Centrality Measures

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# Centrality Measures - PowerPoint PPT Presentation

Centrality Measures. These measure a nodes importance or prominence in the network. The more central a node is in a network the more significant it is to aid in the spread of infection. Walk: A walk is a sequence of nodes connected by edges. Path: A path is a walk with no repeated nodes.

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Centrality Measures

These measure a nodes importance or prominence in the network.

The more central a node is in a network the more significant it is to aid in the spread of infection.

• Walk: A walk is a sequence of nodes connected by edges.
• Path: A path is a walk with no repeated nodes.
• Geodesic: This is the shortest path between two nodes.
• Distance: This is the length of the shortest path between two nodes.
Degree: The degree of a vertex is the number of other vertices to which it is attached.
• Normalized Degree (nDegree): Is the degree divided by the maximum possible degree expressed as a percentage.
• Degree nDegree

------------ ------------

1 3.000 75.000

4 3.000 75.000

3 3.000 75.000

2 2.000 50.000

5 1.000 25.000

• Farness: This is the sum of the lengths of the geodesics to every other node .

(i.e. the sum of the distances to every other every other node).

• Closeness: The reciprocal of farness is closeness.
• Normalized Closeness (nCloseness): Is the closeness divided by the minimum possible farness expressed as a percentage.

nCloseness

------------

1 80.000

2 57.143

3 80.000

4 80.000

5 50.000

Eigenvector: The equation Mx = x can be viewed as a linear transformation that maps a given vector x into a new vector x, where M is the adjacency matrix.The nonzero solutionsof the equation that are obtained by using a value of (known as an eigenvalue) are called the eigenvectors corresponding to that eigenvalue.
• Normalized Eigenvector (nEigenvector): This is the eigenvector divided by the maximum difference possible expressed as a percentage.
• Betweenness: This is a measure of the number of times a node occurs on a geodesic. So, to have a large betweenness centrality, the node must be between many of the nodes via their geodesics.
• Normalized Betweenness (nBetweenness): Is the betweenness divided by the maximum possible betweenness expressed as a percentage.

nBetweenness nEigenvector

------------ ------------

1 16.667 75.954

2 0.000 57.515

3 16.667 75.954

4 50.000 67.140

5 0.000 25.420

Assortative

Random 1

Random 2

Random 3

Scale-Free

nDegree v nCloseness

0.2299

0.9731

0.9531

0.9335

0.8557

nDegree v nBetweenness

0.1461

0.6224

0.7443

0.6791

0.8954

nDegree v nEigenvector

0.9242

0.9673

0.9505

0.9415

0.9352

nCloseness v nBetweenness

0.2466

0.6094

0.6281

0.6628

0.7565

nCloseness v nEigenvector

0.3317

0.9431

0.9554

0.8766

0.9478

nBetweenness v nEigenvector

0.1316

0.4576

0.5464

0.467

0.7694

Correlation

A high correlation tells us there may be an easier way of measuring the centalities on a larger scale project.

For instance, measuring the degree of a farm by observing that farm is much easier than measuring its betweenness or closeness, as we would then have to observe the entire network of farms. So, if measuring degree means we can make assumptions about the values of another centrality then this saves us measuring both centralities. This is providing that we are not essentially measuring the same thing which would inevitably give a high correlation.