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But much of the structure within these systems involves interactions that take place among more than two nodes at once\u2014for example, communication within a group rather than person to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental difference from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.<\/jats:p>","DOI":"10.1073\/pnas.1800683115","type":"journal-article","created":{"date-parts":[[2018,11,9]],"date-time":"2018-11-09T17:19:04Z","timestamp":1541783944000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":470,"title":["Simplicial closure and higher-order link prediction"],"prefix":"10.1073","volume":"115","author":[{"given":"Austin R.","family":"Benson","sequence":"first","affiliation":[{"name":"Department of Computer Science, Cornell University, Ithaca, NY 14853;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5487-9108","authenticated-orcid":false,"given":"Rediet","family":"Abebe","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Cornell University, Ithaca, NY 14853;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2426-6404","authenticated-orcid":false,"given":"Michael T.","family":"Schaub","sequence":"additional","affiliation":[{"name":"Institute for Data, Systems, and Society, Massachusetts Institute of Technology, Cambridge, MA 02139;"},{"name":"Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom;"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Jadbabaie","sequence":"additional","affiliation":[{"name":"Institute for Data, Systems, and Society, Massachusetts Institute of Technology, Cambridge, MA 02139;"},{"name":"Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jon","family":"Kleinberg","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Cornell University, Ithaca, NY 14853;"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2018,11,9]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert R","year":"2002","unstructured":"R Albert, AL Barab\u00e1si, Statistical mechanics of complex networks. 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