$p\in [P]$, i.e., the values of $q_p$ are as close to each other as possible. This corresponds to the class of topologies with the highest delivery latency (see Fig. \[fig:f1\](c) for an example). The topology that requires the minimum delivery latency of $T_{sd} =\frac{K-t}{t+1}$ is when $q_p$ is either $0$ or $K$ for each server, or equivalently, when all the users are connected to the same $\rho$ servers (see Fig. \[fig:f1\](b) for an example).

Next we study the average worst-case normalized delivery latency, where the average is taken over all possible network topologies. As we have seen above, the delivery latency depends on the topology, and for a given topology, the “worst-case” delivery latency refers to the worst-case demand combination when each user requests a different file. Note that, in the worst case, due to the symmetry in the network and the proposed caching and delivery scheme, the latency depends only on the type of the network topology. We further assume that the probability of having any network of the same type is the same.

\[lemma\_prob\] Let $w_i$ be the probability of exactly $i$ users being served by a server; that is, $w_i=Pr\{ q_p=i \}, p\in [P]$. We have $$\begin{aligned}
