This paper examines graph-theoretic properties of existing peer-to-peer architectures and proposes a new infrastructure based on optimal-diameter de Bruijn graphs. Since generalized de Bruijn graphs possess very short average routing distances and high resilience to node failure, they are well suited for structured peer-to-peer networks. Using the example of Chord, CAN, and de Bruijn, we first study routing performance, graph expansion, and clustering properties of each graph. We then examine bisection width, path overlap, and several other properties that affect routing and resilience of peer-to-peer networks. Having confirmed that de Bruijn graphs offer the best diameter and highest connectivity among the existing peer-to-peer structures, we offer a very simple incremental building process that preserves optimal properties of de Bruijn graphs under uniform user joins/departures. We call the combined peer-to-peer architecture ODRI -Optimal Diameter Routing Infrastructure.

To model P2P networks that are commonly faced with high rates of churn and random departure decisions by end-users, this paper investigates the resilience of random graphs to lifetime-based node failure and derives the expected delay before a user is forcefully isolated from the graph and the probability that this occurs within his/her lifetime. Using these metrics, we show that systems with heavy-tailed lifetime distributions are more resilient than those with light-tailed (e.g., exponential) distributions and that for a given average degree, -regular graphs exhibit the highest level of fault tolerance. As a practical illustration of our results, each user in a system with = 100 billion peers, 30-minute average lifetime, and 1-minute node-replacement delay can stay connected to the graph with probability 1 1 using only 9 neighbors. This is in contrast to 37 neighbors required under previous modeling efforts. We finish the paper by observing that many P2P networks are almost surely (i.e., with probability 1 (1)) connected if they have no isolated nodes and derive a simple model for the probability that a P2P system partitions under churn.Index Terms-Lifetime node failure, network disconnection, peer-to-peer networks, stochastic resilience, user isolation.

Previous analytical results on the resilience of unstructured P2P systems have not explicitly modeled heterogeneity of user churn (i.e., difference in online behavior) or the impact of in-degree on system resilience. To overcome these limitations, we introduce a generic model of heterogeneous user churn, derive the distribution of the various metrics observed in prior experimental studies (e.g., lifetime distribution of joining users, joint distribution of session time of alive peers, and residual lifetime of a randomly selected user), derive several closed-form results on the transient behavior of in-degree, and eventually obtain the joint in/out degree isolation probability as a simple extension of the out-degree model in [20].

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Recent research efforts to design better Internet transport protocols combined with scalable Active Queue Management (AQM) have led to significant advances in congestion control. One of the hottest topics in this area is the design of discrete congestion control algorithms that are asymptotically stable under heterogeneous feedback delay and whose control equations do not explicitly depend on the RTTs of end-flows. In this paper, we show that max-min fair congestion control methods with a stable symmetric Jacobian remain stable under arbitrary feedback delay (including heterogeneous directional delays) and that the stability condition of such methods does not involve any of the delays. To demonstrate the practicality of the obtained result, we change the original controller in Kelly's work [14] to become robust under random feedback delay and fixed constants of the control equation. We call the resulting framework Maxmin Kelly Control ( M K C ) and show that it offers smooth sending rate, exponential convergence to efficiency, and fast convergence to fairness, all of which make it appealing for future high-speed networks.