Abstract
Models for generating simple graphs are important in the study of real-world complex networks. A well established example of such a model is the erased configuration model, where each node receives a number of half-edges that are connected to half-edges of other nodes at random, and then self-loops are removed and multiple edges are concatenated to make the graph simple. Although asymptotic results for many properties of this model, such as the limiting degree distribution, are known, the exact speed of convergence in terms of the graph sizes remains an open question. We provide a first answer by analyzing the size dependence of the average number of removed edges in the erased configuration model. By combining known upper bounds with a Tauberian Theorem we obtain upper bounds for the number of removed edges, in terms of the size of the graph. Remarkably, when the degree distribution follows a power-law, we observe three scaling regimes, depending on the power law exponent. Our results provide a strong theoretical basis for evaluating finite-size effects in networks.
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References
Andersson, H.: Limit theorems for a random graph epidemic model. Ann. Appl. Probab. 8, 1331–1349 (1998)
Artzy-Randrup, Y., Stone, L.: Generating uniformly distributed random networks. Phys. Rev. E 72(5), 056708 (2005)
Bingham, N.H., Doney, R.A.: Asymptotic properties of supercritical branching processes i: the galton-watson process. Adv. Appl. Probab. 6, 711–731 (1974)
Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6(4), 489–522 (2011)
Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb. 1(4), 311–316 (1980). http://www.sciencedirect.com/science/article/pii/S0195669880800308
Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124(6), 1377–1397 (2006)
Cooper, C., Dyer, M., Greenhill, C.: Sampling regular graphs and a peer-to-peer network. Comb. Probab. Comput. 16(04), 557–593 (2007)
Del Genio, C.I., Kim, H., Toroczkai, Z., Bassler, K.E.: Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PloS One 5(4), e10012 (2010)
Embrechts, P., Goldie, C.M.: On closure and factorization properties of subexponential and related distributions. J. Aust. Math. Soc. (Series A) 29(02), 243–256 (1980)
van den Esker, H., van der Hofstad, R., Hooghiemstra, G., Znamenski, D.: Distances in random graphs with infinite mean degrees. Extremes 8(3), 111–141 (2005)
Ferreira, S.C., Castellano, C., Pastor-Satorras, R.: Epidemic thresholds of the susceptible-infected-susceptible model on networks: a comparison of numerical and theoretical results. Phys. Rev. E 86(4), 041125 (2012)
van der Hofstad, R.: Random graphs and complex networks. Unpublished manuscript (2007). http://www.win.tue.nl/rhofstad/NotesRGCN.pdf
van der Hofstad, R., Hooghiemstra, G., Van Mieghem, P.: Distances in random graphs with finite variance degrees. Random Struct. Algorithms 27(1), 76–123 (2005)
van der Hofstad, R., Hooghiemstra, G., Znamenski, D.: Distances in random graphs with finite mean and infinite variance degrees. Eurandom (2005)
van der Hoorn, P., Litvak, N.: Convergence of rank based degree-degree correlations in random directed networks. Moscow J. Comb. Number Theor. 4(4), 45–83 (2014). http://mjcnt.phystech.edu/en/article.php?id=92
van der Hoorn, P., Litvak, N.: Phase transitions for scaling of structural correlations in directed networks (2015). arXiv preprint arXiv:1504.01535
Lee, H.K., Shim, P.S., Noh, J.D.: Epidemic threshold of the susceptible-infected-susceptible model on complex networks. Phys. Rev. E 87(6), 062812 (2013)
Maslov, S., Sneppen, K.: Specificity and stability in topology of protein networks. Science 296(5569), 910–913 (2002)
Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6(2–3), 161–180 (1995). http://onlinelibrary.wiley.com/doi/10.1002/rsa.3240060204/full
Molloy, M., Reed, B.: The size of the giant component of a random graph with a given degree sequence. Comb. Probab. Comput. 7(03), 295–305 (1998)
Newman, M.E., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64(2), 026118 (2001). http://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.026118
Schlauch, W.E., Horvát, E.Á., Zweig, K.A.: Different flavors of randomness: comparing random graph models with fixed degree sequences. Soc. Netw. Anal. Min. 5(1), 1–14 (2015)
Tabourier, L., Roth, C., Cointet, J.P.: Generating constrained random graphs using multiple edge switches. J. Exp. Algorithmics (JEA) 16, 1–7 (2011)
Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York (2002)
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van der Hoorn, P., Litvak, N. (2015). Upper Bounds for Number of Removed Edges in the Erased Configuration Model. In: Gleich, D., Komjáthy, J., Litvak, N. (eds) Algorithms and Models for the Web Graph. WAW 2015. Lecture Notes in Computer Science(), vol 9479. Springer, Cham. https://doi.org/10.1007/978-3-319-26784-5_5
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DOI: https://doi.org/10.1007/978-3-319-26784-5_5
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