Betweenness Centrality in the Examples

44

Centralization of Network

If we want to measure the degree to which the graph as a whole is centralized, we look at the dispersion of centrality:

Simple: variance of the individual centrality scores.

gCnCSg

idiDD /))((

1

22

Or, using Freeman’s general formula for centralization (which ranges from 0 to 1):

Centralization of Network

The ration CG :

0CG1

i

stari

stari

Gi

G

cc

cc

)(

)(

max

max

46

Degree Centralization

Freeman: .07Variance: .20

Freeman: 1.0Variance: 3.9

Freeman: .02Variance: .17

Freeman: 0.0Variance: 0.0

47

Homework

Closeness centralization Betweenness centralization

48

Comparison

Comparing across these 3 centrality values•Generally, the 3 centrality types will be positively correlated•When they are not (low) correlated, it probably tells you something interesting about the network.

  Low Degree

Low Closeness

Low Betweenness

High Degree   Embedded in cluster that is far from the rest of the network

Ego's connections are redundant - communication bypasses him/her

High Closeness Key player tied to important/active alters

  Probably multiple paths in the network, ego is near many people, but so are many others

High Betweenness Ego's few ties are crucial for network flow

Very rare cell. Would mean that ego monopolizes the ties from a small number of people to many others.

 

Lab 9: Calculating Centrality

Network>Centrality and Power Degree Closeness Betweenness

Centrality Analysisof the Blogosphere

Data: 2008 2009 2010

51

Results of Degree Centralities

0

4

8

12

16

20

0 25 50 75 100 125 150

in-degree centrality rank

in-d

egre

e ce

ntra

lity

(%

)

2008

2009

2010

0

4

8

12

16

20

0 25 50 75 100 125 150

out-degree centrality rankou

t-de

gree

cen

tral

ity

(%)

2008

2009

2010

52

Results of Closeness Centralities

0

8

16

24

32

40

0 25 50 75 100 125 150

in-closeness centrality rank

in-c

lose

ness

cen

tral

ity

(%)

2008

2009

2010

0

8

16

24

32

40

0 25 50 75 100 125 150

out-closeness centrality rankou

t-cl

osen

ess

cent

rali

ty (

%)

2008

2009

2010

53

Results of Betweenness Centralities

0

2

4

6

8

10

0 25 50 75 100 125 150

betweenness centrality rank

betw

eenn

ess

cent

rali

ty (

%)

2008

2009

2010

54

CentralityComparisonofBlogCommunities

中心性

年度 社群

連入分支中心性

連出分支 中心性

連入接近 中心性

連出接近中心性

中介 中心性

資訊科技 3.29 4.06 12.47 13.20 1.06

美食 2.96 3.40 10.04 10.74 1.13

網路應用 3.94 4.51 13.51 14.10 1.60

社會評論 2.24 2.16 10.03 8.96 1.09

心情日記 3.22 2.65 11.53 9.62 1.06

旅行 4.36 3.13 12.59 10.84 1.35

圖文 4.06 2.81 12.59 13.58 1.03

2008

整體網絡 2.71 2.71 10.06 10.06 1.01

資訊科技 3.36 3.14 11.24 15.07 0.65

美食 2.72 2.44 13.46 7.10 0.78

網路應用 2.85 3.55 9.90 18.23 0.91

社會評論 1.76 1.15 9.22 6.90 0.50

心情日記 2.44 3.07 11.51 12.54 1.11

旅行 2.77 1.67 12.47 7.45 0.30

圖文 5.23 4.99 13.85 15.41 0.71

2009

整體網絡 2.45 2.45 10.01 10.01 0.55

資訊科技 2.53 2.53 8.14 13.98 0.52

美食 1.96 2.07 6.42 7.43 0.34

網路應用 2.52 2.95 7.72 15.74 0.71

社會評論 1.07 0.63 5.03 2.42 0.21

心情日記 2.10 2.10 8.87 9.11 0.88

旅行 2.07 1.63 6.84 6.94 0.25

圖文 4.17 3.40 12.70 8.36 0.33

2010

整體網絡 1.86 1.86 6.84 6.84 0.32

Results of Network Centralization

Centralization

Network

In degreeCentrality

Out degreeCentrality

In closenessCentrality

Out closenessCentrality

BetweennessCentrality

2008/11/1 7.79% 16.21% 20.75% 52.36% 8.29%

2009/9/11 9.00% 12.28% 19.78% 53.03% 5.73%

2010/7/2 7.03% 10.46% 22.38% 55.39% 6.51%

56

The Web’s Bowtie Structure

OUTIN

LargestStrongly

ConnectedComponent

… …

Disconnected Components

Tube

Tendrils

…

57

The Bowtie Structure of the 2008 Blogosphere

Lab 10: Exploring the Bowtie Structure

Excluding disconnected components,we can find out the the bowtie structure of the LWCC:

SCC IN OUT Tendril = LWCC – SCC – IN – OUT

59

Bonacich Power Centrality

Actor’s centrality (prestige) is equal to a function of the prestige of those they are connected to. Thus, actors who are tied to very central actors should have higher prestige/centrality than those who are not.

1)(),( 1 RRIC • is a scaling vector, which is set to normalize the score. • reflects the extent to which you weight the centrality of people ego is tied to.• R is the adjacency matrix (can be valued)• I is the identity matrix (1s down the diagonal) • 1 is a matrix of all ones.

60

The Parameter:

The magnitude of reflects the radius of power. Small values of weight local structure, Larger values weight global structure.

If is positive, then ego has higher centrality when tied to people who are central.

If is negative, then ego has higher centrality when tied to people who are not central.

As approaches zero, you get degree centrality.

61

Example

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 2 3 4 5 6 7

PositiveNegative

62

Two Key Dimensions of Centrality (Borgatti, 2003; 2005)

The key question for centrality is knowing what is flowing through the network. The key features are:

Whether the actor retains the good to pass to others orwhether they pass the good and then loose it.

Whether the key factor for spread is distance ormultiple sources.

Radial Medial

Frequency

Distance

Degree CentralityBon. Power centrality

Closeness Centrality

Betweenness

(empty: but would be an interruption measure based on distance)

Suggested Readings (ftp://163.25.117.117/nplu)

In the directory: / 資訊網絡分析 /Readings 20 Identifying the role that animals play in their social

networks.pdf 21 Graph structure in the Web.pdf

64

Blockmodeling

65

Blockmodeling

Through row column permutation, adjust adjacency matrix to form compact blocks: 0-block 1-block

Social structure emerges from blockmodeling Diagonal block

intra group (community) relationship Off-diagonal block

inter group (community) relationship

66

Boolean Matrix Operations

C = A+B

C = A * B

67

Power Series of Adjacency Matrix

M =

M2 = * =

M3 = ……

4

87

23

51

6

01000000

00001000

01000000

00000000

10010110

00001010

00001101

00000010

01000000

00001000

01000000

00000000

10010110

00001010

00001101

00000010

01000000

00001000

01000000

00000000

10010110

00001010

00001101

00000010

00001000

10010110

00001000

00000000

01001111

10011111

10011110

00001101

68

Reachability Matrix of i Steps

R(M, i) = Mk

= M + M2 + M3 + … + Mi

i

k 1

69

Example of Blockmodeling (1 Step)

4

87

23

51

6

11000000

01001000

01100000

00010000

10011110

00001110

00001111

00000011

clique

R(M, 1)

70

11001000

11011110

01101000

00010000

11011111

10011111

10011111

00001111

Example of Blockmodeling (2 Steps)

4

87

23

51

6

2-clique

R(M, 2)

71

Example of Blockmodeling (3 Steps)

4

87

23

51

6

11011110

11011111

11111110

00010000

11011111

11011111

11011111

10011111

3-clique

R(M, 3)

72

11011111

11011111

11111111

00010000

11011111

11011111

11011111

11011111

Example of Blockmodeling (4 Steps)

10000000

11111111

10111111

10111111

10111111

10111111

10111111

10111111

4

87

23

51

6

4-clique&

LSCC

Bowtie model

R(M, 4)

73

Core/Periphery Structure(Borgatti and Everett, 1999)

Idealdensity matrix

Idealadjacency matrix

0001111

0001111

0001111

0001111

1111111

1111111

1111111

1111111

01

11

C

ED

AB

3

54

6

1

87

2

Hub/Spoke

Core/Periphery

74

Example of Core/Periphery Analysis

3

54

6

1

87

2

0110000

0101100

1010100

0011110

0011100

0111101

0001011

0000011

0001100

0001011

0010101

0010001

1000101

0010111

0000111

0101111

1

32

4

5

87

6

167.0500.0

250.0917.0

CorrelationCoefficient: 0.294

75

Data Collection

Select Top 100 blogs from「部落格觀察」 http://look.urs.tw

Topological parameters

項目

連結網絡

節點數 v

雙向 連結數

ebi

單向 連結數

euni

總連結數 e = (2ebi + euni)

連結密度

d =)1( vv

e

2008年 11月 1日 97 48 156 252 = (2 * 48 + 156) 0.0271

2009年 9月 11日 124 72 230 374 = (2 * 72 + 230) 0.0245

2010年 7月 2日 148 77 252 406 = (2 * 77 + 252) 0.0187

2011年 5月 6日 161 112 305 529 = (2 * 112 + 305) 0.0205

76

77

Analytical Results

Using core/periphery analysis of UCINET Implemented by genetic algorithm

第一次 第二次 第三次 核心/邊陲 分析

連結 網絡

連結密度 矩陣

相關 係數

連結密度 矩陣

相關 係數

連結密度 矩陣

相關 係數

2008年 11月 1日

033.0012.0

027.0436.0 0.344

032.0015.0

013.0518.0 0.317

021.0016.0

033.0265.0 0.311

2009年 9月 11日

027.0011.0

012.0664.0 0.362

019.0013.0

041.0423.0 0.346

013.0013.0

027.0273.0 0.283

2010年 7月 2日

023.0010.0

016.0673.0 0.360

015.0007.0

024.0307.0 0.348

010.0004.0

018.0146.0 0.244

2011年 5月 6日

022.0010.0

021.0360.0 0.327

017.0016.0

012.0576.0 0.398

005.0005.0

037.0075.0 0.202

78

5140

3750

11106

65136

1760

111151

3501

2292

1068

3600

0730

1066

79

Conclusion

Three core communities are found:

Communities

Characteristics

picture article

information technology/

networking application

cuisine/travel

Expansiveness low medium high

Connectedness high medium low

Lab 11: Exploring Core/Periphery Structure

Network>Core/Periphery>…

Suggested Readings (ftp://163.25.117.117/nplu)

In the directory: / 資訊網絡分析 /Readings 22 Models of core-periphery structures.pdf 23 Blockmodels.pdf 24 Social Network Analysis.doc

82

Network positions and social roles:The idea of equivalence

"positions" or "roles" or "social categories" are defined by "relations" among actors.

Two actors have the same "position" or "role" to the extent that their pattern of relationships with other actors is the same.

Three type of equivalence Structural equivalence Automorphic equivalence Regular equivalence

83

Structural Equivalence

Two nodes are said to be exactly structurally equivalent if they have the same relationships to all other nodes.

Seven "structural equivalence classes: {1}, {2}, {3}, {4}, {5, 6}, {7}, and {8, 9}.

1

3 42

765 98

1

3 42

765 98

84

Automorphic Equivalence

The idea of automorphic equivalenceis that sets of actors can be equivalent by being embedded in local structures that have the same patterns of ties -- "parallel" structures.

Five automorphic equivalence classes: {1}, {2, 4}, {3}, {5, 6, 8, 9}, and {7}.

1

3 42

765 98

1

3 42

765 98

85

Regular Equivalence

Two nodes are said to be regularly equivalentif they have the same profile of ties with members of other sets of actors that are also regularly equivalent.

Three regular equivalence classes: {1}, {2, 3, 4}, and {5, 6, 7, 8, 9}.

1

3 42

765 98

1

3 42

765 98

Lab 12: Exploring Equivalence Structure

Network>Roles & Positions>…

Suggested Readings (ftp://163.25.117.117/nplu)

In the directory: / 資訊網絡分析 /Readings 25 Notions of position.pdf 26 Defining and measuring trophic role similarity.pdf