An individual detected as HIV positive in Cuba is asked to provide a list of his/her sexual contacts for the previous 2 years. This allows one to gather detailed information on the spread of the HIV epidemic. Here we study the evolution of the sexual contact graph of detected individuals and also the directed graph of HIV infections. The study covers the Cuban HIV epidemic between the years 1986 and 2004 inclusive and is motivated by an earlier study on the static properties of the network at the end of 2004. We use a variety of advanced graph algorithms to paint a picture of the growth of the epidemic, including an examination of diameters, geodesic distances, community structure and centrality amongst others characteristics. The analysis contrasts the HIV network with other real networks, and graphs generated using the configuration model. We find that the early epidemic starts in the heterosexual population and then grows mainly through MSM (Men having Sex with Men) contact. The epidemic exhibits a giant component which is shown to have degenerate chains of vertices and after 1989, diameters are larger than that expected by the equivalent configuration model graphs. In 1997 there is an significant increase in the detection rate from 73 to 256 detections/year covering mainly MSMs which results in a rapid increase of distances and diameters in the giant component.