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