Graph pattern matching is typically defined in terms of subgraph isomorphism, which makes it an np-complete problem. Moreover, it requires bijective functions, which are often too restrictive to characterize patterns in emerging applications. We propose a class of graph patterns, in which an edge denotes the connectivity in a data graph within a predefined number of hops. In addition, we define matching based on a notion of bounded simulation, an extension of graph simulation. We show that with this revision, graph pattern matching can be performed in cubic-time, by providing such an algorithm. We also develop algorithms for incrementally finding matches when data graphs are updated, with performance guarantees for dag patterns. We experimentally verify that these algorithms scale well, and that the revised notion of graph pattern matching allows us to identify communities commonly found in real-world networks.

Graph Pattern Matching.

Coresets are compact representations of data sets such that models trained on a coreset are provably competitive with models trained on the full data set. As such, they have been successfully used to scale up clustering models to massive data sets. While existing approaches generally only allow for multiplicative approximation errors, we propose a novel notion of lightweight coresets that allows for both multiplicative and additive errors. We provide a single algorithm to construct lightweight coresets for k-means clustering as well as soft and hard Bregman clustering. The algorithm is substantially faster than existing constructions, embarrassingly parallel, and the resulting coresets are smaller. We further show that the proposed approach naturally generalizes to statistical k-means clustering and that, compared to existing results, it can be used to compute smaller summaries for empirical risk minimization. In extensive experiments, we demonstrate that the proposed algorithm outperforms existing data summarization strategies in practice.

Scalable k-Means Clustering via Lightweight Coresets

We consider a generalization of the shortest-path problem: given an alphabet Σ, a graph Gwhose edges are weighted and Σ-labeled, and a regular language L? Σ*, the L-constrained shortest-path problemconsists of finding a shortest path pin Gsuch that the concatenated labels along pform a word of L. This definition allows to model, e. g., many traffic-planning problems. We present extensions of well-known speed-up techniques for the standard shortest-path problem, and conduct an extensive experimental study of their performance with various networks and language constraints. Our results show that depending on the network type, both goal-directed and bidirectional search speed up the search considerably, while combinations of these do not.

Engineering Label-Constrained Shortest-Path Algorithms.

We present MaNIACS, a sampling-based randomized algorithm for computing high-quality approximations of the collection of the subgraph patterns that are frequent in a single, large, vertex-labeled graph, according to the Minimum Node Image-based (MNI) frequency measure. The output of MaNIACS comes with strong probabilistic guarantees, obtained by using the empirical Vapnik-Chervonenkis (VC) dimension, a key concept from statistical learning theory, together with strong probabilistic tail bounds on the difference between the frequency of a pattern in the sample and its exact frequency. MaNIACS leverages properties of the MNI-frequency to aggressively prune the pattern search space, and thus to reduce the time spent in exploring subspaces containing no frequent patterns. In turn, this pruning leads to better bounds to the maximum frequency estimation error, which leads to increased pruning, resulting in a beneficial feedback effect. The results of our experimental evaluation of MaNIACS on real graphs show that it returns high-quality collections of frequent patterns in large graphs up to two orders of magnitude faster than the exact algorithm.

MaNIACS: Approximate Mining of Frequent Subgraph Patterns through Sampling

The sliding window model of computation captures scenarios in which data is arriving continuously, but only the latest $w$ elements should be used for analysis. The goal is to design algorithms that update the solution efficiently with each arrival rather than recomputing it from scratch. In this work, we focus on $k$-clustering problems such as $k$-means and $k$-median. In this setting, we provide simple and practical algorithms that offer stronger performance guarantees than previous results. Empirically, we show that our methods store only a small fraction of the data, are orders of magnitude faster, and find solutions with costs only slightly higher than those returned by algorithms with access to the full dataset.

Sliding Window Algorithms for k-Clustering Problems

A great variety of complex systems ranging from user interactions in communication networks to transactions in financial markets can be modeled as temporal graphs, which consist of a set of vertices and a series of timestamped and directed edges. Temporal motifs in temporal graphs are generalized from subgraph patterns in static graphs which take into account edge orderings and durations in addition to structures. Counting the number of occurrences of temporal motifs is a fundamental problem for temporal network analysis. However, existing methods either cannot support temporal motifs or suffer from performance issues. In this paper, we focus on approximate temporal motif counting via random sampling. We first propose a generic edge sampling (ES) algorithm for estimating the number of instances of any temporal motif. Furthermore, we devise an improved EWS algorithm that hybridizes edge sampling with wedge sampling for counting temporal motifs with 3 vertices and 3 edges. We provide comprehensive analyses of the theoretical bounds and complexities of our proposed algorithms. Finally, we conduct extensive experiments on several real-world datasets, and the results show that our ES and EWS algorithms have higher efficiency, better accuracy, and greater scalability than the state-of-the-art sampling method for temporal motif counting.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=6937&context=sis_research

Efficient Sampling Algorithms for Approximate Temporal Motif Counting

Coresets are important tools to generate concise summaries of massive datasets for approximate analysis. A coreset is a small subset of points extracted from the original point set such that certain geometric properties are preserved with provable guarantees. This paper investigates the problem of maintaining a coreset to preserve the minimum enclosing ball (MEB) for a sliding window of points that are continuously updated in a data stream. Although the problem has been extensively studied in batch and append-only streaming settings, no efficient sliding-window solution is available yet. In this work, we first introduce an algorithm, called AOMEB, to build a coreset for MEB in an append-only stream. AOMEB improves the practical performance of the state-of-the-art algorithm while having the same approximation ratio. Furthermore, using AOMEB as a building block, we propose two novel algorithms, namely SWMEB and SWMEB+, to maintain coresets for MEB over the sliding window with constant approximation ratios. The proposed algorithms also support coresets for MEB in a reproducing kernel Hilbert space (RKHS). Finally, extensive experiments on real-world and synthetic datasets demonstrate that SWMEB and SWMEB+ achieve speedups of up to four orders of magnitude over the state-of-the-art batch algorithm while providing coresets for MEB with rather small errors compared to the optimal ones.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=5621&context=sis_research

Coresets for Minimum Enclosing Balls over Sliding Windows

Graphs are commonly used for representing complex structures such as social relationships, biological interactions, and knowledge bases. In many scenarios, graphs not only represent topological relationships but also store the attributes that denote the semantics associated with their vertices and edges, known as attributed graphs. Attributed graphs can meet demands for a wide range of applications, and thus a variety of queries on attributed graphs have been proposed. However, these diverse types of attributed graph queries have not been systematically investigated yet. In this paper, we provide an extensive survey of several typical types of attributed graph queries. We propose a taxonomy of attributed graph queries based on query inputs and outputs. We summarize the definitions of queries that fall into each category and present a fine-grained classification of queries within each category by analyzing the semantics and algorithmic motivations behind these queries. Moreover, we discuss the insights of how existing studies address the technical challenges of query processing and outline several promising future research directions.

https://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=6546&context=sis_research

A survey of typical attributed graph queries

We study the problem of extracting a small subset of representative items from a large data stream. In many data mining and machine learning applications such as social network analysis and recommender systems, this problem can be formulated as maximizing a monotone submodular function subject to a cardinality constraint $k$. In this work, we consider the setting where data items in the stream belong to one of several disjoint groups and investigate the optimization problem with an additional \emph{fairness} constraint that limits selection to a given number of items from each group. We then propose efficient algorithms for the fairness-aware variant of the streaming submodular maximization problem. In particular, we first give a $ (\frac{1}{2}-\varepsilon) $-approximation algorithm that requires $ O(\frac{1}{\varepsilon} \log \frac{k}{\varepsilon}) $ passes over the stream for any constant $ \varepsilon>0 $. Moreover, we give a single-pass streaming algorithm that has the same approximation ratio of $(\frac{1}{2}-\varepsilon)$ when unlimited buffer sizes and post-processing time are permitted, and discuss how to adapt it to more practical settings where the buffer sizes are bounded. Finally, we demonstrate the efficiency and effectiveness of our proposed algorithms on two real-world applications, namely \emph{maximum coverage on large graphs} and \emph{personalized recommendation}.

Fair and Representative Subset Selection from Data Streams

Coresets are succinct summaries of large datasets such that, for a given problem, the solution obtained from a coreset is provably competitive with the solution obtained from the full dataset. As such, coreset-based data summarization techniques have been successfully applied to various problems, e.g., geometric optimization, clustering, and approximate query processing, for scaling them up to massive data. In this paper, we study coresets for the maxima representation of multidimensional data: Given a set P of points in $ \mathbbR ^d $, where d is a small constant, and an error parameter $ \varepsilon \in (0,1) $, a subset $ Q \subseteq P $ is an $ \varepsilon $-coreset for the maxima representation of P iff the maximum of Q is an $ \varepsilon $-approximation of the maximum of P for any vector $ u \in \mathbbR ^d $, where the maximum is taken over the inner products between the set of points (P or Q) and u. We define a novel minimum $\varepsilon$-coreset problem that asks for an $\varepsilon$-coreset of the smallest size for the maxima representation of a point set. For the two-dimensional case, we develop an optimal polynomial-time algorithm for the minimum $ \varepsilon $-coreset problem by transforming it into the shortest-cycle problem in a directed graph. Then, we prove that this problem is NP-hard in three or higher dimensions and present polynomial-time approximation algorithms in an arbitrary fixed dimension. Finally, we provide extensive experimental results on both real and synthetic datasets to demonstrate the superior performance of our proposed algorithms.

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Yanhao Wang

Papers

Efficient Sampling Algorithms for Approximate Temporal Motif Counting

Coresets for Minimum Enclosing Balls over Sliding Windows

A survey of typical attributed graph queries

Fair and Representative Subset Selection from Data Streams

Minimum Coresets for Maxima Representation of Multidimensional Data