The path length differs for nodes from each level. For a node in level $4,$
we have maximum no. of hops as follows,
Level |
Max. no. of hops |
1 |
3 (3-2-1) |
2 |
3+1 = 4 (3-2-1-2) |
3 |
3 + 2 = 5 (3-2-1-2-3) |
4 |
3 + 3 = 6 (3-2-1-2-3-4) |
So, mean no. of hops for a node in level $4$
$= \dfrac{1.3 + 2.4 + 4.5 + 7.6}{14} =\dfrac{73}{14}$, as we have $1, 2, 4$ and $8$ nodes
respectively in levels $1, 2, 3$ and $4$ and we discard the source one in level $4.$
Similarly, from a level $3$ node we get mean no. of hops,
$= \dfrac{1.2 + 2.3 + 3.4+ 8.5}{14} = \dfrac{60}{14}$
From level $2,$ we get mean no. of hops
$= \dfrac{1.1 +1.2 + 4.3 + 8.4}{14} = \dfrac{47}{14}$
And from level $1,$ we get, mean no. of hops
$= \dfrac {0 + 2.1 + 4.2 + 8.3}{14} = \dfrac{34}{14}$.
So, now we need to find the overall mean no. of hops which will be
$= \dfrac{\text{Sum of mean no. of hops for each node}}{\text{No. of nodes}}$
$= \dfrac{ \dfrac{73}{14} \times 8 + \dfrac{60}{14} \times 4 + \dfrac{47}{14} \times 2 + \dfrac{34}{14} \times 1}{15}$
$= \dfrac{68}{15}$
$= 4.53 $