Vol.:(0123456789) The Review of Socionetwork Strategies (2019) 13:101–121 https://doi.org/10.1007/s12626-019-00054-0 1 3 ARTICLE What Graph Properties Are Constant‑Time Testable? Dense Graphs, Sparse Graphs, and Complex Networks Hiro Ito 1 Received: 29 January 2019 / Accepted: 28 August 2019 / Published online: 25 September 2019 © Springer Japan KK, part of Springer Nature 2019 Abstract In this paper, we survey what graph properties have been found to be constant-time testable at present. How to handle big data is a very important issue in computer science. Developing efcient algorithms for problems on big data is an urgent task. For this purpose, constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. In other words, the running time is invariant regardless of the size of the input. The idea of constant-time algorithms appeared in the 1990s and spread quickly especially in this century. Research on graph algorithms is one of the best studied areas in theoretical computer science. When we study constant-time algorithms on graphs, we have three models that dif- fer in the way that the graphs are represented: the dense-graph model, the bounded- degree model, and the general model. The frst one is used to treat properties on dense graphs, and the other two are used to treat properties on sparse graphs. In this paper, we survey one by one what properties have been found to be constant-time testable in each of the three models. Keywords Constant-time algorithms · Property testing · Graphs · Regularity lemma · Hyperfnite · Universal testers · Complex networks This work is partially supported by CREST, JST, Grant No. JPMJCR1402 and KAKENHI, JSPS, Grant No. 15K11985. * Hiro Ito itohiro@uec.ac.jp 1 School of Informatics and Engineering, The University of Electro-Communications, Tokyo 182-0021, Japan