{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T03:15:38Z","timestamp":1777778138932,"version":"3.51.4"},"reference-count":27,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2016,9,12]],"date-time":"2016-09-12T00:00:00Z","timestamp":1473638400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Information Visualization"],"published-print":{"date-parts":[[2017,7]]},"abstract":"<jats:p>The increasing size of Big Data is often heralded but how data are transformed and represented is also profoundly important to knowledge discovery, and this is exemplified in Big Graph analytics. Much attention has been placed on the scale of the input graph but the product of a graph algorithm can be many times larger than the input. This is true for many graph problems, such as listing all triangles in a graph. Enabling scalable graph exploration for Big Graphs requires new approaches to algorithms, architectures, and visual analytics. A brief tutorial is given to aid the argument for thoughtful representation of data in the context of graph analysis. Then a new algebraic method to reduce the arithmetic operations in counting and listing triangles in graphs is introduced. Additionally, a scalable triangle listing algorithm in the MapReduce model will be presented followed by a description of the experiments with that algorithm that led to the current largest and fastest triangle listing benchmarks to date. Finally, a method for identifying triangles in new visual graph exploration technologies is proposed.<\/jats:p>","DOI":"10.1177\/1473871616666393","type":"journal-article","created":{"date-parts":[[2016,9,12]],"date-time":"2016-09-12T21:39:24Z","timestamp":1473716364000},"page":"157-166","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":15,"title":["Graphing trillions of triangles"],"prefix":"10.1177","volume":"16","author":[{"given":"Paul","family":"Burkhardt","sequence":"first","affiliation":[{"name":"US National Security Agency, USA"}]}],"member":"179","published-online":{"date-parts":[[2016,9,12]]},"reference":[{"issue":"4","key":"bibr1-1473871616666393","first-page":"4","volume":"20","author":"Burkhardt P","year":"2014","journal-title":"The Next Wave"},{"issue":"4","key":"bibr2-1473871616666393","first-page":"11","volume":"20","author":"Burkhardt P","year":"2014","journal-title":"The Next Wave"},{"key":"bibr3-1473871616666393","unstructured":"Burkhardt P, Waring CA. An NSA big graph experiment. Technical Report NSA-RD-2013-056001v1, US National Security Agency, 2013."},{"key":"bibr4-1473871616666393","first-page":"1","volume-title":"Proceedings of the 19th annual IEEE conference on high performance extreme computing","author":"Burkhardt P"},{"key":"bibr5-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1038\/30918"},{"key":"bibr6-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1109\/MCSE.2009.120"},{"key":"bibr7-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1126\/science.298.5594.824"},{"key":"bibr8-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1145\/1839490.1839494"},{"key":"bibr9-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.032093399"},{"key":"bibr10-1473871616666393","doi-asserted-by":"publisher","DOI":"10.21136\/CMJ.1973.101168"},{"key":"bibr11-1473871616666393","first-page":"493","volume-title":"Proceedings of the 1993 ACM\/IEEE Conference on Supercomputing","author":"Barnard ST"},{"key":"bibr12-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.066702"},{"key":"bibr13-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(03)00471-0"},{"key":"bibr14-1473871616666393","first-page":"296","volume-title":"Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation","author":"Le Gall F"},{"key":"bibr15-1473871616666393","volume-title":"Internal NSA conference presentation","author":"Burkhardt P","year":"2013"},{"key":"bibr16-1473871616666393","first-page":"137","volume-title":"Proceedings of the 6th conference on symposium on operating systems design and implementation","author":"Dean J","year":"2004"},{"key":"bibr17-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973075.76"},{"key":"bibr18-1473871616666393","first-page":"374","volume-title":"Algorithms and computation","author":"Goodrich MT"},{"key":"bibr19-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1145\/2304576.2304607"},{"key":"bibr20-1473871616666393","unstructured":"Burkhardt P. Asking hard graph questions. Technical Report NSA-RD-2014-050001v1, US National Security Agency, 2014."},{"key":"bibr21-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1145\/1557019.1557111"},{"key":"bibr22-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2011.12.007"},{"key":"bibr23-1473871616666393","first-page":"607","volume-title":"Proceedings of the 20th international conference on World Wide Web","author":"Suri S"},{"key":"bibr24-1473871616666393","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2008.07.017"},{"key":"bibr25-1473871616666393","unstructured":"Burkhardt P. Graph it! Vis it! Technical report, US National Security Agency, http:\/\/www.vacommunity.org\/egas2015\/papers\/IEEEEGAS2015-PaulBurkhardt.pdf (accessed October 26, 2015)."},{"key":"bibr26-1473871616666393","first-page":"147","volume-title":"Proceedings of the 2009 IEEE symposium on visual analytics science and technology","author":"Wong PC"},{"key":"bibr27-1473871616666393","first-page":"57","volume-title":"Proceedings of the 2015 IEEE Symposium on Large Data Analysis and Visualization","author":"Wong PC"}],"container-title":["Information Visualization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1473871616666393","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/1473871616666393","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/1473871616666393","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T19:18:46Z","timestamp":1777490326000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/1473871616666393"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,12]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,7]]}},"alternative-id":["10.1177\/1473871616666393"],"URL":"https:\/\/doi.org\/10.1177\/1473871616666393","relation":{},"ISSN":["1473-8716","1473-8724"],"issn-type":[{"value":"1473-8716","type":"print"},{"value":"1473-8724","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,12]]}}}