We study the min-cost seed selection problem in online social networks, where the goal is to select a set of seed nodes with the minimum total cost such that the expected number of influenced nodes in the network exceeds a predefined threshold. We propose several algorithms that outperform the previous studies both on the theoretical approximation ratios and on the experimental performance. Under the case where the nodes have heterogeneous costs, our algorithms are the first bi- criteria approximation algorithms with polynomial running time and provable logarithmic performance bounds using a general contagion model. Under the case where the users have uniform costs, our algorithms achieve logarithmic approximation ratio and provable time complexity which is smaller than that of existing algorithms in orders of magnitude. We conduct extensive experiments using real social networks. The experimental results show that, our algorithms significantly outperform the existing algorithms both on the total cost and on the running time, and also scale well to billion-scale networks.
Since @cite_12 , a lot of studies have aimed to design efficient @math SS algorithms. The earlier studies in this line are mostly based on the naive monte-carlo sampling method (e.g., @cite_21 @cite_18 ), and more recent work @cite_17 @cite_23 @cite_19 @cite_22 has leveraged more advanced sampling methods @cite_20 @cite_5 to reduce the time complexity, such as the RR-set sampling method. Besides the RR-set sampling method, Cohen @cite_15 have proposed a min-hash sketch based method for @math SS, which is also used in @cite_1 . Some variations of the @math SS problem have also been studied in @cite_0 @cite_22 @cite_7 . However, all these studies belong to the category of influence maximization (IM) algorithms.
@cite_18 @cite_22 @cite_7 @cite_21 @cite_1 @cite_0 @cite_19 @cite_23 @cite_5 @cite_15 @cite_20 @cite_12 @cite_17
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Abstract We discuss ways of combining rejection sampling and importance sampling methods in Monte Carlo computations and demonstrate their usefulness in updating dynamic systems. Specifically, we propose the rejection controlled sequential importance sampling (RC-SIS) algorithm, which is designed to simultaneously reduce Monte Carlo variation and retain independent samples in sequential importance sampling. The proposed method is demonstrated by three examples taken from econometrics, hierarchical Bayes analysis, and digital telecommunications. They all show significant improvements over previous results. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . We revisit a problem introduced by Bharat and Broder almost a decade ago: how to sample random pages from the corpus of documents indexed by a search engine, using only the search engine’s public interface? Such a primitive is particularly useful in creating objective benchmarks for search engines. The technique of Bharat and Broder suffers from a well-recorded bias: it favors long documents. In this paper we introduce two novel sampling algorithms: a lexicon-based algorithm and a random walk algorithm. Our algorithms produce biased samples, but each sample is accompanied by a weight, which represents its bias. The samples, in conjunction with the weights, are then used to simulate near-uniform samples. To this end, we resort to four well-known Monte Carlo simulation methods: rejection sampling, importance sampling, the Metropolis-Hastings algorithm, and the Maximum Degree method. The limited access to search engines force our algorithms to use bias weights that are only “approximate”. We characterize analytically the effect of approximate bias weights on Monte Carlo methods and conclude that our algorithms are guaranteed to produce near-uniform samples from the search engine’s corpus. Our study of approximate Monte Carlo methods could be of independent interest. Experiments on a corpus of 2.4 million documents substantiate our analytical findings and show that our algorithms do not have significant bias towards long documents. We use our algorithms to collect fresh comparative statistics about the corpora of the Google, MSN Search, and Yahoo! search engines. We propose a new Monte Carlo method, based on dynamic importance sampling, to estimate the probability that a given set of nodes is connected in a graph (or network) where each link is failed with a given probability. The method generates the link states one by one, using a sampling strategy that approximates an ideal zero-variance importance sampling scheme. The approximation is based on minimal cuts in subgraphs. In an asymptotic rare-event regime where failure probability becomes very small, we prove that the relative error of our estimator remains bounded, and even converges to 0 under additional conditions, when the unreliability of individual links converges to 0. The empirical performance of the new sampling scheme is illustrated by examples.