The Confluent Capacity of the Internet: Congestion vs. Dilation

Using shortest paths, the Internet scales very poorly with respect to congestion [2]. Two main reasons for using shortest paths are dilation (or delay) and size of routing tables. As the Internet grows, the small size of routing tables is important for scaling, but it does not require shortest paths. As long as the paths are confluent, the routing table size is unchanged. In this paper we study the confluent capacity of the Internet. We use the preferential attachment model [5] for the Internet, and all-pair uniform demand for the traffic pattern. Our main theoretical result is that the confluent congestion1 is within a logarithmic factor of the optimal splittable congestion and can be achieved using a simple randomized and distributed scheme called Locally Independent Rounding Algorithm (LIRA). We reinforce this result experimentally by employing simulations to demonstrate that for almost all instances the confluent congestion is (nearly) equal to the splittable congestion. Thus we conclude that the Internet scales well using confluent paths. We combine known results on expanders and the expansion properties of the preferential attachment model to show that for almost all Internet-like networks, we can find a confluent flow that simultaneously achieves O(log n)- approximate congestion and O(1)-approximate dilation. We confirm, using simulations, the intuition that confluence does not come at the cost of dilation.