SNA 3A: Centrality

Lada Adamic

is counting the edges enough?

Stanford Social Web (ca. 1999)

network of personal homepages at Stanford

Y

X

Y

X

Y X

Y

X

indegree

In each of the following networks, X has higher centrality than Y according to a particular measure

outdegree betweenness closeness

different notions of centrality

Y

X

review: indegree

trade in petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data

Quiz Q:

!  Which countries have high indegree (import petroleum and petroleum products from many others) ! Saudi Arabia ! Japan !  Iraq ! USA ! Venezuela

review: outdegree

Y

X

Nepal

Guyana

Ethiopia

Mauritius

Mali

Lebanon

Barbados

Haiti

Cambodia

Suriname

Guadeloupe

Mauritania

Fiji

Costa Rica

Cote Divoire

Bahamas

Jordan

Angola

Nigeria

CanadaUSA

Argentina

Brazil

MexicoJapan

IranIraq Kuwait

Oman

Saudi Arabia

Untd Arab Em

China HK SAR

Korea Rep. MalaysiaSingapore

Thailand

China

Belgium-Lux

France,Monac

GermanyItalyNetherlands

Spain

UK

SwedenRussian Fed

Australia

Indonesia

Poland

Algeria

Portugal

Libya

Jamaica

Panama

Malta

India

South Africa

VenezuelaColombia

Trinidad Tbg

Bahrain

Norway

Egypt

Gabon

Guatemala

Qatar

Afghanistan

Viet NamTaiwan

Myanmar

Sri Lanka

Pakistan

Nicaragua

Korea D P Rp

Guinea

Cuba

Bangladesh

Senegal

trade in petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data

Quiz Q:

!  Which country has low outdegree but exports a significant quantity (thickness of the edges represents $$ value of export) of petroleum products ! Saudi Arabia ! Japan !  Iraq ! USA ! Venezuela

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trade in crude petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data

Undirected degree, e.g. nodes with more friends are more central.

Assumption: the connections that your friend has don't matter, it is what they can do directly that does (e.g. go have a beer with you, help you build a deck...)

putting numbers to it

divide degree by the max. possible, i.e. (N-1)

normalization

Freeman�s general formula for centralization (can use other metrics, e.g. gini coefficient or standard deviation):

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CD =CD (n

*) −CD (i)[ ]i=1

g∑[(N −1)(N − 2)]

How much variation is there in the centrality scores among the nodes?

maximum value in the network

centralization: skew in distribution

CD = 0.167

CD = 0.167

CD = 1.0

degree centralization examples

example financial trading networks

high in-centralization: one node buying from many others

low in-centralization: buying is more evenly distributed

real-world examples

In what ways does degree fail to capture centrality in the following graphs?

what does degree not capture?

Stanford Social Web (ca. 1999)

network of personal homepages at Stanford

Brokerage not captured by degree

Y

X

constraint

constraint

! intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops?

Y X

betweenness: capturing brokerage

€

CB (i) = g jk (i) /g jkj<k∑

Where gjk = the number of shortest paths connecting jk gjk(i) = the number that actor i is on.

Usually normalized by:

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CB' (i) = CB (i ) /[(n −1)(n − 2) /2]

number of pairs of vertices excluding the vertex itself

betweenness: definition

betweenness on toy networks ! non-normalized version:

betweenness on toy networks ! non-normalized version:

A B C E D

!  A lies between no two other vertices !  B lies between A and 3 other vertices: C, D, and E !  C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E)

!  note that there are no alternate paths for these pairs to take, so C gets full credit

betweenness on toy networks ! non-normalized version:

betweenness on toy networks ! non-normalized version:

A B

C

E

D

!  why do C and D each have betweenness 1?

!  They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: !  ½+½ = 1

Quiz Question

! What is the betweenness of node E?

E

Lada�s old Facebook network: nodes are sized by degree, and colored by betweenness.

betweenness: example

Quiz Q:

! Find a node that has high betweenness but low degree

Quiz Q:

! Find a node that has low betweenness but high degree

closeness

! What if it�s not so important to have many direct friends?

! Or be �between� others

! But one still wants to be in the �middle� of things, not too far from the center

need not be in a brokerage position

Y X

Y

X

Y

X

Y

X

Closeness is based on the length of the average shortest path between a node and all other nodes in the network

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Cc (i) = d(i, j)j=1

N

∑#

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' ( (

−1

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CC' (i) = (CC (i)) /(N −1)

Closeness Centrality:

Normalized Closeness Centrality

closeness: definition

Cc' (A) =

d(A, j)j=1

N

∑

N −1

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((((

−1

=1+ 2+3+ 4

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=104

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−1

= 0.4

A B C E D

closeness: toy example

closeness: more toy examples

Quiz Q:

Which node has relatively high degree but low closeness?