EXERCISE 1.1  A Weighted Betweenness Measure

 

Consider the following variation on the betweenness measure in (1.1). Any given shortest path between two families is weighted by inverse of the number of intermediate

 

 

 

 

nodes on that path. For instance, the shortest path between the Ridolfi and Albizzi involves two links and the Medici are the only family that lies between them on that path. In contrast, between the Ridolfi and the Ginori the shortest path is three links and there are two families, the Medici and Albizzi, that lie between the Ridolfi and Ginori on that path.

More specifically, let l!ij   be the length of the shortest path between nodes i and j and let Wk(ij) = Pk(ij)/(l!ij  - 1), (setting l!ij  = oo and Wk(ij) = 0 if i and j  are not connected).  Then the weighted betweenness measure for a given node k be defined by

 

 

W B =                     Wk(ij)/P (ij)

k                                (n - 1)(n - 2)/2

 

 

                                   (1.5)

 

ij:i/ j,kE/{i,j}

 

where we take the convention that   W k (ij ) = 0/0 = 0 if there are no paths connecting i

and j.

Show that

 

•   W Bk > 0 if and only if k has more than one link in a network and some of k's neighbors are not linked to each other,

 

•   W Bk = 1 for the center node in a star network that includes all nodes (with

n 2" 3), and

•   W Bk 1 unless k is the center node in a star network that contains all nodes.

 

Calculate this measure for the the network pictured in Figure 1.3 for nodes 5 and

6.

Contrast this measure with the betweenness measure in (1.1).