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1. 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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2. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams

   Bera, Suman K; Chakrabarti, Amit (March 2017, 34th Symposium on Theoretical Aspects of Computer Science)

   We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles, obtaining a number of new upper and lower bounds. For the important special case of counting triangles, we give a $$4$$-pass, $$(1\pm\varepsilon)$$-approximate, randomized algorithm that needs at most $$\widetilde{O}(\varepsilon^{-2}\cdot m^{3/2}/T)$$ space, where $$m$$ is the number of edges and $$T$$ is a promised lower bound on the number of triangles. This matches the space bound of a very recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of $$\Omega(\min\{m^{3/2}/T, m/\sqrt{T}\})$$, applicable at essentially all densities $$\Omega(n) \le m \le O(n^2)$$. We also prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013). Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problems are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused (Manjunath et al., ESA 2011; Kane et al., ICALP 2012). We use Tur{\'a}n graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary. 

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3. Fast Triangle Counting

   https://doi.org/10.1109/HPEC58863.2023.10363539  

   Bader, David A (September 2023, IEEE)

   Listing and counting triangles in graphs is a key algorithmic kernel for network analyses including community detection, clustering coefficients, k-trusses, and triangle centrality. We design and implement a new serial algorithm for triangle counting that performs competitively with the fastest previous approaches on both real and synthetic graphs, such as those from the Graph500 Benchmark and the MIT/Amazon/IEEE Graph Challenge. The experimental results use the recently-launched Intel Xeon Platinum 8480+ and CPU Max 9480 processors. 

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4. Fast Triangle Counting

   Bader, David (September 2023, The 27th Annual IEEE High Performance Extreme Computing Conference (HPEC))

   Listing and counting triangles in graphs is a key algorithmic kernel for network analyses including community detection, clustering coefficients, k-trusses, and triangle centrality. We design and implement a new serial algorithm for triangle counting that performs competitively with the fastest previous approaches on both real and synthetic graphs, such as those from the Graph500 Benchmark and the MIT/Amazon/IEEE Graph Challenge. The experimental results use the recently-launched Intel Xeon Platinum 8480+ and CPU Max 9480 processors. 

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5. Fast Approximate Counting of Cycles

   https://doi.org/10.4230/LIPIcs.ICALP.2024.37  

   Censor-Hillel, Keren; Even, Tomer; Vassilevska_Williams, Virginia (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal. 

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