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Limit theorems for assortativity and clustering in null models for scale-free networks

Published online by Cambridge University Press:  03 December 2020

Remco van der Hofstad
Affiliation:
Eindhoven University of Technology
Pim van der Hoorn
Affiliation:
Eindhoven University of Technology and Northeastern University
Nelly Litvak
Affiliation:
Eindhoven University of Technology and University of Twente
Clara Stegehuis
Affiliation:
University of Twente
Corresponding
E-mail address:

Abstract

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

Type
Original Article
Copyright
© Applied Probability Trust 2020

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References

Angel, O., van der Hofstad, R. and Holmgren, C. (2019). Limit laws for self-loops and multiple edges in the configuration model. Ann. Inst. H. Poincaré Prob. Statist. 55, 15091530.CrossRefGoogle Scholar
Bannink, T., van der Hofstad, R. and Stegehuis, C. (2018). Switch chain mixing times and triangle counts in simple random graphs with given degrees. J. Complex Networks 7, 210225.CrossRefGoogle Scholar
Bingham, N. H. and Doney, R. A. (1974). Asymptotic properties of supercritical branching processes I: the Galton–Watson process. Adv. Appl. Prob. 6, 711731.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Boguñá, M. and Pastor-Satorras, R. (2003). Class of correlated random networks with hidden variables. Phys. Rev. E 68, 036112.CrossRefGoogle ScholarPubMed
Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1, 311316.CrossRefGoogle Scholar
Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 13771397.CrossRefGoogle Scholar
Catanzaro, M., Boguñá, M. and Pastor-Satorras, R. (2005). Generation of uncorrelated random scale-free networks. Phys. Rev. E 71, 27103.CrossRefGoogle ScholarPubMed
Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proc. Nat. Acad. Sci. USA 99, 1587915882.CrossRefGoogle Scholar
Colomer-de Simon, P. and Boguñá, M. (2012). Clustering of random scale-free networks. Phys. Rev. E 86, 026120.CrossRefGoogle ScholarPubMed
Federico, L. and van der Hofstad, R. (2017). Critical window for connectivity in the configuration model. Combinatorics Prob. Comput. 26, 660680.CrossRefGoogle Scholar
Gao, J., van der Hofstad, R., Southall, A. and Stegehuis, C. (2018). Counting triangles in power-law uniform random graphs. Preprint. Available at https://arxiv.org/abs/1812.04289.Google Scholar
Van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.CrossRefGoogle Scholar
Van der Hofstad, R., Hooghiemstra, G. and van Mieghem, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27, 76123.CrossRefGoogle Scholar
Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007). Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Prob. 12, 703766.10.1214/EJP.v12-420CrossRefGoogle Scholar
Van der Hofstad, R., Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Stegehuis, C. (2017). Local clustering in scale-free networks with hidden variables. Phys. Rev. E 95, 022307.CrossRefGoogle ScholarPubMed
Van der Hofstad, R. and Litvak, N. (2014). Degree–degree dependencies in random graphs with heavy-tailed degrees. Internet Math. 10, 287334.CrossRefGoogle Scholar
Van der Hofstad, R., van Leeuwaarden, J. S. H. and Stegehuis, C. (2017). Optimal subgraph structures in scale-free networks. To appear in Ann. Appl. Prob.Google Scholar
Van der Hofstad, R., van Leeuwaarden, J. S. H. and Stegehuis, C. (2018). Triadic closure in configuration models with unbounded degree fluctuations. J. Statist. Phys. 173, 746774.CrossRefGoogle ScholarPubMed
Van der Hoorn, P. (2016). Asymptotic analysis of network structures: degree–degree correlations and directed paths. Doctoral Thesis, University of Twente.Google Scholar
Van der Hoorn, P., Lippner, G. and Krioukov, D. (2018). Sparse maximum-entropy random graphs with a given power-law degree distribution. J. Statist. Phys. 173, 806844.CrossRefGoogle Scholar
Van der Hoorn, P. and Litvak, N. (2015). Phase transitions for scaling of structural correlations in directed networks. Phys. Rev. E 92, 022803.CrossRefGoogle ScholarPubMed
Van der Hoorn, P. and Litvak, N. (2015). Upper bounds for number of removed edges in the erased configuration model. In Algorithms and Models for the Web Graph: WAW 2015, Springer, Cham, pp. 5465.10.1007/978-3-319-26784-5_5CrossRefGoogle Scholar
Van der Hoorn, P. and Olvera-Cravioto, M. (2018). Typical distances in the directed configuration model. Ann. Appl. Prob. 28, 17391792.CrossRefGoogle Scholar
Janson, S. (2009). On percolation in random graphs with given vertex degrees. Electron. J. Prob. 14, 86118.10.1214/EJP.v14-603CrossRefGoogle Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics Prob. Comput. 18, 205225.CrossRefGoogle Scholar
Janson, S. (2014). The probability that a random multigraph is simple. II. J. Appl. Prob. 51, 123137.CrossRefGoogle Scholar
Karamata, J. (1962). Some theorems concerning slowly varying functions. Tech. Rep., University of Wisconsin–Madison Mathematics Research Center.CrossRefGoogle Scholar
Newman, M. E. J. (2002). Assortative mixing in networks. Phys. Rev. Lett. 89, 208701.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003). Mixing patterns in networks. Phys. Rev. E 67, 026126.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167256.CrossRefGoogle Scholar
Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118.10.1103/PhysRevE.64.026118CrossRefGoogle ScholarPubMed
Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 5975.CrossRefGoogle Scholar
Ostroumova Prokhorenkova, L. (2016). Global clustering coefficient in scale-free weighted and unweighted networks. Internet Math. 12, 5467.CrossRefGoogle Scholar
Ostroumova Prokhorenkova, L. and Samosvat, E. (2014). Global clustering coefficient in scale-free networks. In Algorithms and Models for the Web Graph: WAW 2014, Springer, Cham, pp. 4758.CrossRefGoogle Scholar
Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E 70, 066117.CrossRefGoogle ScholarPubMed
Pierce, R. D. (1997). Application of the positive alpha-stable distribution. In Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics, 1997, IEEE, Banff, pp. 420424.CrossRefGoogle Scholar
Riordan, O. (2012). The phase transition in the configuration model. Combinatorics Prob. Comput. 21, 265299.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall/CRC, Boca Raton.Google Scholar
Stegehuis, C. (2019). Degree correlations in scale-free random graph models. J. Appl. Prob. 56, 672700.CrossRefGoogle Scholar
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of small-world networks. Nature 393, 440442.CrossRefGoogle Scholar
Whitt, W. (2006). Stochastic-Process Limits. Springer, New York.Google Scholar
Wormald, N. C. (1980). Some problems in the enumeration of labelled graphs. Bull. Austral. Math. Soc. 21, 159160.CrossRefGoogle Scholar
Yao, D., van der Hoorn, P. and Litvak, N. (2018). Average nearest neighbor degrees in scale-free networks. Internet Math. J. 1, 38 pp.Google Scholar

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