{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T16:38:18Z","timestamp":1783183098133,"version":"3.54.6"},"reference-count":70,"publisher":"Association for Computing Machinery (ACM)","issue":"9","license":[{"start":{"date-parts":[[2024,10,14]],"date-time":"2024-10-14T00:00:00Z","timestamp":1728864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Knowl. Discov. Data"],"published-print":{"date-parts":[[2024,11,30]]},"abstract":"Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all.<\/jats:p>\n \n Given a simple, undirected graph\n \n \\(G=(V,E)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n with\n \n \\(n=|V|\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n vertices and\n \n \\(m=|E|\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n edges, let\n \n \\(\\triangle(v)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and\n \n \\(\\triangle(G)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n denote the respective triangle counts of\n \n \\(v\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and\n \n \\(G\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . Let\n \n \\(N(v)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n be the neighborhood set of\n \n \\(v\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . Respectively,\n \n \\(N_{\\triangle}(v)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and\n \n \\(N_{\\triangle}[v]=\\{v\\}\\cup N_{\\triangle}(v)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n denote the set of neighbors that are in triangles with\n \n \\(v\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and the closed set including\n \n \\(v\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . Then the triangle centrality for a vertex\n \n \\(v\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n is\n \n \\(\\begin{align*}TC(v)=\\frac{\\frac{1}{3}\\sum_{u\\in N_{\\triangle}[v]}\\triangle(u)+\\sum_{w\\in\\{N(v)\\setminus N_{\\triangle}(v)\\}}\\triangle(w)}{\\triangle(G)}.\\end{align*}\\)<\/jats:tex-math>\n <\/jats:disp-formula>\n <\/jats:p>\n We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures.<\/jats:p>\n \n We introduce optimal algorithms that compute triangle centrality in\n \n \\(O(m\\overline{\\delta})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n time and\n \n \\(O(m+n)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n space, where\n \n \\(\\overline{\\delta}\\leq O(\\sqrt{m})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n is the\n average degeneracy<\/jats:italic>\n introduced by Burkhardt, Faber, and Harris (2020). In practical applications,\n \n \\(\\overline{\\delta}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n is much smaller than\n \n \\(\\sqrt{m}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes\n \n \\(O(\\log n)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n time using\n \n \\(O(m\\sqrt{m})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n CREW PRAM processors. In MapReduce, we show it takes four rounds using\n \n \\(O(m\\sqrt{m})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in\n \n \\(O(m\\overline{\\delta})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n time on sparse graphs.\n <\/jats:p>","DOI":"10.1145\/3685677","type":"journal-article","created":{"date-parts":[[2024,7,31]],"date-time":"2024-07-31T17:16:54Z","timestamp":1722446214000},"page":"1-34","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["Triangle Centrality"],"prefix":"10.1145","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5472-2963","authenticated-orcid":false,"given":"Paul","family":"Burkhardt","sequence":"first","affiliation":[{"name":"Research Directorate, National Security Agency, Fort Meade, MD, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2024,10,14]]},"reference":[{"key":"e_1_3_4_2_2","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1145\/1134271.1134277","volume-title":"Proceedings of the 3rd International Workshop on Link Discovery","author":"Adamic L. 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