The degree distribution is one of the most fundamental graph properties of interest for real-world graphs. It has been widely observed in numerous domains that graphs typically have a tailed or scale-free degree distribution. While the average degree is usually quite small, the variance is quite high and there are vertices with degrees at all scales. We focus on the problem of approximating the degree distribution of a large streaming graph, with small storage. We design an algorithm headtail, whose main novelty is a new estimator of infrequent degrees using truncated geometric random variables. We give a mathematical analysis of headtail and show that it has excellent behavior in practice. We can process streams will millions of edges with storage less than 1 and get extremely accurate approximations for all scales in the degree distribution. We also introduce a new notion of Relative Hausdorff distance between tailed histograms. Existing notions of distances between distributions are not suitable, since they ignore infrequent degrees in the tail. The Relative Hausdorff distance measures deviations at all scales, and is a more suitable distance for comparing degree distributions. By tracking this new measure, we are able to give strong empirical evidence of the convergence of headtail.
Finding frequent items, aka heavy hitters," is a classic problem in the data stream model. Cormode and Hadjieleftheriou @cite_0 compare three of the most important algorithms: the algorithm @cite_38 @cite_42 @cite_36 , the algorithm @cite_32 , and the algorithm @cite_23 . Other popular algorithms such as CountSketch @cite_7 and CountMin @cite_18 enable frequent items to be identified when the frequency of an item may be incremented and decremented. For large degrees, these approaches will give accurate results, but the error term dwarfs the degree at smaller scales. We demonstrate this empirically in Section . Much work has been done in approximating frequency moments @cite_10 @cite_15 @cite_22 @cite_0 , but they do not give an estimate for multiple scales. Nor has this work been implemented in practice for large data sets.
@cite_38 @cite_18 @cite_22 @cite_7 @cite_36 @cite_42 @cite_32 @cite_0 @cite_23 @cite_15 @cite_10
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The problem of finding heavy hitters and approximating the frequencies of items is at the heart of many problems in data stream analysis. It has been observed that several proposed solutions to this problem can outperform their worst-case guarantees on real data. This leads to the question of whether some stronger bounds can be guaranteed. We answer this in the positive by showing that a class of counter-based algorithms (including the popular and very space-efficient Frequent and SpacesSaving algorithms) provides much stronger approximation guarantees than previously known. Specifically, we show that errors in the approximation of individual elements do not depend on the frequencies of the most frequent elements, but only on the frequency of the remaining tail. This shows that counter-based methods are the most space-efficient (in fact, space-optimal) algorithms having this strong error bound. This tail guarantee allows these algorithms to solve the sparse recovery problem. Here, the goal is to recover a faithful representation of the vector of frequencies, f. We prove that using space O(k), the algorithms construct an approximation f* to the frequency vector f so that the L1 error ppf−pf*p1 is close to the best possible error minf′ pf′ − fp1, where f′ ranges over all vectors with at most k non-zero entries. This improves the previously best known space bound of about O(k log n) for streams without element deletions (where n is the size of the domain from which stream elements are drawn). Other consequences of the tail guarantees are results for skewed (Zipfian) data, and guarantees for accuracy of merging multiple summarized streams. We present a 1-pass algorithm for estimating the most frequent items in a data stream using limited storage space. Our method relies on a data structure called a COUNT SKETCH, which allows us to reliably estimate the frequencies of frequent items in the stream. Our algorithm achieves better space bounds than the previously known best algorithms for this problem for several natural distributions on the item frequencies. In addition, our algorithm leads directly to a 2-pass algorithm for the problem of estimating the items with the largest (absolute) change in frequency between two data streams. To our knowledge, this latter problem has not been previously studied in the literature. This paper considers the framework of the so-called "market basket problem", in which a database of transactions is mined for the occurrence of unusually frequent item sets. In our case, "unusually frequent" involves estimates of the frequency of each item set divided by a baseline frequency computed as if items occurred independently. The focus is on obtaining reliable estimates of this measure of interestingness for all item sets, even item sets with relatively low frequencies. For example, in a medical database of patient histories, unusual item sets including the item "patient death" (or other serious adverse event) might hopefully be flagged with as few as 5 or 10 occurrences of the item set, it being unacceptable to require that item sets occur in as many as 0.1 of millions of patient reports before the data mining algorithm detects a signal. Similar considerations apply in fraud detection applications. Thus we abandon the requirement that interesting item sets must contain a relatively large fixed minimal support, and adopt a criterion based on the results of fitting an empirical Bayes model to the item set counts. The model allows us to define a 95 Bayesian lower confidence limit for the "interestingness" measure of every item set, whereupon the item sets can be ranked according to their empirical Bayes confidence limits. For item sets of size J > 2, we also distinguish between multi-item associations that can be explained by the observed J(J-1) 2 pairwise associations, and item sets that are significantly more frequent than their pairwise associations would suggest. Such item sets can uncover complex or synergistic mechanisms generating multi-item associations. This methodology has been applied within the U.S. Food and Drug Administration (FDA) to databases of adverse drug reaction reports and within AT&T to customer international calling histories. We also present graphical techniques for exploring and understanding the modeling results. As data mining techniques are being increasingly applied to non-traditional domains, existing approaches for finding frequent itemsets cannot be used as they cannot model the requirement of these domains. An alternate way of modeling the objects in these data sets is to use graphs. Within that model, the problem of finding frequent patterns becomes that of discovering subgraphs that occur frequently over the entire set of graphs.The authors present a computationally efficient algorithm for finding all frequent subgraphs in large graph databases. We evaluated the performance of the algorithm by experiments with synthetic datasets as well as a chemical compound dataset. The empirical results show that our algorithm scales linearly with the number of input transactions and it is able to discover frequent subgraphs from a set of graph transactions reasonably fast, even though we have to deal with computationally hard problems such as canonical labeling of graphs and subgraph isomorphism which are not necessary for traditional frequent itemset discovery.