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