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How to determine if a random graph with a fixed degree sequence has a giant component

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Abstract

For a fixed degree sequence \({\mathcal {D}}=(d_1,\ldots ,d_n)\), let \(G({\mathcal {D}})\) be a uniformly chosen (simple) graph on \(\{1,\ldots ,n\}\) where the vertex i has degree \(d_i\). In this paper we determine whether \(G({\mathcal {D}})\) has a giant component with high probability, essentially imposing no conditions on \({\mathcal {D}}\). We simply insist that the sum of the degrees in \({\mathcal {D}}\) which are not 2 is at least \(\lambda (n)\) for some function \(\lambda \) going to infinity with n. This is a relatively minor technical condition, and when \({\mathcal {D}}\) does not satisfy it, both the probability that \(G({\mathcal {D}})\) has a giant component and the probability that \(G({\mathcal {D}})\) has no giant component are bounded away from 1.

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Notes

  1. A degree sequence is feasible if there is a graph with the given degree sequence.

  2. A component is cyclic if it is a cycle and non-cyclic if it is not.

  3. Here and throughout the paper, when we say we suppress a vertex u of degree 2, this means we delete u and we add an edge between its neighbours. Observe that this may create loops and multiple edges, so the resulting object might not be a simple graph.

  4. As it is standard, we use order and size to denote the number of vertices and the number of edges of a graph, respectively.

  5. Random graphs with special degree sequences had been studied earlier (see, e.g. [22, 33]).

  6. Note that some of these results give a more precise description on the order of the largest component. Our results only deal with the existential question.

  7. Their result gives convergence in probability of the proportion of vertices in the giant component and they also consider the case \(Q(\mathfrak {D})=0\).

  8. They also proved some results on the distribution of the order of the largest component and also consider the case \(Q(\mathfrak {D})=0\).

  9. Observe that \(X_\ell \) follows a hypergeometric distribution.

  10. In fact, the Molloy–Reed result does not discuss the case \(Q(\mathfrak {D})=0\).

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Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

    Felix Joos & Guillem Perarnau

  2. Institute of Optimization and Operations Research, Ulm University, Ulm, Germany

    Dieter Rautenbach

  3. CNRS, Nice, France

    Bruce Reed

  4. Kawarabayashi Large Graph Project, NII, Tokyo, Japan

    Bruce Reed

  5. IMPA, Rio de Janeiro, Brazil

    Bruce Reed

Authors
  1. Felix Joos
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  2. Guillem Perarnau
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  3. Dieter Rautenbach
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  4. Bruce Reed
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Corresponding author

Correspondence to Guillem Perarnau.

Additional information

Felix Joos was supported by the EPSRC, Grant No. EP/M009408/1.

Guillem Perarnau was supported by the European Research Council, ERC Grant Agreement No. 306349.

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Joos, F., Perarnau, G., Rautenbach, D. et al. How to determine if a random graph with a fixed degree sequence has a giant component. Probab. Theory Relat. Fields 170, 263–310 (2018). https://doi.org/10.1007/s00440-017-0757-1

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  • DOI: https://doi.org/10.1007/s00440-017-0757-1

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