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Critical Window for Connectivity in the Configuration Model

  • LORENZO FEDERICO (a1) and REMCO VAN DER HOFSTAD (a1)

Abstract

We identify the asymptotic probability of a configuration model CM n (d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

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[1] Arratia, R. and Gordon, L. (1989) Tutorial on large deviations for the binomial distribution. Bull. Math. Biol. 51 125131.
[2] Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.
[3] van der Hofstad, R. (2017) Random Graphs and Complex Networks, Vol. 1, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.
[4] Janson, S. (2008) The largest component in a subcritical random graph with a power law degree distribution. Ann. Appl. Probab. 18 16511668.
[5] Janson, S. (2009) The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205225.
[6] Janson, S. (2014) The probability that a random multigraph is simple, II. J. Appl. Probab. 51A 123137.
[7] Janson, S. and Luczak, M. J. (2009) A new approach to the giant component problem. Random Struct. Alg. 34 197216.
[8] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience.
[9] Łuczak, T. (1992) Sparse random graphs with a given degree sequence. In Random Graphs (Poznań 1989), Vol. 2, Wiley-Interscience, pp. 165182.
[10] Molloy, M. and Reed, B. (1995) A critical point for random graphs with a given degree sequence. In Proc. Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science: Random Graphs '93 (Poznań 1993), Random Struct. Alg. 6 161179.
[11] Molloy, M. and Reed, B. (1998) The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295305.
[12] Wormald, N. C. (1981) The asymptotic connectivity of labelled regular graphs. J. Combin. Theory Ser. B 31 156167.

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Critical Window for Connectivity in the Configuration Model

  • LORENZO FEDERICO (a1) and REMCO VAN DER HOFSTAD (a1)

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