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Transcript of Lab 4: Working with NetDraw Opening a dataset File>Open Editing node attributes Transform>Node...
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}
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Network Data Collection
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5
12
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3
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… …
… …
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
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12
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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
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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
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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
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Structural Hole
29
Structural Holes
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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
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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
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Formula of Betweenness Centrality
betweenness centrality:
normalized betweenness centrality:
3
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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
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Degree Centralization
Freeman: .07Variance: .20
Freeman: 1.0Variance: 3.9
Freeman: .02Variance: .17
Freeman: 0.0Variance: 0.0
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Homework
Closeness centralization Betweenness centralization
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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
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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
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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
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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
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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%
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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
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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.
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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.
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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
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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
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Blockmodeling
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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
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Boolean Matrix Operations
C = A+B
C = A * B
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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
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Reachability Matrix of i Steps
R(M, i) = Mk
= M + M2 + M3 + … + Mi
i
k 1
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Example of Blockmodeling (1 Step)
4
87
23
51
6
11000000
01001000
01100000
00010000
10011110
00001110
00001111
00000011
clique
R(M, 1)
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11001000
11011110
01101000
00010000
11011111
10011111
10011111
00001111
Example of Blockmodeling (2 Steps)
4
87
23
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6
2-clique
R(M, 2)
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Example of Blockmodeling (3 Steps)
4
87
23
51
6
11011110
11011111
11111110
00010000
11011111
11011111
11011111
10011111
3-clique
R(M, 3)
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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)
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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
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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
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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
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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
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5140
3750
11106
65136
1760
111151
3501
2292
1068
3600
0730
1066
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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
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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
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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
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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
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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