We investigate the maintenance of overlay networks under massive churn where an adversary may churn a constant fraction an of nodes over the course of O(log(n)) rounds. In particular, the adversary has an almost up-to-date information of the network topology as it can observe an only slightly outdated topology that is at least 2 rounds old. Other than that, we only have the provably minimal restriction that new nodes can only join the network via nodes that have taken part in the network for at least one round. Our contributions are as follows: First, we show that it is impossible to maintain a connected topology if the adversary has up-to-date information about the nodes' connections. As our main result, we present an algorithm that constructs a new overlay — completely independent of all previous overlays — every 2 rounds. Furthermore, each node sends and receives only O(log^3(n)) messages in each round. As part of our solution, we propose the Linearized DeBruijn Swarm (LDS), a highly churn resistant overlay, which will be maintained by the algorithm. However, our approaches can be transferred to a variety of classical P2P topologies where nodes are mapped into the [0, 1)-interval.
Last, the @math framework presented in @cite_6 probably bears the greatest resemblance with our work. In @math the nodes maintain a logical overlay resembling a butterfly network. To ensure robustness each of the butterfly's virtual nodes is simulated by @math nodes. The key difference between our work and @math is the adversary's lateness. Just as @cite_2 @math assumes the adversary to be ( @math , @math )-late, but in return allows the churn to be as high as @math in @math rounds. However, other than @cite_2 @math allows the adversary to be immediate.
@cite_6 @cite_2
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Learning low-dimensional representations of networks has proved effective in a variety of tasks such as node classification, link prediction and network visualization. Existing methods can effectively encode different structural properties into the representations, such as neighborhood connectivity patterns, global structural role similarities and other high-order proximities. However, except for objectives to capture network structural properties, most of them suffer from lack of additional constraints for enhancing the robustness of representations. In this paper, we aim to exploit the strengths of generative adversarial networks in capturing latent features, and investigate its contribution in learning stable and robust graph representations. Specifically, we propose an Adversarial Network Embedding (ANE) framework, which leverages the adversarial learning principle to regularize the representation learning. It consists of two components, i.e., a structure preserving component and an adversarial learning component. The former component aims to capture network structural properties, while the latter contributes to learning robust representations by matching the posterior distribution of the latent representations to given priors. As shown by the empirical results, our method is competitive with or superior to state-of-the-art approaches on benchmark network embedding tasks. We consider a system of @math servers inter-connected by some underlying graph topology @math . Tasks arrive at the various servers as independent Poisson processes of rate @math . Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in @math . Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case @math is a clique. Since the servers are exchangeable in that case, the queue length process is quite tractable, and it has been proved that for any @math , the fraction of servers with two or more tasks vanishes in the limit as @math . For an arbitrary graph @math , the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph @math is said to be @math -optimal or @math -optimal when the occupancy process on @math is equivalent to that on a clique on an @math -scale or @math -scale, respectively. We prove that if @math is an Erd o s-R 'enyi random graph with average degree @math , then it is with high probability @math -optimal and @math -optimal if @math and @math as @math , respectively. This demonstrates that optimality can be maintained at @math -scale and @math -scale while reducing the number of connections by nearly a factor @math and @math compared to a clique, provided the topology is suitably random. It is further shown that if @math contains @math bounded-degree nodes, then it cannot be @math -optimal. In addition, we establish that an arbitrary graph @math is @math -optimal when its minimum degree is @math , and may not be @math -optimal even when its minimum degree is @math for any @math . We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy @math is improved. We introduce small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases due to risk-aversion. We adopt the model of @math -modifiable routing games, a variant of routing games with restricted tolls. We prove that computing the best @math -enforceable flow is @math -hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured @math -enforceable flow whose @math improves fast as @math increases. We show that the complexity of computing such a @math -enforceable flow is determined by the complexity of computing a Nash flow of the original game. Moreover, we prove that the @math of this flow is best possible in the worst-case, in the sense that there are instances where ii¾?the best @math -enforceable flow has the same @math , and iii¾?considering more flexible modifications does not lead to any further improvement. We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol.