Centrality Measures

1
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.

2

• 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

• 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 solutions of 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

4
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.

5
Random
6
Scale-free
7
Assortative
8
Lattice