Skip to main content Accessibility help
×
Home

Degree correlations in scale-free random graph models

  • Clara Stegehuis (a1)

Abstract

We study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$ , with $\tau$ the degree exponent.

Copyright

Corresponding author

*Current address: Twente University, The Netherlands. Email address: c.stegehuis@utwente.nl

References

Hide All
[1] Albert, R., Jeong, H. and Barabási, A.-L. (1999). Internet: diameter of the world-wide web. Nature 401, 130131.
[2] Barabási, A.-L. (2016). Network Science. Cambridge University Press.
[3] Barrat, A., Barthélemy, M., Pastor-Satorras, R. and Vespignani, A. (2004). The architecture of complex weighted networks. Proc. Natl Acad. Sci. USA 101, 37473752.
[4] Bhamidi, S., Dhara, S., van der Hofstad, R. and Sen, S. (2017). Universality for critical heavy-tailed network models: metric structure of maximal components. Available at arXiv:1703.07145.
[5] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2017). Universality for first passage percolation on sparse random graphs. Ann. Probab. 45, 25682630.
[6] Bhamidi, S., Sen, S. and Wang, X. (2017). Continuum limit of critical inhomogeneous random graphs. Probab. Theory Related Fields 169, 565641.
[7] Bode, M., Fountoulakis, N. and Müller, T. (2015). On the largest component of a hyperbolic model of complex networks. Electron. J. Combin. 22, P3.24.
[8] Bode, M., Fountoulakis, N. and Müller, T. (2016). The probability of connectivity in a hyperbolic model of complex networks. Random Structures Algorithms 49, 6594.
[9] Boguñá, M. and Pastor-Satorras, R. (2003). Class of correlated random networks with hidden variables. Phys. Rev. E 68, 036112.
[10] Boguñá, M., Pastor-Satorras, R. and Vespignani, A. (2003). Absence of epidemic threshold in scale-free networks with degree correlations. Phys. Rev. Lett. 90, 028701.
[11] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin . 1, 311316.
[12] Bringmann, K., Keusch, R. and Lengler, J. (2015). Sampling geometric inhomogeneous random graphs in linear time. In 25th European Symposium on Algorithms (ESA 2017) (Leibniz International Proceedings in Informatics 87), art. 20. Leibniz-Zentrum für Informatik, Schloss Dagstuhl.
[13] Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124, 13771397.
[14] Candellero, E. and Fountoulakis, N. (2014). Clustering and the hyperbolic geometry of complex networks. In Algorithms and Models for the Web Graph (WAW 2014) (Lecture Notes in Computer Science 8882), pp. 112. Springer, Cham.
[15] Catanzaro, M., Boguñá, M. and Pastor-Satorras, R. (2005). Generation of uncorrelated random scale-free networks. Phys. Rev. E 71, 027103.
[16] Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proc. Natl Acad. Sci. USA 99, 1587915882.10.1073/pnas.252631999
[17] Colomer-de Simon, P. and Boguñá, M. (2012). Clustering of random scale-free networks. Phys. Rev. E 86, 026120.
[18] van den Esker, H., van der Hofstad, R. and Hooghiemstra, G. (2008). Universality for the distance in finite variance random graphs. J. Stat. Phys. 133, 169202.
[19] Faloutsos, M., Faloutsos, P. and Faloutsos, C. (1999). On power-law relationships of the internet topology. In ACM SIGCOMM Computer Communication Review, vol. 29, pp. 251262. ACM.
[20] Friedrich, T. and Krohmer, A. (2015). Cliques in hyperbolic random graphs. In INFOCOM Proceedings 2015, pp. 15441552. IEEE.
[21] Gradshteyn, I. and Ryzhik, I. (2015). Table of Integrals, Series, and Products. Elsevier.
[22] Gugelmann, L., Panagiotou, K. and Peter, U. (2012). Random hyperbolic graphs: degree sequence and clustering. In ICALP Proceedings 2012, Part II, pp. 573585. Springer, Berlin and Heidelberg.
[23] van der Hofstad, R. (2017). Random Graphs and Complex Networks, vol. 1. Cambridge University Press.
[24] van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27, 76123.
[25] van der Hofstad, R., van der Hoorn, P., Litvak, N. and Stegehuis, C. (2017). Limit theorems for assortativity and clustering in the configuration model with scale-free degrees. Available at arxiv:1712.08097.
[26] 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, 022307.10.1103/PhysRevE.95.022307
[27] van der Hofstad, R., van Leeuwaarden, J. S. H. and Stegehuis, C. (2017). Optimal subgraph structures in scale-free configuration models. Available at arXiv:1709.03466.
[28] 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) (Lecture Notes in Computer Science 9479), pp. 5465. Springer, Cham.
[29] Janson, S. and Luczak, M. J. (2007). A simple solution to the k-core problem. Random Structures Algorithms 30, 5062.
[30] Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N. and Barabási, A.-L. (2000). The large-scale organization of metabolic networks. Nature 407, 651654.
[31] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguná, M. (2010). Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106.
[32] Leskovec, J. and Krevl, A. (2014). SNAP Datasets: Stanford large network dataset collection. Available at http://snap.stanford.edu/data.
[33] Maslov, S., Sneppen, K. and Zaliznyak, A. (2004). Detection of topological patterns in complex networks: correlation profile of the internet. Phys. A 333, 529540.10.1016/j.physa.2003.06.002
[34] Mayo, M., Abdelzaher, A. and Ghosh, P. (2015). Long-range degree correlations in complex networks. Comput. Social Networks 2, 4.
[35] Newman, M. E. J. (2002). Assortative mixing in networks. Phys. Rev. Lett. 89, 208701.
[36] Ostilli, M. (2014). Fluctuation analysis in complex networks modeled by hidden-variable models: necessity of a large cutoff in hidden-variable models. Phys. Rev. E 89, 022807.
[37] Park, J. and Newman, M. E. J. (2003). Origin of degree correlations in the internet and other networks. Phys. Rev. E 68, 026112.
[38] Pastor-Satorras, R., Vázquez, A. and Vespignani, A. (2001). Dynamical and correlation properties of the internet. Phys. Rev. Lett. 87, 258701.
[39] Ravasz, E. and Barabási, A.-L. (2003). Hierarchical organization in complex networks. Phys. Rev. E 67, 026112.
[40] Serrano, M. Á. and Boguñá, M. (2006). Percolation and epidemic thresholds in clustered networks. Phys. Rev. Lett. 97, 088701.
[41] Stegehuis, C., van der Hofstad, R., van Leeuwaarden, J. S. H. and Janssen, A. J. E. M. (2017). Clustering spectrum of scale-free networks. Phys. Rev. E 96, 042309.
[42] Vázquez, A. (2003). Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. Phys. Rev. E 67, 056104.
[43] Vázquez, A., Pastor-Satorras, R. and Vespignani, A. (2002). Large-scale topological and dynamical properties of the internet. Phys. Rev. E 65, 066130.
[44] Whitt, W. (2006). Stochastic-Process Limits. Springer, New York.
[45] Yao, D., van der Hoorn, P. and Litvak, N. (2018). Average nearest neighbor degrees in scale-free networks. Internet Mathematics 2018. doi:10.24166/im.02.2018.

Keywords

MSC classification

Degree correlations in scale-free random graph models

  • Clara Stegehuis (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed