Consider a ring $$caveman network$,$ like the one in Fig. $2.$ In a caveman network, there are cliques $($i$.$e$.,$ fully

connected communities of nodes$).$ We arrange the cliques in a ring so that each clique has individual edges

to its two neighboring cliques. Suppose that there are a total of k cliques $($where k is even$),$ which each have

$\backslash$alpha nodes in them. Consider the following two network partitions: $(1)$ a partition into k communities in which

each of the k cliques is assigned to its own community; and $(2)$ a partition with k$/2$ communities, which each

consist of two neighboring cliques. Let Q$1$ denote the modularity of the first partition, and let Q$2$ denote the

modularity of the second partition.