# Combinatorial Miller–Hagberg Algorithm for Randomization of Dense Networks

## Abstract

We propose a slightly revised Miller–Hagberg (MH) algorithm that efficiently generates a random network from a given expected degree sequence. The revision was to replace the approximated edge probability between a pair of nodes with a combinatorically calculated edge probability that better captures the likelihood of edge presence especially, where edges are dense. The computational complexity of this combinatorial MH algorithm is still in the same order as the original one. We evaluated the proposed algorithm through several numerical experiments. The results demonstrated that the proposed algorithm was particularly good at accurately representing high-degree nodes in dense, heterogeneous networks. This algorithm may be a useful alternative to other more established network randomization methods, given that the data are increasingly becoming larger and denser in today’s network science research.

### References

- 1.Hakimi, S.L.: On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. Soc. Ind. Appl. Math.
**10**(3), 496–506 (1962)MathSciNetCrossRefMATHGoogle Scholar - 2.Bender, E.A., Canfield, E.R.: The asymptotic number of labeled graphs with given degree sequences. J. Comb. Theory Ser. A
**24**(3), 296–307 (1978)MathSciNetCrossRefMATHGoogle Scholar - 3.Newman, M.E.: The structure and function of complex networks. SIAM Rev.
**45**(2), 167–256 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar - 4.Bayati, M., Kim, J.H., Saberi, A.: A sequential algorithm for generating random graphs. Algorithmica
**58**(4), 860–910 (2010)MathSciNetCrossRefMATHGoogle Scholar - 5.Miller, J., Hagberg, A.: Efficient generation of networks with given expected degrees. Algorithms and Models for the Web Graph (WAW 2011), pp. 115–126. Springer, Berlin (2011)Google Scholar
- 6.Chung, F., Lu, L.: Connected components in random graphs with given expected degree sequences. Ann. Comb.
**6**(2), 125–145 (2002)MathSciNetCrossRefMATHGoogle Scholar - 7.Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys.
**124**(6), 1377–1397 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar - 8.van der Hofstad, R.: Critical behavior in inhomogeneous random graphs. Random Struct. Algorithms
**42**(4), 480–508 (2013)MathSciNetCrossRefMATHGoogle Scholar - 9.Del Genio, C.I., Gross, T., Bassler, K.E.: All scale-free networks are sparse. Phys. Rev. Lett.
**107**(17), 178701 (2011)ADSCrossRefGoogle Scholar - 10.Leskovec, J., McAuley, J.J.: Learning to discover social circles in ego networks. In: Advances in Neural Information Processing Systems, pp. 539–547 (2012)Google Scholar
- 11.Holme, P., Saramäki, J.: Temporal networks. Phys. Rep.
**519**(3), 97–125 (2012)ADSCrossRefGoogle Scholar - 12.Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. J. Complex Netw.
**2**(3), 203–271 (2014)CrossRefGoogle Scholar - 13.Zamani Esfahlani, F., Sayama, H.: A percolation-based thresholding method with applications in functional connectivity analysis (submitted to CompleNet 2018, under review)Google Scholar
- 14.Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthr. Res.
**33**(4), 452–473 (1977)Google Scholar - 15.Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science
**286**(5439), 509–512 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar