We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a $c$-closed graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to weakly $c$-closed graphs, which only require a vertex deletion ordering that avoids pairs of nonadjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks with thousands of vertices tend to be weakly $c$-closed for modest values of $c$.


  1. graph algorithms
  2. social networks
  3. fixed-parameter tractability

MSC codes

  1. 68Q27
  2. 68W40
  3. 68R10

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E. C. A. Bhaskara, M. Charikar, U. Feige, and A. Vijayaraghavan, Detecting high log-densities: An $O(n ^{1/4})$ approximation for densest $k$-subgraph, in Proceedings of the 2010 ACM Symposium on Theory of Computing, ACM, New York, 2010, pp. 201--210.
A. Abraham and R. Sandler, An Introduction to Finite Projective Planes, Dover, Mineola, NY, 2015.
R. Albert, H. Jeong, and A.-L. Barabási, Error and attack tolerance of complex networks, Nature, 406 (2000), pp. 378--382.
N. Alon, S. Arora, R. Manokaran, D. Moshkovitz, and O. Weinstein, Inapproximability of Densest $\kappa$-Subgraph from Average Case Hardness, unpublished manuscript, 2011.
J. Balogh, R. Morris, and W. Samotij, Independent sets in hypergraphs, J. Amer. Math. Soc., 28 (2015), pp. 669--709.
A.-L. Barabasi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), pp. 509--512, https://doi.org/10.1126/science.286.5439.509.
M. Bloznelis and V. Kurauskas, Clustering function: Another view on clustering coefficient, J. Complex Netw., 4 (2016), pp. 61--86.
M. Bloznelis and V. Kurauskas, Large cliques in sparse random intersection graphs, Electron. J. Combin., 24 (2017), P2.5.
P. Borassi, M. Crescenzi, and L. Trevisan, An Axiomatic and an Average-Case Analysis of Algorithms and Heuristics for Metric Properties of Graphs, preprint, https://arxiv.org/abs/1604.01445, 2016.
P. Brach, M. Cygan, J. Ła̧cki, and P. Sankowski, Algorithmic complexity of power law networks, in Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2016, pp. 1306--1325, https://doi.org/10.1137/1.9781611974331.ch91.
A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener, Graph structure in the Web, Comput. Netw., 33 (2000), pp. 309--320.
D. Chakrabarti and C. Faloutsos, Graph mining: Laws, generators, and algorithms, ACM Comput. Surv., 38 (2006), 2, https://doi.org/10.1145/1132952.1132954.
D. Chakrabarti, Y. Zhan, and C. Faloutsos, R-MAT: A recursive model for graph mining, in Proceedings of the Fourth SIAM International Conference on Data Mining, 2004, pp. 442--446, https://doi.org/10.1137/1.9781611972740.43.
F. Chung and L. Lu, The average distances in random graphs with given expected degrees, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 15879--15882, https://doi.org/10.1073/pnas.252631999.
F. Chung and L. Lu, Connected components in random graphs with given degree sequences, Ann. Comb., 6 (2002), pp. 125--145, https://doi.org/10.1007/PL00012580.
A. Conte, R. De Virgilio, A. Maccioni, M. Patrignani, and R. Torlone, Finding all maximal cliques in very large social networks, in Proceedings of the 19th International Conference on Extending Database Technology (EDBT), 2016, pp. 173--184.
M. Cygan, F. V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh, Parameterized Algorithms, Springer, Cham, 2015.
N. Du, B. Wu, L. Xu, B. Wang, and X. Pei, A parallel algorithm for enumerating all maximal cliques in complex network, in Proceedings of the Sixth IEEE International Conference on Data Mining - Workshops (ICDMW'06), 2006, pp. 320--324.
V. Dujmović, G. Fijavž, G. Joret, T. Sulanke, and D. R. Wood, On the maximum number of cliques in a graph embedded in a surface, European J. Combin., 32 (2011), pp. 1244--1252.
D. Eppstein, M. Löffler, and D. Strash, Listing all maximal cliques in sparse graphs in near-optimal time, in ISAAC 2010: Algorithms and Computation, Springer, Berlin, Heidelberg, 2010, pp. 403--414, https://doi.org/10.1007/978-3-642-17517-6_36.
D. Eppstein and D. Strash, Listing all maximal cliques in large sparse real-world graphs, in Experimental Algorithms, Lecture Notes in Comput. Sci. 6630, Springer, Berlin, Heidelberg, 2011, pp. 364--375.
E. M. Eschen, C. T. Hoàng, J. P. Spinrad, and R. Sritharan, On graphs without a $C_4$ or a diamond, Discrete Appl. Math., 159 (2011), pp. 581--587.
M. Faloutsos, P. Faloutsos, and C. Faloutsos, On power-law relationships of the Internet topology, in Proceedings of SIGCOMM, 1999, pp. 251--262.
A. Ferrante, G. Pandurangan, and K. Park, On the hardness of optimization in power law graphs, in Proceedings of Conference on Computing and Combinatorics, Lecture Notes in Comput. Sci. 4598, Springer, Berlin, 2007, pp. 417--427.
A. Ferrante, G. Pandurangan, and K. Park, On the hardness of optimization in power-law graphs, Theoret. Comput. Sci., 393 (2008), pp. 220--230, https://doi.org/10.1016/j.tcs.2007.12.007.
S. Fortunato, Community detection in graphs, Phys. Rep., 486 (2010), pp. 75--174.
J. Fox and F. Wei, On the Number of Cliques in Graphs with a Forbidden Subdivision or Immersion, preprint, https://arxiv.org/abs/1606.06810, 2016.
J. Fox and F. Wei, On the number of cliques in graphs with a forbidden minor, J. Combin. Theory Ser. B, 126 (2017), pp. 175--197, https://doi.org/10.1016/j.jctb.2017.04.004.
Z. Füredi and M. Simonovits, The history of degenerate (bipartite) extremal graph problems, in Erdös Centennial, Springer, Berlin, Heidelberg, 2013, pp. 169--264, https://doi.org/10.1007/978-3-642-39286-3_7.
L. Ga̧sieniec, M. Kowaluk, and A. Lingas, Faster multi-witnesses for Boolean matrix multiplication, Inform. Process. Lett., 109 (2009), pp. 242--247.
M. Girvan and M. E. J. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 7821--7826, https://doi.org/10.1073/pnas.122653799.
R. Gupta, T. Roughgarden, and C. Seshadhri, Decompositions of triangle-dense graphs, SIAM J. Comput., 45 (2016), pp. 197--215, https://doi.org/10.1137/140955331.
M. Jha, C. Seshadhri, and A. Pinar, Path sampling: A fast and provable method for estimating 4-vertex subgraph counts, in Proceedings of the 24th International Conference on World Wide Web, 2015, pp. 495--505.
J. M. Kleinberg, Navigation in a small world, Nature, 406 (2000), p. 845.
J. M. Kleinberg, The small-world phenomenon: An algorithmic perspective, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, AMC, New York, 2000, pp. 163--170.
J. M. Kleinberg, Small-world phenomena and the dynamics of information, in Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, 2002, pp. 431--438.
R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal, Stochastic models for the Web graph, in Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2000, pp. 57--65, http://doi.ieeecomputersociety.org/10.1109/SFCS.2000.892065.
J. Leskovec, D. Chakrabarti, J. M. Kleinberg, C. Faloutsos, and Z. Ghahramani, Kronecker graphs: An approach to modeling networks, J. Mach. Learn. Res., 11 (2010), pp. 985--1042, http://jmlr.csail.mit.edu/papers/v11/leskovec10a.html.
J. Leskovec and A. Krevl, SNAP Datasets: Stanford Large Network Dataset Collection, http://snap.stanford.edu/data, June 2014.
J. Leskovec, K. Lang, A. Dasgupta, and M. Mahoney, Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters, Internet Math., 6 (2008), pp. 29--123.
H. Lin, C. Amanatidis, M. Sideri, R. M. Karp, and C. H. Papadimitriou, Linked decompositions of networks and the power of choice in Polya urns, in Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 993--1002.
M. Mitzenmacher, J. Pachocki, R. Peng, C. Tsourakakis, and S. Xu, Scalable large near-clique detection in large-scale networks via sampling, in Proceedings of the 21st AMC SIGKDD International Conference on Knowledge Discovery and Data Mining, 2015, pp. 815--824, https://doi.org/10.1145/2783258.2783385.
A. Montanari and A. Saberi, The spread of innovations in social networks, Proc. Natl. Acad. Sci. USA, 107 (2010), pp. 20196--20201.
J. Moon and L. Moser, On cliques in graphs, Israel J. Math., 3 (1965), pp. 23--28.
M. E. J. Newman, The structure of scientific collaboration networks, Proc. Natl. Acad. Sci. USA, 98 (2001), pp. 404--409, https://doi.org/10.1073/pnas.98.2.404.
M. E. J. Newman, Properties of highly clustered networks, Phys. Rev. E, 68 (2003), 026121, https://doi.org/10.1103/PhysRevE.68.026121.
M. E. J. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E, 74 (2006), 036104, https://doi.org/10.1103/PhysRevE.74.036104.
V. Nikiforov, The number of cliques in graphs of given order and size, Trans. Amer. Math. Soc., 363 (2011), pp. 1599--1618.
V. Raman and S. Saket, Short cycles make $W$-hard problems hard: FPT algorithms for $W$-hard problems in graphs with no short cycles, Algorithmica, 52 (2008), pp. 203--225.
A. A. Razborov, On the minimal density of triangles in graphs, Combin. Probab. Comput., 17 (2008), pp. 603--618, https://doi.org/10.1017/S0963548308009085.
C. Reiher, The clique density theorem, Ann. of Math. (2), 184 (2016), pp. 683--707, https://doi.org/10.4007/annals.2016.184.3.1.
R. A. Rossi, D. F. Gleich, and A. H. Gebremedhin, Parallel maximum clique algorithms with applications to network analysis, SIAM J. Sci. Comput., 37 (2015), pp. C589--C616, https://doi.org/10.1137/14100018X.
A. Sala, L. Cao, C. Wilson, R. Zablit, H. Zheng, and B. Y. Zhao, Measurement-calibrated graph models for social network experiments, in Proceedings of the 19th International Conference on World Wide Web, ACM, 2010, pp. 861--870, https://doi.org/10.1145/1772690.1772778.
A. E. Sar\iyüce, C. Seshadhri, A. P\inar, and Ü. V. Çatalyürek, Finding the hierarchy of dense subgraphs using nucleus decompositions, in Proceedings of the 24th International Conference on World Wide Web, Geneva, Switzerland, 2015, pp. 927--937, https://doi.org/10.1145/2736277.2741640.
D. Saxton and A. Thomason, Hypergraph containers, Invent. Math., 201 (2015), pp. 925--992, https://doi.org/10.1007/s00222-014-0562-8.
C. Seshadhri, A. Pinar, and T. G. Kolda, Triadic measures on graphs: The power of wedge sampling, in Proceedings of the 2013 SIAM International Conference on Data Mining, SIAM, Philadelphia, 2013, pp. 10--18, https://doi.org/10.1137/1.9781611972832.2.
E. Tomita and T. Kameda, An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments, J. Global Optim., 37 (2007), pp. 95--111.
E. Tomita, A. Tanaka, and H. Takahashi, The worst-case time complexity for generating all maximal cliques and computational experiments, Theoret. Comput. Sci., 363 (2006), pp. 28--42, https://doi.org/10.1016/j.tcs.2006.06.015.
C. Tsourakakis, The k-clique densest subgraph problem, in Proceedings of the 24th International Conference on World Wide Web, Geneva, Switzerland, 2015, pp. 1122--1132, https://doi.org/10.1145/2736277.2741098.
C. Tsourakakis, F. Bonchi, A. Gionis, F. Gullo, and M. Tsiarli, Denser than the densest subgraph: Extracting optimal quasi-cliques with quality guarantees, in Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2013, pp. 104--112.
S. Tsukiyama, M. Ide, H. Ariyoshi, and I. Shirakawa, A new algorithm for generating all the maximal independent sets, SIAM J. Comput., 6 (1977), pp. 505--517, https://doi.org/10.1137/0206036.
J. Ugander, L. Backstrom, and J. Kleinberg, Subgraph frequencies: Mapping the empirical and extremal geography of large graph collections, in Proceedings of the 22nd International Conference on World Wide Web, 2013, pp. 1307--1318.
J. Ugander, B. Karrer, L. Backstrom, and C. Marlow, The Anatomy of the Facebook Social Graph, preprint, https://arxiv.org/abs/1111.4503, 2011.
D. Watts and S. Strogatz, Collective dynamics of `small-world' networks, Nature, 393 (1998), pp. 440--442, https://doi.org/10.1038/30918.

Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 448 - 464
ISSN (online): 1095-7111


Submitted: 29 August 2018
Accepted: 6 January 2020
Published online: 27 April 2020


  1. graph algorithms
  2. social networks
  3. fixed-parameter tractability

MSC codes

  1. 68Q27
  2. 68W40
  3. 68R10



Funding Information

Packard Fellowship
Alfred P. Sloan Foundation https://doi.org/10.13039/100000879
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1352121
National Science Foundation https://doi.org/10.13039/100000001 : CCF-1524062
National Science Foundation https://doi.org/10.13039/100000001 : CCF-1319080, CCF-1740850
National Science Foundation https://doi.org/10.13039/100000001

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