Cohesion
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Cohesion
Dyadic and Whole Network
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Dyadic vs Whole Network
• Dyadic cohesion refers to pairwise social closeness
• Whole network measures can be– Averages of dyadic cohesion– Measures not easily reducible to dyadic
measures
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Measures of Group Cohesion• Density & Average degree• Average Distance and Diameter• Number of components• Fragmentation• Distance-weighted Fragmentation• Cliques per node• Connectivity• Centralization• Core/Peripheriness• Transitivity (clustering coefficient)
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Density• Number of ties, expressed as percentage of the number
of ordered/unordered pairs
Low Density (25%)Avg. Dist. = 2.27
High Density (39%)Avg. Dist. = 1.76
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Help With the Rice Harvest
Data from Entwistle et al
Village 1
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Help With the Rice Harvest
Which village is more likely to survive?
Village 2Data from Entwistle et al
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Average Degree• Average number of
links per person• Is same as
density*(n-1), where n is size of network– Density is just
normalized avg degree – divide by max possible
• Often more intuitive than density
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Density 0.14Avg Deg 4
Density 0.47Avg Deg 4
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Average Distance
• Average geodesic distance between all pairs of nodes
avg. dist. = 1.9 avg. dist. = 2.4
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Diameter
• Maximum distance
Diameter = 3 Diameter = 3
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Fragmentation Measures
• Component ratio• F measure of fragmentation• Distance-weighted fragmentation DF
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I1
I3
W1
W2
W3
W4
W5
W6
W7
W8
W9
S1
S2
S4
• No. of components divided by number of nodes
Component ratio = 3/14 = 0.21
Component Ratio
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F Measure of Fragmentation
• Proportion of pairs of nodes that are unreachable from each other
• If all nodes reachable from all others (i.e., one component), then F = 0
• If graph is all isolates, then F = 1
rij = 1 if node i can reach node j by a path of any lengthrij = 0 otherwise
)1(
21
−−=∑>
nn
rF ji
ij
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Computation Formula for F Measure
• No ties across components, and all reachable within components, hence can express in terms of size of components
)1(
)1(1
−
−−=∑
nn
ssF k
kk
Sk = size of kth component
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Computational ExampleGames Data
I1
I3
W1
W2
W3
W4
W5
W6
W7
W8
W9
S1
S2
S4 = 14/(132*131) = F
Comp Size Sk(Sk-1)1 1 02 1 03 12 132
14 132
0.2747
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Heterogeneity/Concentration
• Sum of squared proportion of nodes falling in each component, where sk gives size of kthcomponent:
• Maximum value is 1-1/n• Can be normalized by dividing by 1-1/n. If we
do, we obtain the F measure
)1(
)1(1
−
−−=∑
nn
ssF k
kk
2
1 ∑ ⎟⎠⎞
⎜⎝⎛−=
k
k
nsH
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Heterogeneity Example
Comp Size Prop Prop^21 1 0.0714 0.00512 1 0.0714 0.00513 12 0.8571 0.7347
14 1.0000 0.7449
Games Data
I1
I3
W1
W2
W3
W4
W5
W6
W7
W8
W9
S1
S2
S4
Heterogeneity = 0.255
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Distance-Weighted Fragmentation
• Use average of the reciprocal of distance– letting 1/∞ = 0
• Bounds– lower bound of 0 when every pair is adjacent to every
other (entire network is a clique)– upper bound of 1 when graph is all isolates
)1(
121
−−=∑>
nnd
F ji ijD
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Connectivity
• Line connectivity λ is the minimum number of lines that must be removed to discon-nect network
• Node connectivity κ is minimum number of nodes that must be removed to discon-nect network
ST
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Core/Periphery Structures
• Does the network consist of a single group (a core) together with hangers-on (a periphery), or
• are there multiple sub-groups, each with their own peripheries?
C/P struct.
Clique struct.
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Kinds of CP/Models
• Partitions vs. subgraphs– just as in cohesive subgroups
• Discrete vs. continuous– classes, or– coreness
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A Core/Periphery Structure
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Blocked/PermutedAdjacency Matrix
C O R E P E R I P H E R Y
C O R E
- 1 1 1 1 - 1 1 1 1 - 1 1 1 1 -
1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1
P E R I P H E R Y
1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1
- 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 -
• Core-core is 1-block• Core-periphery are (imperfect) 1-blocks• Periphery-periphery is 0-block
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Idealized BlockmodelC O R E P E R I P H E R Y
C O R E - 1 1 1 1 - 1 1 1 1 - 1 1 1 1 -
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
P E R I P H E R Y
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 - 0 0 0 0 0 0 -
δ i ji ji f c C O R E o r c C O R E
o t h e r w i s e=
= =⎧⎨⎩
⎫⎬⎭
10
ci = class (core or periphery) that node i is assigned to
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Partitioning a Data Matrix
• Given a graphmatrix, we can randomly assign nodes to either core or periphery
• Search for partition that resembles the ideal
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Assessing Fit to Data
aij = cell in data matrixci = class (core or periphery) that node i is
assigned to
• A Pearson correlation coefficient r(A,D) is b tt
δ i ji ji f c C O R E o r c C O R E
o t h e r w i s e=
= =⎧⎨⎩
⎫⎬⎭
10
ρ δ= ∑ a i j i ji j,
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Alternative ImagesCore Periphery
C - 1 1 1 1 0 0 0 0 0o 1 - 1 1 1 0 0 0 0 0r 1 1 - 1 1 0 0 0 0 0e 1 1 1 - 1 0 0 0 0 0
1 1 1 1 - 0 0 0 0 0
P 0 0 0 0 0 - 0 0 0 0e 0 0 0 0 0 0 - 0 0 0r 0 0 0 0 0 0 0 - 0 0i 0 0 0 0 0 0 0 0 - 0
0 0 0 0 0 0 0 0 0 -
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Alternative ImagesCore Periphery
C - 1 1 1 1 - - - - -o 1 - 1 1 1 - - - - -r 1 1 - 1 1 - - - - -e 1 1 1 - 1 - - - - -
1 1 1 1 - - - - - -
P - - - - - - 0 0 0 0e - - - - - 0 - 0 0 0r - - - - - 0 0 - 0 0I - - - - - 0 0 0 - 0
- - - - - 0 0 0 0 -
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Continuous Model
• Xij ~ CiCj– Strength or probability of tie between node i
and node j is function of product of corenessof each
– Central players are connected to each other– Peripheral players are connected only to core
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Dim 2 ┌───────────┴───────────┴───────────┴───────────┴───────────┴─────────┐│ ││ ││ │
1.85 ┤ ├│ ││ ││ 0 ││ ││ ││ │
1.04 ┤ ├│ ││ ││ 1 ││ 1 ││ ││ 0 │
0.23 ┤ 1 3 ├│ ││ 2 18 3 2 ││ 6 3 ││ 3 3 ││ 2 ││ 0 │
-0.57 ┤ 1 ├│ ││ ││ 1 ││ ││ ││ │
-1.38 ┤ ├│ ││ ││ 0 ││ ││ ││ │└───────────┬───────────┬───────────┬───────────┬───────────┬─────────┘
-1.39 -0.63 0.12 0.88 1.64 Dim 1
Figure 4. MDS of core/perip
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10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8
Month
Group Morale
Core/Periphery-ness
Study by Jeff Johnson of a South Pole scientific team over 8 months
C/P structure seems to affect morale
CP Structures & Morale
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Centralization
• Degree to which network revolves around a single node
Carter admin.Year 1
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Transitivity
• Proportion of triples with 3 ties as a proportion of triples with 2 or more ties– Aka the clustering coefficient
T
A
B C
DE
{C,T,E} is a transitive triple, but {B,C,D} is not. {A,D,T} is not counted at all.
cc = 12/26 = 46.15%
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Dyadic Cohesion
• Adjacency– Strength of tie
• Distance– Length of shortest path between two nodes
• Multiplexity– Number of ties of different relations linking
two nodes• Number of paths linking two nodes
Average is density
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Classifying Cohesion
Cohesion
Distance- Length of paths
Frequency- Number of paths
Adjacencydensity
connectivity