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Sudden Emergence of a Giantk-Core in a Random Graph
Open accessAbstract
Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdős–Rényi random graphG(n, m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork⩾3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=minλ>0 λ/πk(λ) andπk(λ)=P{Poisson(λ)⩾k−1}. In particular,c3≈3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ≈pk(λk) n. Hereλkis the minimum point ofλ/πk(λ) andpk(λk)=P{Poisson(λk)⩾k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one.
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Cited by
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Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. F. (2008) · Reviews of Modern Physics · doi:10.1103/revmodphys.80.1275 · Cited by 2,220
A model of Internet topology using k-shell decomposition
Carmi, S., Havlin, S., Kirkpatrick, S., Shavitt, Y., & Shir, E. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0701175104 · Cited by 760
k-Core Organization of Complex Networks
Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. F. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.040601 · Cited by 709
Frieze, A., & Karoński, M. (2015) · doi:10.1017/cbo9781316339831 · Cited by 576
Understanding brain networks and brain organization
Pessoa, L. (2014) · Physics of Life Reviews · doi:10.1016/j.plrev.2014.03.005 · Cited by 464
“8 One way to evaluate this is to use k-shell decomposition [131].”
Method
Metadata
- Authors
- Pittel, B., Spencer, J., & Wormald, N.
- Venue
- Journal of Combinatorial Theory, Series B
- Publisher
- Elsevier BV
- Published
- 1996-05
- Volume
- 67
- Issue
- 1
- Pages
- 111-151
- ISSN
- 0095-8956 (ISSN-L)
- OpenAlex
- W2065663455
- Cited by
- 437
- References
- 0
- FWCI
- 6.91
- Type
- journal article
- Language
- en
- Open access
- hybrid
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- Pittel, B., Spencer, J., & Wormald, N. (1996). Sudden Emergence of a Giantk-Core in a Random Graph. Journal of Combinatorial Theory, Series B. https://doi.org/10.1006/jctb.1996.0036
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