Constructing the Overlay Network by Tuning Link Weights Huijuan Wang and Piet Van Mieghem Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands Email: {H.Wang, P.VanMieghem}@ewi.tudelft.nl. Abstract— When transport in networks follows the shortest paths, the union of all shortest path trees G∪spt can be regarded as the “transport overlay network”. Overlay networks such as peer-to-peer networks or virtual private networks can be considered as a subgraph of G∪spt . We construct two types of G ∪spt : (a) G ∪spt(α) where α is the extreme value index of polynomial link weights and (b) G ∪spt(ρ) where ρ is the correlation coefficient of the 2-dimensional correlated uniformly distributed link weights in QoS routing. By tuning the extreme value index α of polynomial link weights, a phase transition occurs around a critical extreme value index α c of the link weight distribution. If α>α c , transport in the network traverses many links whereas for α<α c , all transport flows over a critical backbone: the Minimum Spanning Tree (MST). In QoS routing with 2-dimensional link weights, as we decrease the correlation coefficient ρ from 1 to −1, the overlay G ∪spt becomes denser, and is equal to the substrate when ρ = −1. With the Erdös-Rényi random graph as the underlying topology, we show that the overlay G ∪spt(ρ) is also close to an Erdös-Rényi random graph G p (N ), an observation with potential for mobile and wireless ad-hoc networks. The existence of such a controllable transition in the overlay structure may allow network operators to steer and balance flows in their network. I. I NTRODUCTION Routing in communication networks is based on shortest paths(or the best approximation due to e.g. the distracting influence of BGP) between any two nodes of the network. The resources of a network are most efficiently used when traffic follows shortest path [16]. Even for the Internet, it is a reasonable assumption, since roughly 80% of the routes seems to correspond to shortest paths. In this paper, we study the overlay G ∪spt formed by the union of all shortest path trees SPT in a graph G (N,L) with N nodes and L links, where a SPT is the union of the shortest paths from one node to all the other nodes. The relation between the overlay G ∪spt and the underlying graph or substrate G (N,L) is shown in Figure 1. The overlay G ∪spt can be regarded as the “transport overlay network” on top of the network topology or substrate. In the Internet, for example, traffic is carried along the overlay G ∪spt , composed of a fraction of the links in the underlying network, which is just the maximal part of the Internet that we can actually observe by traceroute measurements. The importance of overlay networks is believed to grow in the future. One example of an overlay network is peer- to-peer networks [21] with n distributed systems sharing resources such as content, CPU cycles and storage, where n is smaller than the number of nodes N in the underlying Underlying Topology G(N,L) Overlay Network G Uspt Link weight distribution e.g. i w [0,1) [1, ) () 1 1 w x x Fx x α ∈ ∈∞ = + Fig. 1. The relation between the overlay network and the underlying topology. network. The peer-to-peer overlay network can be regarded as a union of paths connecting these n nodes. Another type of overlay network is a virtual private network (VPN), a private network that uses a public network (usually the Internet or the telephony network) to connect remote sites or users together. The physical networks traversed by both the peer-to- peer and the VPN overlay networks are a subgraph of G ∪spt . The robustness in such overlay networks, the persistence of epidemics [2] and the vulnerability to node failures and attacks [10] are depending on structural properties of G ∪spt that are studied in this paper. The overlay G ∪spt , not the substrate, determines the net- work’s performance: any link removed in G ∪spt will definitely impact at least those flows of traffic that pass over that link. Here we show that, instead of changing the infrastructure of a network [22], the overlay network G ∪spt can be controlled by varying the link weight structure. Current best-effort routing simply computes appropriate paths based on a single, relatively static measure (e.g. the delay, the monetary cost, etc.). Several quality-of-service (QoS) based networking frameworks (e.g., IntServ, DiffServ, MPLS) have been extensively investigated. QoS routing takes into account multiple measures including both the applications requirements’ and the availability of network resources. We present two ways of constructing the overlay G ∪spt : (a) one by changing the single link weight per link as in best-effort routing and (b) another by changing the link weight vector assigned to each link as in QoS routing [13]. The paper is outlined as follows. First, in Section II,