$2.$ Define the path length as being equal to the number of links the path goes through $($known as the hop length$).$ For a connected undirected network with $1,000$ nodes and an average nodal degree of $2,$ design network topologies that will give the minimum and the maximum average shortest path length between all the node pairs, respectively. Calculate the corresponding average shortest path lengths for these two network topologies, respectively. If exact calculation of the average shortest path length is too difficult without running a program on a computer, you may give as$-$tight$-$aspossible lower and upper bounds of the values. Note that we do not allow parallel edges between two nodes and we assume that every edge is bidirectional. Results got by a simulation program will not be accepted.

$(25$ marks$)$