Abstract
This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.
Similar content being viewed by others
References
Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Bansal, S., Khandelwal, S., Meyers, L.A.: Exploring biological network structure with clustered random networks. BMC Bioinform. 10, 405 (2009)
Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Sci. 286, 509–512 (1999)
Barabási, A.-L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Phys. A. 272, 173–187 (1999)
Barabási, A.-L., Albert, R., Jeong, H.: The diameter of the world wide web. Nat. 401, 130–131 (1999)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: structure and dynamics. Phys. Rep. 424(45), 175–308 (2006)
Bollobás, B., Riordan, O.M.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks: From the Genome to the Internet, pp. 1–34 (2003)
Bollobás, B., Riordan, O.M., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Struct. Algorithms 18(3), 279–290 (2001)
Borgs, C., Brautbar, M., Chayes, J., Khanna, S., Lucier, B.: The power of local information in social networks. In: Internet and Network Economics. LNCS, vol. 7695, pp. 406–419 (2012)
Buchanan, A., Walteros, J.L., Butenko, S., Pardalos, P.M.: Solving maximum clique in sparse graphs: an \(O(nm+n2^{d/4})\) algorithm for \(d\)-degenerate graphs. Optim. Lett. 8(5), 1611–1617 (2014)
Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124(6), 1377–1397 (2006)
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web. Comput. Netw. 33(16), 309–320 (2000)
Buckley, P.G., Osthus, D.: Popularity based random graph models leading to a scale-free degree sequence. Discret. Math. 282, 53–63 (2004)
Catanzaro, M., Caldarelli, G., Pietronero, L.: Assortative model for social networks. Phys. Rev. E. 70, 037101 (2004)
Costa, L. da F., Rodrigues, F.A., Travieso, G., Boas, P.R.U.: Characterization of complex networks: a survey of measurements. Adv. Phys. 56, 167–242 (2007)
Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Assortative model for social networks. Phys. Rev. Lett. 85, 4633 (2000)
Drinea, E., Enachescu, M., Mitzenmacher, M.: Variations on random graph models for the web, technical report. Harvard University, Department of Computer Science (2001)
Eggemann, N., Noble, S.D.: The clustering coefficient of a scale-free random graph. Discret. Appl. Math. 159(10), 953–965 (2011)
Erdős, P., Gallai, T.: Graphs with given degrees of vertices. Mat. Lapok 11, 264–274 (1960)
Faloutsos, M., Faloutsos, P., Faloutsos, Ch.: On power-law relationships of the Internet topology. In: Proc. SIGCOMM’99 (1999)
Girvan, M., Newman, M.E.: Community structure in social and biological networks. Proc. Natl. Acad Sci. 99(12), 7821–7826 (2002)
Grechnikov, E.A.: An estimate for the number of edges between vertices of given degrees in random graphs in the Bollobás–Riordan model. Mosc. J. Comb. Number Theory 1(2), 40–73 (2011)
Grechnikov, E.A.: The degree distribution and the number of edges between vertices of given degrees in the Buckley–Osthus model of a random web graph. J. Internet Math. 8, 257–287 (2012)
Holme, P., Kim, B.J.: Growing scale-free networks with tunable clustering. Phys. Rev. E 65(2), 026107 (2002)
Krot, A., Ostroumova Prokhorenkova, L.: Local clustering coefficient in generalized preferential attachment models. In: Algorithms and Models for the Web Graph. LNCS, vol. 9479, pp. 15–28 (2015)
Leskovec, J.: Dynamics of Large Networks, ProQuest (2008)
Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6, 161–179 (1995)
Móri, T.F.: The maximum degree of the Barabási–Albert random tree. Comb. Probab. Comput. 14, 339–348 (2005)
Móri, T.F.: On random trees. In: Studia Sci. Math. Hungar., vol. 39, p. 143155 (2003)
Newman, M.E.J.: Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002)
Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46(5), 323–351 (2005)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)
Opsahl, T., Panzarasa, P.: Clustering in weighted networks. Soc. Netw. 31(2), 155–163 (2009)
Ostroumova Prokhorenkova, L., Samosvat, E.: Global clustering coefficient in scale-free networks. In: Algorithms and Models for the Web Graph. LNCS, vol. 8882, pp. 47–58 (2014)
Ostroumova Prokhorenkova, L.: Global clustering coefficient in scale-free weighted and unweighted networks. Internet. Math. 12(1–2), 54–67 (2016)
Ostroumova, L., Ryabchenko, A., Samosvat, E.: Generalized preferential attachment: tunable power-law degree distribution and clustering coefficient. In: Algorithms and Models for the Web Graph. LNCS, vol. 8305, pp. 185–202 (2013)
Pastor-Satorras, R., Vázquez, A., Vespignani, A.: Dynamical and correlation properties of the Internet. Phys. Rev. Lett. 87, N25, 258701 (2001)
Ravasz, E., Barabási, A.-L.: Hierarchical organization in complex networks. Phys. Rev. E 67(2) (2003)
Serrano, M.A., Boguñá, M.: Clustering in complex networks. I. General formalism. Phys. Rev. E 74, 056114 (2006)
Serrano, M.A., Boguñá, M.: Clustering in complex networks. II. Percolation properties. Phys. Rev. E 74, 056115 (2006)
Vázquez, A., Pastor-Satorras, R., Vespignani, A.: Large-scale topological and dynamical properties of the Internet. Phys. Rev. E 65, 066130 (2002)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)
Zhou, T., Yan, G., Wang, B.-H.: Maximal planar networks with large clustering coefficient and power-law degree distribution journal. Phys. Rev. E 71(4), 46141 (2005)
Acknowledgments
This work was supported by the Grant of RFBR No. 15-01-03530.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ostroumova Prokhorenkova, L. General results on preferential attachment and clustering coefficient. Optim Lett 11, 279–298 (2017). https://doi.org/10.1007/s11590-016-1030-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-016-1030-8