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1. Sampling Methods for Counting Temporal Motifs

   https://doi.org/10.1145/3289600.3290988  

   Liu, Paul; Benson, Austin R.; Charikar, Moses (February 2019, Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining)

   Pattern counting in graphs is fundamental to several network sci- ence tasks, and there is an abundance of scalable methods for estimating counts of small patterns, often called motifs, in large graphs. However, modern graph datasets now contain richer structure, and incorporating temporal information in particular has become a key part of network analysis. Consequently, temporal motifs, which are generalizations of small subgraph patterns that incorporate temporal ordering on edges, are an emerging part of the network analysis toolbox. However, there are no algorithms for fast estimation of temporal motifs counts; moreover, we show that even counting simple temporal star motifs is NP-complete. Thus, there is a need for fast and approximate algorithms. Here, we present the first frequency estimation algorithms for counting temporal motifs. More specifically, we develop a sampling framework that sits as a layer on top of existing exact counting algorithms and enables fast and accurate memory-efficient estimates of temporal motif counts. Our results show that we can achieve one to two orders of magnitude speedups over existing algorithms with minimal and controllable loss in accuracy on a number of datasets. 

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2. Massively Parallel Algorithms for Small Subgraph Counting

   Biswas, Amartya Shankha; Eden, Talya; Liu, Quanquan C.; Rubinfeld, Ronitt Slobodan (January 2022, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1+ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max{(√ m, n²/m)} space per machine and assuming T = Ω(√{m/n}). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_δ(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_δ(√{log n}) rounds, O(n^δ) space per machine and O(mα³) total space. In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems. 

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3. Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

   N. Pashanasangi, C.Seshadhri (August 2021, SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) 2021)

   null (Ed.)

   Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)-temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine. 

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4. Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

   Pashanasangi, Noujan; Seshadhri, C. (January 2021, KDD)

   null (Ed.)

   Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)-temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine. 

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5. GraphZip: Mining Graph Streams using Dictionary-based Compression

   Packer, Charles; Holder, Lawrence B (August 2017, SIGKDD Workshop on Mining and Learning in Graphs (MLG))

   A massive amount of data generated today on platforms such as social networks, telecommunication networks, and the internet in general can be represented as graph streams. Activity in a network’s underlying graph generates a sequence of edges in the form of a stream; for example, a social network may generate a graph stream based on the interactions (edges) between different users (nodes) over time. While many graph mining algorithms have already been developed for analyzing relatively small graphs, graphs that begin to approach the size of real-world networks stress the limitations of such methods due to their dynamic nature and the substantial number of nodes and connections involved. In this paper we present GraphZip, a scalable method for mining interesting patterns in graph streams. GraphZip is inspired by the Lempel-Ziv (LZ) class of compression algorithms, and uses a novel dictionary-based compression approach to discover maximally- compressing patterns in a graph stream. We experimentally show that GraphZip is able to retrieve complex and insightful patterns from large real-world graphs and artificially-generated graphs with ground truth patterns. Additionally, our results demonstrate that GraphZip is both highly efficient and highly effective compared to existing state-of-the-art methods for mining graph streams. 

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