centrality spring 2012. why do we care? diffusion (practices, information, disease) structure,...
TRANSCRIPT
Centrality
Spring 2012
Why do we care?
• Diffusion (practices, information, disease)• Structure, status, prestige• Seeing, perspective, worldview• Power as relational, constraints as relational• Network location as dependent variable– Explaining outcomes– Supporting strategic “networking”
Example: 2-Step Flow of Communication*
• Micro- macro- link in communications theory– Lazarsfeld on mass media and voting (1940s)– high centrality nodes – opinion leaders – mediate broadcast info flow
• later (Lazarsfeld & Katz (1955)) formalized as two-step flow of communication model: mass media messages filtered through more-exposed central members of social groups.
*Remix of http://www.soc.umn.edu/~knoke/pages/SOC8412.htm
The Question
What Vertices are Most Important?
Everyday Understandings
• Important = prominent• Important = admired• Important = linchpin• Important = listened to• Important = in the know• Important = gate keeper• Important = involved
TranslationsOrdinary Description Possible Network Interpretation
prominent Vertex is “visible” to many other vertices
admired Vertex is “chosen” by many other vertices
listened to Vertex is “received” by many other vertices
in the know Vertex is short distance from many sources of information
linchpin Vertex irreplaceable part
gate keeper Vertex stands between one part of graph and another
involved Vertex connected to many parts of graph
A Simple NetworkA B C D E F G
A - 1 1 1 0 1 0
B - 0 1 1 1 0
C - 1 1 1 0
D - 1 0 0
E - 1 0
F - 1
G -
DEGREEcentrality
𝐶𝐷 (𝑣 𝑖 )=𝑘𝑖
Degree Centrality can Fail to Differentiate
CD
A 4
B 4
C 4
D 4
E 4
F 4
G 1
A B C D E F G
A - 1 1 1 0 1 0
B - 0 1 1 1 0
C - 1 1 1 0
D - 1 0 0
E - 1 0
F - 1
G -
Degree Centrality Can Mislead
CLOSENESScentrality
𝐶𝑐 (𝑣 𝑖 )=1/∑𝑗=1
𝑛
𝑑(𝑣 𝑖𝑣 ,𝑣 𝑗)
Closeness Centrality• Closeness = 1/total distance to other vertices
Compare Two Graphs
• What is the problem here?• How would you fix it?
Compute Closeness Centrality of a Vertex
𝐶𝑐 ( 𝐴 )= 11+1+1
=0.33 𝐶𝑐 ( 𝐴 )= 11+1+1+1+1
=0.2
Normalization
• Adjusting a formula to take into account things like graph size
• Usually by “mapping” values to (0…1) or -1…+1
• For closeness centrality:
• Where n is number of vertices in the graph
Compare Two Graphs
𝐶 ′𝑐 ( 𝐴 )= 31+1+1
=1 𝐶 ′𝑐 ( 𝐴 )= 51+1+1+1+1
=1
Intuitively, both blue vertices should have the same closenesscentrality since both are 1 step away from all other vertices.
BETWEENNESScentrality
Betweenness Centrality
• Fraction of shortest paths that include vertexA B C D E F G
A - 1,1 1,1 1,1 2,4 1,1 2,1
B - 2,4 1,1 1,1 1,1 2,1
C - 1,1 1,1 1,1 2,1
D - 1,1 2,4 3,4
E - 1,1 2,1
F - 1,1
G -
Betweenness Centrality
• Fraction of shortest paths that include vertex
A B C D E F G
A - 1,1 1,1 1,1 2,4 1,1 2,1
B - 2,4 1,1 1,1 1,1 2,1
C - 1,1 1,1 1,1 2,1
D - 1,1 2,4 3,4
E - 1,1 2,1
F - 1,1
G - = 0.75
1 shortest path of 4 goes through A
1 shortest path of 4 goes through A
1 shortest path of 4 goes through A
Example: Calculate betweenness centrality of vertex A
Normalizing Betweenness
• Middle vertices should have same CB?
• Since number of paths vertex COULD be on is (n-1)(n-2)/2 we can use this as our denominator
Calculate Cb(F)
A B C D E F G
A - 1 1 1 4 1 1
B - 4 1 1 1 1
C - 1 1 1 1
D - 1 4 4
E - 1 1
F - 1
G -
Vertex Centrality Comparison
• Usually centrality metrics positively correlated• When not, something interesting going on
Low Degree Low Closeness Low Betweenness
HighDegree
Ego embedded in cluster that is far from the rest of the network
Ego's connections are redundant - communication bypasses him/her
High Closeness
Ego tied to important or 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.
Information Centrality
• Betweenness only uses geodesic paths• Information can also flow on longer paths• Sometimes we hear it through the grapevine
• While betweenness focuses just on the geodesic, information centrality focuses on how information might flow through many different paths, weighted by strength of tie and distance. (Moody)
Information Centrality
Chapter 2 Resistance Distance, Information Centrality, Node Vulnerability and Vibrations in Complex Networks by Ernesto Estrada and Naomichi Hatano
Diagrams by J Moody, Duke U.
EIGENVECTORcentrality
Consider this Example
• The two red nodes have similaramounts of “local” centrality,but different amounts of “global”centrality.
Power/Eigenvector Centrality
• Weakness of degree centrality – it counts your neighbors but not whether or not they count
• Basic ideaego’s centrality is function of neighbors’ centrality
C(ego) = f (C(ego’s neighbors) )
Algorithm
• Assume all vertices have centrality, C = 1• Recalculate C by summing C of neighbors• Repeat the process– Each time we are “taking into account” the
centralities of yet another “layer” of the vertices around us
1
1
1
1
1
1
1
1
1
1
11 1
1
1
1
1
1
1
1 1
11
2
2
3
2
3
2
2
2
2
2
22 2
2
5
5
4
4
2
2 2
44
6
6
7
6
13
6
6
6
7
6
77 6
6
13
13
10
10
7
7 7
189
15
15
22
15
33
16
16
16
20
16
2020 16
16
52
52
36
36
20
20 20
4625
40
40
58
40
126
52
52
52
72
52
7272 52
52
139
139
94
94
72
72 72
20992
• Consider the xy coordinate plane where aline from (0,0) to (x,y) is the vector
• And consider the matrix
• What does this matrix “do” to the vector ?
(x,y)
x
y
1 ½
0 1
x
y
0
1
Matrix Multiplication as Distortion
1 ½
0 1
0
1
½
1
1 ½
0 1
1
0
1
0
BUT
So, what is an Eigenvector?
Eigenvector
• Adjacency matrix redistributes vertex contents
• Some vector of contents is in equilibrium
• These are the eigenvector centralities
What is an Eigenvector?
• Consider a graph & its 5x5 adjacency matrix, A
And then consider a vector, x…
• a 5x1 vector of values, one for each vertex in the graph. In this case, we've used the degree centrality of each vertex.
What happens when…
• …we multiply the vector x by the matrix A?
• The result, of course, is another 5x1 vector.
Axx diffuses the vertex values
• Look at first element of resulting vector• The 1s in the A matrix "pick up" values of each
vertex to which the first vertex is connected • Result value is sum of values of these vertices.
Intuitiveness Visible on Rearrangment