Network Matrix and Graph

Network Matrix and Graph
Network Size
Network size – a number of actors (nodes)
in a network, usually denoted as k or n
•Size is critical for the structure of social
relations due to limited capacity of a single
actor for maintaining ties
•In smaller networks, actors are very likely to
be connected to each other, while in larger
networks connecting to everyone else
becomes increasingly difficult
•For directed (asymmetric) network, the
number of possible ties is k*(k-1), for
undirected (symmetric) it is k*(k-1)/2
•The number of possible relationships grows
exponentially as the number of actors
increases linearly and so does network
complexity
Network size=3
N of all possible ties=6
Network size=6
N of all possible ties = 30
Network size=6
N of all possible ties = 30
Network Density
Density - proportion of all possible ties that are actually present
• Sum of existing ties divided by the number of all possible ties
a1
a2
a1
1
a3
a4
a5
a6
sum
1
1
1
1
5
1
1
0
0
3
1
0
0
3
0
0
3
1
2
a2
1
a3
1
1
a4
1
1
1
a5
1
0
0
0
a6
1
0
0
0
1
Density=18/(6*5)=18/30=.6
In UCINET: Network  Cohesion  Density  Density Overall
2
18
Network Density
Density informs about the speed at which information
or resources diffuse among the nodes and the extent
to which actors have high levels of social capital
and/or social constraint
Density=.6
Density=1.0
Components
Component – a part of a network that is connected within, but
disconnected from other parts of a network
Isolate – a single actor disconnected from the rest of the network
Disconnected components mean that network consists of separate subpopulations with no flow of information, resources, influence, etc. between
them
For directed graphs, there are two types of components: 1) a weak
component is a set of nodes that are connected, regardless of the
direction of ties 1) a strong component requires that there be a directed
path from A to B in order for the two to be in the same component
Three components (1,2,3,4), (5,6,7,8),(9)
One isolate (9).
In UCINET for binary data: Network  Regions  Components  Simple graphs
(For valued graphs: Network  Regions  Components  Valued graphs )
Reachability
Reachability tells us whether two actors are connected or not by way of
either a direct or an indirect pathways of any length
•If some actors in a network cannot reach others:
– there is the potential for disruption of the flow of information and
resources
– this may indicate that the population we are studying is really
composed of more than one sub-populations
In UCINET: Network  Cohesion  Reachability
a1 is reachable only from a4,a6
a2 is reachable only from a1, a4 and a6
a6 is unreachable to the nodes in
the larger component. Everyone in
the connected component is
reachable to each other.
Geodesic Distance
A walk is a sequence of actors and relations that begins and ends with actors
Geodesic distance - the number of relations in the shortest possible walk from
one actor to another
• The geodesic path (or paths, as there can be more than one) is often the
"optimal" or most "efficient" connection between two actors
Diameter - the largest geodesic distance in the (connected) network
In UCINET: Network  Cohesion  Geodesic Distances
Geodesic Distances
123456
aaaaaa
- - - -- 1 a1 0 1 2 1 x 1 5
2 a2 x 0 x x x x 0
3 a3 x x 0 x x x 0
4 a4 2 3 1 0 x 1 7
5 a5 x x x 1 0 x 1
6 a6 1 2 1 2 x 0 6
SUM
19
(x means no path)
Diameter is 3
The geodesic distance from a1to a2=1, from a1 to a3=2, from a1 to a4=1,
from a1 to a6=1 and there is no geodesic distance from a1 to a5.
The average geodesic among reachable pairs is (19/13=)1.462.
Blocks, Cutpoints, and Bridges
Cut-point - a node, removal of which
would break up a network into
disconnected parts
Blocks – parts into which cut-points
divide a network
Bridge – a tie between two nodes,
removal of which would break up a
network into disconnected parts
Cut-points may act as brokers among
otherwise disconnected groups
Cut-points and bridges are network’s
weak spots vulnerable to disruptions in
the flow of information, resources, and
influence
No cut-points, no bridges
a1 is a cut-point, no bridges
Two blocks: (a5, a6) and (a2, a3, a4)
1 and 2 are cut-points a tie
Tie between 1 and 2 is a bridge
In UCINET for binary data: Network  Regions  Bi-components
Centrality
• Look at the positions of actor A
in these three networks
• In which network does actor A
have more favorable and less
favorable position?
• Actor D?
• How equally is centrality
distributed in these networks?
“Line” network
"Star” network
“Circle” network
Degree Centrality
• More choices/alternatives means more
opportunities and less
dependence/constraint
• The more ties actor have, the more
power they may have
• Degree centrality is the number of
connections an actor (node) has
• When ties are directed, we calculate total
number of ties sent (out-degree) and
ties received (in-degree)
• Out-degree typically indicates influence,
in-degree indicates prestige or popularity
In UCINET: Network  Centrality and Power
Degree
Select whether to treat data as symmetric or not.
A has 6 connections,
B,C, D, E, F, G have 1 connection
A B,C, D, E, F, G have 2
connections
Closeness Centrality
•
•
•
•
•
•
•
•
Degree centrality does not take into account indirect
ties an actor has
Closeness centrality emphasizes the distance of an
actor to all others in the network
The closer one is to others in the network, the more
favored is that actor’s position
Farness is the sum of the geodesic distances from
each ego to all others in the network “Closeness" is
the reciprocal of farness = 1/farness
“nCloseness” is Closeness times the number of
alters
Degree centrality measures one’s local position,
while closeness centrality measures position globally
Closeness is indefinite for disconnected nodes
Closeness is meaningful only for a connected
network
In UCINET: Network  Centrality and Power 
Closeness (old)
Select whether to treat data as symmetric or not. Interpret
results cautiously if a network is disconnected.
Farness: A gets in 1 step to each of
the other 6, 6*1=6
The rest get to A in 1 step and to the
other 5 in 2, 1+5*2=11
Closeness: A (1/6)=.167
nCloseness: A 100*(7-1)*1/6=100
The rest 100*(7-1)*1/11=54.545.
Farness: 2*1+2*2+2*3=12
Closeness: 1/12=.08
nCloseness: 100*(7-1)*1/12=50
Betweenness Centrality
• The more people depend on an actor
to make connections with others, the
more power that actor has
• Betweenness centrality is the extent
to which an actor falls on the
geodesic paths between other pairs
of actors in the network
All of them are on 3 geodesic paths.
E.g. A is between B&G, B&F and C&G.
A is in all geodesic paths between any pair of
the others. There are 6 others and 6*5/2=15
ways you can get from any of the 6 to any of
the other 5, so for A=15. For the rest 0.
In UCINET: Network  Centrality and Power Freeman Betweenness  Node Betweenness
Centrality vs. Centralization
• Centrality is a characteristic of an actor’s position in a network
• Centralization is a characteristic of a network
• Centralization indicates:
– how unequal the distribution of centrality is in a network or
– how much variance there is in the distribution of centrality in a network
• Centrality is a micro-level measure
• Centralization is a macro-level measure
– There are as many centralization measures as centrality measures
• To find centralization,
– a. you must find the most central actor C*
– b. take its centrality score and subtract the centrality score of each of
the others, Ci, from it
– c. add up the differences: Σ(C*-Ci)
– d. then divide this by what this sum would be under the largest possible
centralization (if the network was a star) : Max Σ(C*-Ci)
– e. multiply this by 100 to turn it into a percentage
• Centralization= 100* Σ(C*-Ci) / Max Σ(C*-Ci)
Centralization (symmetric matrices)
•
•
Note: all three networks are symmetric
Star: 100%, Circle: 0% on all measures of centralization.
• Degree Centralization
•
•
•
•
For the third network: The most central actor is a1
Σ(C*-Ci)=(5-3)+(5-3)+(5-3)+(5-2)+(5-2)=2+2+2+3+3=12
Max. centralization: one actor has 5 ties, all the rest have 1.
For symmetric matrices:
–
•
•
Max. Degree Centralization=(N-1)*N-2)
Max Σ(C*-Ci)=(5-1)+(5-1)+(5-1)+(5-1)+(5-1)=5*4=20
Centralization=100*12/20=60 or 60%
1
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2
3
Degree NrmDegree Share
------------ ----------- -----------1 a1 (C*) 5.000
100.000
0.278
2 a2
3.000
60.000
0.167
3 a3
3.000
60.000
0.167
4 a4
3.000
60.000
0.167
5 a5
2.000
40.000
0.111
6 a6
2.000
40.000
0.111
Network Centralization = 60.00%
•
In UCINET: Network  Centrality and Power Degree Select whether to treat data as symmetric or not.
Centralization (symmetric matrices)
• Closeness Centralization
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•
•
•
•
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The most central actor is a1.
– Σ(C*-Ci)=(100-71.429)+(100-71.429)+(100-71.429)+(10062.5)+(100-62.5)=3*28.571+2*37.5=160.713
Max. centralization:
one actor gets to all others in one step:
– sum of geodesic distance or farness=(n-1)*1,
– nCloseness=100*(n-1)*1/(n-1)*1=100,
all the others will get to central actor in one step and everyone else in 2
steps:
– sum of geodesic distance or farness=1+2(n-2),
– nCloseness=100*(n-1)*1/(1+2(n-2))=100*5/9=55.55556
Max Σ(C*-Ci)= (100-55.5556)+(100-55.5556)+(100-55.5556)+(10055.5556)+(100-55.5556)=5*44.4444=222.2222
Centralization=100*160.713/222.2222=72.32 or 72.32%
•
1
2
•
•
•
•
•
•
•
•
Farness nCloseness
----------------------1 a1 (C*) 5.000
100.000
2 a2
7.000
71.429
3 a3
7.000
71.429
4 a4
7.000
71.429
5 a5
8.000
62.500
6 a6
8.000
62.500
•
Network Centralization = 72.32%
In UCINET: Network  Centrality and Power  Closeness (old)
Select whether to treat data as symmetric or not.
Centralization (symmetric matrices)
• Betweenness centralization
•
•
•
•
•
•
The most central actor is a1.
– Σ(C*-Ci)=(6-0)+(6-0)+(6-0)+(6-0)+(6-0)=5*6=30
Max. centralization:
one actor gets in between all the others, no matter how you pair them:
– The others can be paired in (n-1)*(n-2)/2 ways so the central actor will
get a score of (6-1)*(6-2)/2=10
The others are not between any pair of actors
Max Σ(C*-Ci)= (10-0)+(10-0)+(10-0)+(10-0)+(10-0)=50
Centralization=100*30/50=60 or 60%
•
(It is a coincidence that Degree and Betweenness Centralization takes the same value
in this particular case.)
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1
2
Betweenness nBetweenness
------------ -----------1 a1 (C*) 6
60
2 a2
0
0
3 a3
0
0
4 a4
0
0
5 a5
0
0
6 a6
0
0
• Network Centralization Index = 60.00%
In UCINET: Network  Centrality and Power Freeman Betweenness  Node Betweenness
Centralization symmetric vs. asymmetric
matrices
•
•
The same matrix will give different centralization scores depending on whether it is analyzed as symmetric or
asymmetric. Asymmetric matrices will have an Indegree and an Outdegree measure of centralization.
Take the Star matrix. As a symmetric matrix, it has a maximum centralization score (100%). But if we analyze it
as a matrix of asymmetric relationships that just happen to show a symmetric pattern it is less centralized. Take
Degree Centralization in a Star matrix where the asymmetric relationships happen to be symmetrical:
•
OutDegree
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InDegree NrmOutDeg NrmInDeg
------------ ------------ ------------ -----------1A
6.000
6.000
100.000
100.000
2B
1.000
1.000
16.667
16.667
3C
1.000
1.000
16.667
16.667
4D
1.000
1.000
16.667
16.667
5E
1.000
1.000
16.667
16.667
6F
1.000
1.000
16.667
16.667
7G
1.000
1.000
16.667
16.667
•
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Network Centralization (Outdegree) = 83.333%
Network Centralization (Indegree) = 83.333%
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Now, centralization is not 100%. It is because the Maximum Centralization is different. In this case, the Max.
Centralization (Outdegree) occurs when only one actor (A) has out-ties and none of the others do. (The same
goes for Indegree centralization: it reaches maximum when only one actor has in-ties and none of the others do.)
•
100*(6*(6-1))/6*(6-0)=100*30/36=83.333
•
If only A has out-ties then it will have no in-ties. In that case, the Max. Indegree Centralization would be close to
zero as 6 out of 7 actors (B—G) would have exactly the same number of in-ties (1 coming from A) and A would
have none (0).
•
•
Outdegree 100*(6*(6-0)/6*(6-0)=100
Indegree 100*(5*(1-1)+(1-0))/6*(6-0)=100*1/36= 2.778%