Network Centrality: Measuring Importance in Networks, Slides of Data Communication Systems and Computer Networks

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Lecture 5:

Network centrality

Measures and Metrics

• Knowing the structure of a network, we can calculate

various useful quantities or measures that capture particular features of the network topology.

• basis of most of such measures are from social network analysis
• So far,
• Degree distribution, Average path length, Density
• Centrality
• Degree, Eigenvector, Katz, PageRank, Hubs, Closeness,

Betweenness, ….

• Several other graph metrics
  • Clustering coefficient, Assortativity, Modularity, …

2

network centrality

• Which nodes are most ‘central’?
• Definition of ‘central’ varies by context/purpose
• Local measure:
  • degree
• Relative to rest of network:
  • closeness, betweenness, eigenvector (Bonacich power

centrality), Katz, PageRank, …

• How evenly is centrality distributed among nodes?
  • Centralization, hubs and autthorities, …

4

centrality: who’s important based on their

network position

indegree

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

a particular measure

outdegree betweenness closeness

5

He who has many friends is most important.

degree centrality (undirected)

When is the number of connections the best centrality measure?

o people who will do favors for you

o people you can talk to (influence set, information access, …)

o influence of an article in terms of citations (using in-degree)

7

degree: normalized degree centrality

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

8

Extensions of undirected degree centrality - prestige

• degree centrality
  • indegree centrality
    • a paper that is cited by many others has high prestige
    • a person nominated by many others for a reward has

high prestige

10

Freeman’s general formula for centralization:

(can use other metrics, e.g. gini coefficient or standard deviation)

C (^) D =

C (^) D ( n

[ )^ −^ C^ D ( i )]

i = 1

g

∑

[( N −1)( N − 2)]

centralization: how equal are the nodes?

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

maximum value in the network

11

degree centralization examples

example financial trading networks

high centralization : one node trading with many others

low centralization : trades are more evenly distributed

13

when degree isn’t everything

• ability to broker between groups
• likelihood that information originating

anywhere in the network reaches you… 14

In what ways does degree fail to capture centrality in the

following graphs?

betweenness: another centrality

measure

• intuition : how many pairs of individuals would

have to go through you in order to reach one

another in the minimum number of hops?

• who has higher betweenness, X or Y?

16

Y X

C B ( i ) = g jk ( i ) / g jk

j < k

Where g jk = the number of geodesics connecting j-k , and

g jk = the number that actor i is on.

Usually normalized by:

C B

'

( i ) = C B ( i ) /[( n −1)( n − 2) /2]

number of pairs of vertices excluding the vertex itself

betweenness centrality: definition

17

betweenness of vertex i

paths between j and k that pass through i

all paths between j and k

directed graph: (N-1)*(N-2)

betweenness on toy networks

• non-normalized version:

19

betweenness on toy networks

• non-normalized version:

20

broker