This problem is from Networks: An Introduction by Newmann.
Consider the following variation on the small-world model. Again we have a ring of n vertices in which each is connected to its c nearest neighbors, where c is even. And again a shortcut is added to the network with probability p for each edge around the ring, but now instead of connecting random vertex pairs, each shortcut connects a random vertex to the same single hub vertex in the center of the network:
This model could be, for example, a model of a (one-dimensional) world connected together by a bus or train (the central vertex) whose stops are represented by the shortcuts. Show that the mean distance between two vertices in this network in the limit of large n is $$ l = \frac{2(c{^2}p + 1)}{c^2p} $$ (which is a constant, independent of n).