2023-226

 

Imagine a Cayley Tree, constructed starting from a central node of degree \( k \). Each node at distance \( d \) from the central node has degree \( k \), until the nodes reachable at distance \( P \) that have degree one, as shown in figure above. Consider \( k+1 \) communities, with communities corresponding to the subtrees rooted at the children of the central node, and the central node by itself being a community. What is the modularity of this partition if \( k=3 \) and \( P=4 \)? Round it up to three decimal places.

1. 0.523
2. 0.621
3. 0.785
4. 0.891
5. None of the above.

Original idea by: Germán Darío Buitrago Salazar