The k-Core and Branching Processes

@article{Riordan2008TheKA,
  title={The k-Core and Branching Processes},
  author={Oliver Riordan},
  journal={Combinatorics, Probability and Computing},
  year={2008},
  volume={17},
  pages={111 - 136}
}
  • O. Riordan
  • Published 3 November 2005
  • Mathematics
  • Combinatorics, Probability and Computing
The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ/n), and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an… 
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The Size of the Giant Joint Component in a Binomial Random Double Graph
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References

SHOWING 1-10 OF 29 REFERENCES
A simple solution to the k-core problem
TLDR
This work recovers the result by Pittel, Spencer and Wormald on the existence and size of a k-core in G(n,p) and G( n,m), based on the properties of empirical distributions of independent random variables, and leads to simple proofs.
The Small Giant Component in Scale-Free Random Graphs
  • O. Riordan
  • Computer Science
    Combinatorics, Probability and Computing
  • 2005
TLDR
The phase transition in four ‘scale-free’ random graph models is studied, obtaining upper and lower bounds on the size of the giant component when there is one, and the extremely slow rate of growth just above the phase transition is determined.
The cores of random hypergraphs with a given degree sequence
  • C. Cooper
  • Mathematics
    Random Struct. Algorithms
  • 2004
We study random r‐uniform n vertex hypergraphs with fixed degree sequence d = (d1…,dn), maximum degree Δ = o(n1/24) and total degree θn, where θ is bounded. We give the size, number of edges and
Anomalous percolation properties of growing networks.
TLDR
The anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking is described and exact results for the size and the distribution of vertices among connected components are obtained.
Sudden Emergence of a Giantk-Core in a Random Graph
TLDR
These proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one, and demonstrate that, unlike the 2-core, when ak- core appears for the first time it is very likely to be giant, of size ?pk(?k)n.
The phase transition in the uniformly grown random graph has infinite order
TLDR
The main result is that for c = 1/4 + ε, ε > 0, the giant component in Gn(c) has order exp ` −Θ(1/ √ ε) ́ n, and the phase transition in the bond percolation on Gn(1) has infinite order.
Are randomly grown graphs really random?
TLDR
It is concluded that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.
A Random Graph Model for Power Law Graphs
TLDR
A random graph model is proposed which is a special case of sparserandom graphs with given degree sequences which satisfy a power law and involves only a small number of parameters, called logsize and log-log growth rate, which capture some universal characteristics of massive graphs.
Encores on Cores
We give a new derivation of the threshold of appearance of the $k$-core of a random graph. Our method uses a hybrid model obtained from a simple model of random graphs based on random functions, and
A Critical Point for Random Graphs with a Given Degree Sequence
TLDR
It is shown that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if λ0, λ1… which sum to 1, then almost surely all components in such graphs are small.
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