We study the stability of random scale-free networks to degree-dependent attacks. We present analytical and numerical results to compute the critical fraction p_{c} of nodes that need to be removed for destroying the network under this attack for different attack parameters. We study the effect of different defense strategies, based on the addition of a constant number of links on network robustness. We test defense strategies based on adding links to either low degree, middegree or high degree nodes. We find using analytical results and simulations that the middegree nodes defense strategy leads to the largest improvement to the network robustness against degree-based attacks. We also test these defense strategies on an internet autonomous systems map and obtain similar results.

In recent years there has been a growing interest in developing "streaming algorithms" for efficient processing and querying of continuous data streams. These algorithms seek to provide accurate results while minimizing the required storage and the processing time, at the price of a small inaccuracy in their output. A fundamental query of interest is the intersection size of two big data streams. This problem arises in many different application areas, such as network monitoring, database systems, data integration and information retrieval. In this paper we develop a new algorithm for this problem, based on the Maximum Likelihood (ML) method. We show that this algorithm outperforms all known schemes and that it asymptotically achieves the optimal variance. 1. For the first time, we present a complete analysis of the statistical performance (bias and variance) of the above three schemes.2. We find the optimal (minimum) variance of any unbiased set intersection estimator.3. We present and analyze a new unbiased estimator, based on the Maximum Likelihood (ML) method, which outperforms the above three schemes.The rest of the paper is organized as follows. Section 2 discusses previous work and presents the three previously known schemes. Section 3 presents our new Maximum Likelihood (ML) estimator. It also shows that the new scheme achieves optimal variance and that it outperforms the three known schemes. Section 4 analyzes the statistical performance (bias and variance) of the three known schemes. Section 5 presents simulation results confirming that the new ML estimator outperforms the three known schemes. Finally, Section 6 concludes the paper.

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Cardinality estimation algorithms receive a stream of elements whose order might be arbitrary, with possible repetitions, and return the number of distinct elements. Such algorithms usually seek to minimize the required storage and processing at the price of inaccuracy in their output. Real-world applications of these algorithms are required to process large volumes of monitored data, making it impractical to collect and analyze the entire input stream. In such cases, it is common practice to sample and process only a small part of the stream elements. This paper presents and analyzes a generic algorithm for combining every cardinality estimation algorithm with a sampling process. We show that the proposed sampling algorithm does not affect the estimator's asymptotic unbiasedness, and we analyze the sampling effect on the estimator's variance.

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