Sample Complexity Bounds for Influence Maximization

Influence maximization (IM) is the problem of finding for a given s ? 1 a set S of |S|=s nodes in a network with maximum influence. With stochastic diffusion models, the influence of a set S of seed nodes is defined as the expectation of its reachability over simulations, where each simulation specifies a deterministic reachability function. Two well-studied special cases are the Independent Cascade (IC) and the Linear Threshold (LT) models of Kempe, Kleinberg, and Tardos [Kempe et al., 2003]. The influence function in stochastic diffusion is unbiasedly estimated by averaging reachability values over i.i.d. simulations. We study the IM sample complexity: the number of simulations needed to determine a (1-?)-approximate maximizer with confidence 1-?. Our main result is a surprising upper bound of O(s ? ?^{-2} ln (n/?)) for a broad class of models that includes IC and LT models and their mixtures, where n is the number of nodes and ? is the number of diffusion steps. Generally ? ? n, so this significantly improves over the generic upper bound of O(s n ?^{-2} ln (n/?)). Our sample complexity bounds are derived from novel upper bounds on the variance of the reachability that allow for small relative error for influential sets and additive error when influence is small. Moreover, we provide a data-adaptive method that can detect and utilize fewer simulations on models where it suffices. Finally, we provide an efficient greedy design that computes an (1-1/e-?)-approximate maximizer from simulations and applies to any submodular stochastic diffusion model that satisfies the variance bounds

Space-Efficient Estimation of Statistics Over Sub-Sampled Streams

In many stream monitoring situations, the data arrival rate is so high that it is not even possible to observe each element of the stream. The most common solution is to subsample the data stream and use the sample to infer properties and estimate aggregates of the original stream. However, in many cases, the estimation of aggregates on the original stream cannot be accomplished through simply estimating them on the sampled stream, followed by a normalization. We present algorithms for estimating frequency moments, support size, entropy, and heavy hitters of the original stream, through a single pass over the sampled stream

Load shedding in network monitoring applications

Monitoring and mining real-time network data streams are crucial operations for managing and operating data networks. The information that network operators desire to extract from the network traffic is of different size, granularity and accuracy depending on the measurement task (e.g., relevant data for capacity planning and intrusion detection are very different). To satisfy these different demands, a new class of monitoring systems is emerging to handle multiple and arbitrary monitoring applications. Such systems must inevitably cope with the effects of continuous overload situations due to the large volumes, high data rates and bursty nature of the network traffic. These overload situations can severely compromise the accuracy and effectiveness of monitoring systems, when their results are most valuable to network operators. In this thesis, we propose a technique called load shedding as an effective and low-cost alternative to over-provisioning in network monitoring systems. It allows these systems to handle efficiently overload situations in the presence of multiple, arbitrary and competing monitoring applications. We present the design and evaluation of a predictive load shedding scheme that can shed excess load in front of extreme traffic conditions and maintain the accuracy of the monitoring applications within bounds defined by end users, while assuring a fair allocation of computing resources to non-cooperative applications. The main novelty of our scheme is that it considers monitoring applications as black boxes, with arbitrary (and highly variable) input traffic and processing cost. Without any explicit knowledge of the application internals, the proposed scheme extracts a set of features from the traffic streams to build an on-line prediction model of the resource requirements of each monitoring application, which is used to anticipate overload situations and control the overall resource usage by sampling the input packet streams. This way, the monitoring system preserves a high degree of flexibility, increasing the range of applications and network scenarios where it can be used. Since not all monitoring applications are robust against sampling, we then extend our load shedding scheme to support custom load shedding methods defined by end users, in order to provide a generic solution for arbitrary monitoring applications. Our scheme allows the monitoring system to safely delegate the task of shedding excess load to the applications and still guarantee fairness of service with non-cooperative users. We implemented our load shedding scheme in an existing network monitoring system and deployed it in a research ISP network. We present experimental evidence of the performance and robustness of our system with several concurrent monitoring applications during long-lived executions and using real-world traffic traces.Postprint (published version