Abstract In this work, we present the first efficient and practical algorithms for estimating the number of triangles in a graph stream using predictions . Our algorithms combine waiting room sampling and uniform sampling schemes with a predictor for the heaviness of edges, that is, the number of triangles in which an edge is involved. As a result, our algorithms are fast, provide guarantees on the amount of memory used, and exploit the additional information provided by the predictor to produce highly accurate estimates. We also propose a simple and domain-independent predictor, based on the degree of nodes, that can be easily computed with one pass on a stream of edges when the stream is available beforehand. Our analytical results show that, when the predictor provides useful information on the heaviness of edges, it leads to estimates with reduced variance compared to the state-of-the-art, even when the predictions are far from perfect. Our experimental results show that, when analyzing a single graph stream, our algorithms are faster than the state-of-the-art for a given memory budget, while providing significantly more accurate estimates. Even more interestingly, when sequences of hundreds of graph streams are analyzed, our algorithm significantly outperforms the state-of-the-art using our simple degree-based predictor built by analyzing only the first graph of the sequence. We also present a method to dynamically update the degree-based predictor to maintain high-quality predictions as the streams in the sequence are processed, leading to improved estimates.

In this work, we present the first efficient and practical algorithm for estimating the number of triangles in a graph stream using predictions. Our algorithm combines waiting room sampling and reservoir sampling with a predictor for the heaviness of edges, that is, the number of triangles in which an edge is involved. As a result, our algorithm is fast, provides guarantees on the amount of memory used, and exploits the additional information provided by the predictor to produce highly accurate estimates. We also propose a simple and domain-independent predictor, based on the degree of nodes, that can be easily computed with one pass on a stream of edges when the stream is available beforehand. Our analytical results show that, when the predictor provides useful information on the heaviness of edges, it leads to estimates with reduced variance compared to the state-of-the-art, even when the predictions are far from perfect. Our experimental results show that, when analyzing a single graph stream, our algorithm is faster than the state-of-the-art for a given memory budget, while providing significantly more accurate estimates. Even more interestingly, when sequences of hundreds of graph streams are analyzed, our algorithm significantly outperforms the state-of-the-art using our simple degree-based predictor built by analyzing only the first graph of the sequence.

Learning interpretable models has become a major focus of machine learning research, given the increasing prominence of machine learning in socially important decision-making. Among interpretable models, rule lists are among the best-known and easily interpretable ones. However, finding optimal rule lists is computationally challenging, and current approaches are impractical for large datasets.

Finding dense subnetworks, with density based on edges or more complex structures, such as subgraphs or k-cliques, is a fundamental algorithmic problem with many applications. While the problem has been studied extensively in static networks, much remains to be explored for temporal networks.

Abstract Many algorithms have been proposed to learn local graphical structures around target variables of interest from observational data, focusing on two sets of variables. The first one, called Parent–Children (PC) set, contains all the variables that are direct causes or consequences of the target while the second one, known as Markov boundary (MB), is the minimal set of variables with optimal prediction performances of the target. In this paper we introduce two novel algorithms for the PC and MB discovery tasks with rigorous guarantees on the Family-Wise Error Rate (FWER), that is, the probability of reporting any false positive in output. Our algorithms use Rademacher averages, a key concept from statistical learning theory, to properly account for the multiple-hypothesis testing problem arising in such tasks. Our evaluation on simulated data shows that our algorithms properly control for the FWER, while widely used algorithms do not provide guarantees on false discoveries even when correcting for multiple-hypothesis testing. Our experiments also show that our algorithms identify meaningful relations in real-world data.

Significant pattern mining is a fundamental task in mining transactional data, requiring to identify patterns significantly associated with the value of a given feature, the target. In several applications, such as biomedicine, basket market analysis, and social networks, the goal is to discover patterns whose association with the target is defined with respect to an underlying population, or process, of which the dataset represents only a collection of observations, or samples. A natural way to capture the association of a pattern with the target is to consider its statistical significance , assessing its deviation from the (null) hypothesis of independence between the pattern and the target. While several algorithms have been proposed to find statistically significant patterns, it remains a computationally demanding task, and for complex patterns such as subgroups, no efficient solution exists. We present FSR, an efficient algorithm to identify statistically significant patterns with rigorous guarantees on the probability of false discoveries. FSR builds on a novel general framework for mining significant patterns that captures some of the most commonly considered patterns, including itemsets, sequential patterns, and subgroups. FSR uses a small number of resampled datasets, obtained by assigning i.i.d. labels to each transaction, to rigorously bound the supremum deviation of a quality statistic measuring the significance of patterns. FSR builds on novel tight bounds on the supremum deviation that require to mine a small number of resampled datasets, while providing a high effectiveness in discovering significant patterns. As a test case, we consider significant subgroup mining, and our evaluation on several real datasets shows that FSR is effective in discovering significant subgroups, while requiring a small number of resampled datasets.

“Sim Sala Bim!” —Silvan, https://en.wikipedia.org/wiki/Silvan_(illusionist) Betweenness centrality is a popular centrality measure with applications in several domains and whose exact computation is impractical for modern-sized networks. We present SILVAN , a novel, efficient algorithm to compute, with high probability, accurate estimates of the betweenness centrality of all nodes of a graph and a high-quality approximation of the top- k betweenness centralities. SILVAN follows a progressive sampling approach and builds on novel bounds based on Monte Carlo Empirical Rademacher Averages, a powerful and flexible tool from statistical learning theory. SILVAN relies on a novel estimation scheme providing non-uniform bounds on the deviation of the estimates of the betweenness centrality of all the nodes from their true values and a refined characterisation of the number of samples required to obtain a high-quality approximation. Our extensive experimental evaluation shows that SILVAN extracts high-quality approximations while outperforming, in terms of number of samples and accuracy, the state-of-the-art approximation algorithm with comparable quality guarantees.

Frequent subgraph mining is a fundamental task in the analysis of collections of graphs. While several exact approaches have been proposed, it remains computationally challenging on large graph datasets due to its inherent link to the subgraph isomorphism problem and the huge number of candidate patterns even for fairly small subgraphs.In this work, we study two statistical learning measures of complexity, VC-dimension and Rademacher averages, for subgraphs, and derive efficiently computable bounds for both. We show how such bounds can be applied to devise efficient sampling-based approaches for rigorously approximating the solution of the frequent subgraph mining problem. We also show that our bounds can be used for true frequent subgraph mining, which requires to identify subgraphs generated with probability above a given threshold from an unknown generative process using samples from such process. Our extensive experimental evaluation on real datasets shows that our bounds lead to efficiently computable, high-quality approximations for both applications.

Abstract The mining of time series data has applications in several domains, and in many cases the data are generated by networks, with time series representing paths on such networks. In this work, we consider the scenario in which the dataset, i.e., a collection of time series, is generated by an unknown underlying network, and we study the problem of mining statistically significant paths, which are paths whose number of observed occurrences in the dataset is unexpected given the distribution defined by some features of the underlying network. A major challenge in such a problem is that the underlying network is unknown, and, thus, one cannot directly identify such paths. We then propose caSPiTa, an algorithm to mine statistically significant paths in time series data generated by an unknown and underlying network that considers a generative null model based on meaningful characteristics of the observed dataset, while providing guarantees in terms of false discoveries. Our extensive evaluation on pseudo-artificial and real data shows that caSPiTa is able to efficiently mine large sets of significant paths, while providing guarantees on the false positives.

Abstract Pattern mining is a fundamental data mining task with applications in several domains. In this work, we consider the scenario in which we have a sequence of datasets generated by potentially different underlying generative processes, and we study the problem of mining statistically robust patterns, which are patterns whose probabilities of appearing in transactions drawn from such generative processes respect well-defined conditions. Such conditions define the patterns of interest, describing the evolution of their probabilities through the datasets in the sequence, which may, for example, increase, decrease, or stay stable, through the sequence. Due to the stochastic nature of the data, one cannot identify the exact set of the statistically robust patterns by analyzing a sequence of samples, i.e., the datasets, taken from the generative processes, and has to resort to approximations. We then propose gRosSo, an algorithm to find rigorous approximations of the statistically robust patterns that do not contain false positives or false negatives with high probability. We apply our framework to the mining of statistically robust sequential patterns and statistically robust itemsets. Our extensive evaluation on pseudo-artificial and real data shows that gRosSo provides high-quality approximations for the problem of mining statistically robust sequential patterns and statistically robust itemsets.

“I’m an MC still as honest” – Eminem, Rap God We present MCRapper , an algorithm for efficient computation of Monte-Carlo Empirical Rademacher Averages (MCERA) for families of functions exhibiting poset (e.g., lattice) structure, such as those that arise in many pattern mining tasks. The MCERA allows us to compute upper bounds to the maximum deviation of sample means from their expectations, thus it can be used to find both (1) statistically-significant functions (i.e., patterns) when the available data is seen as a sample from an unknown distribution, and (2) approximations of collections of high-expectation functions (e.g., frequent patterns) when the available data is a small sample from a large dataset. This flexibility offered by MCRapper is a big advantage over previously proposed solutions, which could only achieve one of the two. MCRapper uses upper bounds to the discrepancy of the functions to efficiently explore and prune the search space, a technique borrowed from pattern mining itself. To show the practical use of MCRapper , we employ it to develop an algorithm TFP-R for the task of True Frequent Pattern (TFP) mining, by appropriately computing approximations of the negative and positive borders of the collection of patterns of interest, which allow an effective pruning of the pattern space and the computation of strong bounds to the supremum deviation. TFP-R gives guarantees on the probability of including any false positives (precision) and exhibits higher statistical power (recall) than existing methods offering the same guarantees. We evaluate MCRapper and TFP-R and show that they outperform the state-of-the-art for their respective tasks.

Counting the number of occurrences of small connected subgraphs, called temporal motifs, has become a fundamental primitive for the analysis of temporal networks, whose edges are annotated with the time of the event they represent. One of the main complications in studying temporal motifs is the large number of motifs that can be built even with a limited number of vertices or edges. As a consequence, since in many applications motifs are employed for exploratory analyses, the user needs to iteratively select and analyze several motifs that represent different aspects of the network, resulting in an inefficient, time-consuming process. This problem is exacerbated in large networks, where the analysis of even a single motif is computationally demanding. As a solution, in this work we propose and study the problem of simultaneously counting the number of occurrences of multiple temporal motifs, all corresponding to the same (static) topology (e.g., a triangle). Given that for large temporal networks computing the exact counts is unfeasible, we propose odeN, a sampling-based algorithm that provides an accurate approximation of all the counts of the motifs. We provide analytical bounds on the number of samples required by odeN to compute rigorous, probabilistic, relative approximations. Our extensive experimental evaluation shows that odeN enables the approximation of the counts of motifs in temporal networks in a fraction of the time needed by state-of-the-art methods, and that it also reports more accurate approximations than such methods.

The mining of time series data has applications in several domains, and in many cases the data are generated by networks, with time series representing paths on such networks. In this work, we consider the scenario in which the dataset, i.e., a collection of time series, is generated by an unknown underlying network, and we study the problem of mining statistically significant paths, which are paths whose number of observed occurrences in the dataset is unexpected given the distribution defined by some features of the underlying network. A major challenge in such a problem is that the underlying network is unknown, and, thus, one cannot directly identify such paths. We then propose CASPITA, an algorithm to mine statistically significant paths in time series data generated by an unknown and underlying network that considers a generative null model based on meaningful characteristics of the observed dataset, while providing guarantees in terms of false discoveries. Our extensive evaluation on pseudo-artificial and real data shows that CASPITA is able to efficiently mine large sets of significant paths, while providing guarantees on the false positives.

An important step in cluster analysis is the evaluation of the quality of a given clustering through structural measures of goodness.Measures that do not require additional information for their evaluation (but the clustering itself), called internal measures, are commonly used because of their generality.The most widely used internal measure is the silhouette coefficient, whose naïve computation requires a quadratic number of distance calculations, unfeasible for massive datasets.Surprisingly, there are no known general methods to efficiently approximate the silhouette coefficient of a clustering with rigorously provable high accuracy.In this paper, we present the first scalable algorithm to compute such a rigorous approximation for the evaluation of clusterings based on any metric distances.Our algorithm approximates the silhouette coefficient within a mere additive error O (ε) with probability 1 -δ using a very small number of distance calculations, for any fixed ε, δ ∈ (0, 1).We also provide a distributed implementation of the algorithm using the MapReduce model, which runs in constant rounds and requires only sublinear local space at each worker, thus making our estimation approach applicable to big data scenarios.An extensive experimental evaluation provides evidence that our algorithm returns highly accurate silhouette estimates, unlike competing heuristics, while running in a fraction of the time required by the exact computation.

The identification and counting of small graph patterns, called network motifs, is a fundamental primitive in the analysis of networks, with application in various domains, from social networks to neuroscience. Several techniques have been designed to count the occurrences of motifs in static networks, with recent work focusing on the computational challenges provided by large networks. Modern networked datasets contain rich information, such as the time at which the events modeled by the networks edges happened, which can provide useful insights into the process modeled by the network. The analysis of motifs in temporal networks, called temporal motifs, is becoming an important component in the analysis of modern networked datasets. Several methods have been recently designed to count the number of instances of temporal motifs in temporal networks, which is even more challenging than its counterpart for static networks. Such methods are either exact, and not applicable to large networks, or approximate, but provide only weak guarantees on the estimates they produce and do not scale to very large networks. In this work we present an efficient and scalable algorithm to obtain rigorous approximations of the count of temporal motifs. Our algorithm is based on a simple but effective sampling approach, which renders our algorithm practical for very large datasets. Our extensive experimental evaluation shows that our algorithm provides estimates of temporal motif counts which are more accurate than the state-of-the-art sampling algorithms, with significantly lower running time than exact approaches, enabling the study of temporal motifs, of size larger than the ones considered in previous works, on billion edges networks.

Pattern mining is a fundamental data mining task with applications in several domains. In this work, we consider the scenario in which we have a sequence of datasets generated by potentially different underlying generative processes, and we study the problem of mining statistically robust patterns, which are patterns whose probabilities of appearing in transactions drawn from such generative processes respect well defined conditions. Such conditions define the patterns of interest, describing the evolution of their probabilities through the datasets in the sequence, which may, for example, increase, decrease, or stay stable, through the sequence. Due to the stochastic nature of the data, one cannot identify the exact set of the statistically robust patterns analyzing a sequence of samples, i.e., the datasets, taken from the generative processes, and has to resort to approximations. We then propose GRosSo, an algorithm to find a rigorous approximation of the statistically robust patterns that does not contain false positives with high probability. We apply our framework to the mining of statistically robust sequential patterns. Our extensive evaluation on pseudo-artificial and real data shows that GRosSo provides high-quality approximations for the problem of mining statistically robust sequential patterns.

We present MCRapper, an algorithm for efficient computation of Monte-Carlo Empirical Rademacher Averages (MCERA) for families of functions exhibiting poset (e.g., lattice) structure, such as those that arise in many pattern mining tasks. The MCERA allows us to compute upper bounds to the maximum deviation of sample means from their expectations, thus it can be used to find both statistically-significant functions (i.e., patterns) when the available data is seen as a sample from an unknown distribution, and approximations of collections of high-expectation functions (e.g., frequent patterns) when the available data is a small sample from a large dataset. This feature is a strong improvement over previously proposed solutions that could only achieve one of the two. MCRapper uses upper bounds to the discrepancy of the functions to efficiently explore and prune the search space, a technique borrowed from pattern mining itself. To show the practical use of MCRapper, we employ it to develop an algorithm TFP-R for the task of True Frequent Pattern (TFP) mining. TFP-R gives guarantees on the probability of including any false positives (precision) and exhibits higher statistical power (recall) than existing methods offering the same guarantees. We evaluate MCRapper and TFP-R and show that they outperform the state-of-the-art for their respective tasks.

We present MiSoSouP, a suite of algorithms for extracting high-quality approximations of the most interesting subgroups, according to different popular interestingness measures, from a random sample of a transactional dataset. We describe a new formulation of these measures as functions of averages, that makes it possible to approximate them using sampling. We then discuss how pseudodimension, a key concept from statistical learning theory, relates to the sample size needed to obtain an high-quality approximation of the most interesting subgroups. We prove an upper bound on the pseudodimension of the problem at hand, which depends on characteristic quantities of the dataset and of the language of patterns of interest. This upper bound then leads to small sample sizes. Our evaluation on real datasets shows that MiSoSouP outperforms state-of-the-art algorithms offering the same guarantees, and it vastly speeds up the discovery of subgroups w.r.t. analyzing the whole dataset.

The identification of significant patterns, defined as patterns whose frequency significantly deviates from what is expected under a suitable null model of the data, is a key data mining task with application in several areas. We present PROMISE, an algorithm for identifying significant sequential patterns while guaranteeing that the probability that one or more false discoveries are reported in output (i.e., the Family-Wise Error Rate - FWER) is less than a user-defined threshold. PROMISE employs the Westfall-Young method to correct for multiple hypothesis testing, a more powerful method than the commonly used Bonferroni correction. PROMISE crucially hinges on the generation of (random) permuted datasets with features similar to the input dataset, for which we provide two efficient strategies. We also provide a rigorous analysis of one of such strategies, which is based on a properly defined swap operation, proving a rigorous bound on the number of swaps it requires. The results of our experimental evaluation show that PROMISE is an efficient method that allows the discovery of statistically significant sequential patterns from transactional datasets while properly controlling for false discoveries.

We present SPuManTE, an efficient algorithm for mining significant patterns from a transactional dataset. SPuManTE controls the Family-wise Error Rate: it ensures that the probability of reporting one or more false discoveries is less than an user-specified threshold. A key ingredient of SPuManTE is UT, our novel unconditional statistical test for evaluating the significance of a pattern, that requires fewer assumptions on the data generation process and is more appropriate for a knowledge discovery setting than classical conditional tests, such as the widely used Fisher's exact test. Computational requirements have limited the use of unconditional tests in significant pattern discovery, but UT overcomes this issue by obtaining the required probabilities in a novel efficient way. SPuManTE combines UT with recent results on the supremum of the deviations of pattern frequencies from their expectations, founded in statistical learning theory. This combination allows SPuManTE to be very efficient, while also enjoying high statistical power. The results of our experimental evaluation show that SPuManTE allows the discovery of statistically significant patterns while properly accounting for uncertainties in patterns' frequencies due to the data generation process.

The availability of massive datasets has highlighted the need of computationally efficient and statistically-sound methods to extracts patterns while providing rigorous guarantees on the quality of the results, in particular with respect to false discoveries. In this tutorial we survey recent methods that properly combine computational and statistical considerations to efficiently mine statistically reliable patterns from large datasets. We start by introducing the fundamental concepts in statistical hypothesis testing, including conditional and unconditional tests, which may not be familiar to everyone in the data mining community. We then explain how the computational and statistical challenges in pattern mining have been tackled in different ways. Finally, we describe the application of these methods in areas such as market basket analysis, subgraph mining, social networks analysis, and cancer genomics.

The extraction of patterns displaying significant association with a class label is a key data mining task with wide application in many domains. We study a variant of the problem that requires to mine the top-k statistically significant patterns, thus providing tight control on the number of patterns reported in output. We develop TopKWY, the first algorithm to mine the top-k significant patterns while rigorously controlling the family-wise error rate of the output and provide theoretical evidence of its effectiveness. TopKWY crucially relies on a novel strategy to explore statistically significant patterns and on several key implementation choices, which may be of independent interest. Our extensive experimental evaluation shows that TopKWY enables the extraction of the most significant patterns from large datasets which could not be analyzed by the state-of-the-art. In addition, TopKWY improves over the state-of-the-art even for the extraction of all significant patterns.

We present MiSoSouP, a suite of algorithms for extracting high-quality approximations of the most interesting subgroups, according to different interestingness measures, from a random sample of a transactional dataset. We describe a new formulation of these measures that makes it possible to approximate them using sampling. We then discuss how pseudodimension, a key concept from statistical learning theory, relates to the sample size needed to obtain an high-quality approximation of the most interesting subgroups. We prove an upper bound on the pseudodimension of the problem at hand, which results in small sample sizes. Our evaluation on real datasets shows that MiSoSouP outperforms state-of-the-art algorithms offering the same guarantees, and it vastly speeds up the discovery of subgroups w.r.t. analyzing the whole dataset.

An uncertain graph 𝒢 = (V, E, p : E → (0, 1]) can be viewed as a probability space whose outcomes (referred to as possible worlds ) are subgraphs of 𝒢 where any edge e ε E occurs with probability p ( e ), independently of the other edges. These graphs naturally arise in many application domains where data management systems are required to cope with uncertainty in interrelated data, such as computational biology, social network analysis, network reliability, and privacy enforcement, among the others. For this reason, it is important to devise fundamental querying and mining primitives for uncertain graphs. This paper contributes to this endeavor with the development of novel strategies for clustering uncertain graphs. Specifically, given an uncertain graph 𝒢 and an integer k , we aim at partitioning its nodes into k clusters, each featuring a distinguished center node, so to maximize the minimum/average connection probability of any node to its cluster's center, in a random possible world. We assess the NP-hardness of maximizing the minimum connection probability, even in the presence of an oracle for the connection probabilities, and develop efficient approximation algorithms for both problems and some useful variants. Unlike previous works in the literature, our algorithms feature provable approximation guarantees and are capable to keep the granularity of the returned clustering under control. Our theoretical findings are complemented with several experiments that compare our algorithms against some relevant competitors, with respect to both running-time and quality of the returned clusterings.

Frequent Itemsets (FIs) mining is a fundamental primitive in data mining. It requires to identify all itemsets appearing in at least a fraction θ of a transactional dataset D. Often though, the ultimate goal of mining D is not an analysis of the dataset per se, but the understanding of the underlying process that generated it. Specifically, in many applications D is a collection of samples obtained from an unknown probability distribution π on transactions, and by extracting the FIs in D one attempts to infer itemsets that are frequently (i.e., with probability at least θ) generated by π, which we call the True Frequent Itemsets (TFIs). Due to the inherently stochastic nature of the generative process, the set of FIs is only a rough approximation of the set of TFIs, as it often contains a huge number of false positives, i.e., spurious itemsets that are not among the TFIs. In this work we design and analyze an algorithm to identify a threshold such that the collection of itemsets with frequency at least in contains only TFIs with probability at least 1 – δ, for some user-specified δ. Our method uses results from statistical learning theory involving the (empirical) VC-dimension of the problem at hand. This allows us to identify almost all the TFIs without including any false positive. We also experimentally compare our method with the direct mining of at frequency θ and with techniques based on widely-used standard bounds (i.e., the Chernoff bounds) of the binomial distribution, and show that our algorithm outperforms these methods and achieves even better results than what is guaranteed by the theoretical analysis.

As advances in technology allow for the collection, storage, and analysis of vast amounts of data, the task of screening and assessing the significance of discovered patterns is becoming a major challenge in data mining applications. In this work, we address significance in the context of frequent itemset mining. Specifically, we develop a novel methodology to identify a meaningful support threshold s* for a dataset, such that the number of itemsets with support at least s* represents a substantial deviation from what would be expected in a random dataset with the same number of transactions and the same individual item frequencies. These itemsets can then be flagged as statistically significant with a small false discovery rate.