Question

7.7 Consider an undirected tree of $n$ nodes. Suppose that a particular node in the tree has degree $k$, so that its removal would divide the tree into $k$ disjoint regions, and suppose that the sizes of those regions are $n_1 dots n_k$. a) Show that the unnormalized betweenness centrality $x$ of this node, as defined in Eq. (7.24), is $$x = n^2 - sum_{m=1}^{k} n_m^2$$ b) Hence, or otherwise, calculate the betweenness of the $i$th node from the end of a "line graph" of $n$ nodes, i.e., $n$ nodes in a row like this:

          7.7 Consider an undirected tree of $n$ nodes. Suppose that a particular node in the
tree has degree $k$, so that its removal would divide the tree into $k$ disjoint regions, and
suppose that the sizes of those regions are $n_1 dots n_k$.
a) Show that the unnormalized betweenness centrality $x$ of this node, as defined in
Eq. (7.24), is
$$x = n^2 - sum_{m=1}^{k} n_m^2$$
b) Hence, or otherwise, calculate the betweenness of the $i$th node from the end of a
"line graph" of $n$ nodes, i.e., $n$ nodes in a row like this:

7.7 Consider an undirected tree of n nodes. Suppose that a particular node in the
tree has degree k, so that its removal would divide the tree into k disjoint regions, and
suppose that the sizes of those regions are n1 dots nk.
a) Show that the unnormalized betweenness centrality x of this node, as defined in
Eq. (7.24), is

x = n^2 - summ=1^k nm^2

b) Hence, or otherwise, calculate the betweenness of the ith node from the end of a
"line graph" of n nodes, i.e., n nodes in a row like this:

Added by Jennifer I.

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