On the robustness of random k-cores

@article{Sato2014OnTR,
  title={On the robustness of random k-cores},
  author={Cristiane M. Sato},
  journal={Eur. J. Comb.},
  year={2014},
  volume={41},
  pages={163-182}
}
The stripping process can be slow: Part I
TLDR
It is shown that the number of iterations that the parallel k-stripping process applied to a hypergraph H can go up to some power of n, as long as c approaches c_{r,k} sufficiently fast.
K-regular Subgraphs near the K-core Threshold of a Random Graph
Loose Cores and Cycles in Random Hypergraphs
Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structure which mirrors the close relationship between cycles and 2-cores in graphs. We prove that
Core forging and local limit theorems for the k-core of random graphs
How does the core sit inside the mantle?
TLDR
A multi-type Galton-Watson branching process is derived that describes precisely how the k-core is embedded into the random graph for any $k\geq3$ and any fixed average degree $d=np>d_k$.
Law of large numbers for the SIR epidemic on a random graph with given degrees
TLDR
The main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.

References

SHOWING 1-10 OF 30 REFERENCES
The k-Core and Branching Processes
  • O. Riordan
  • Mathematics
    Combinatorics, Probability and Computing
  • 2008
TLDR
A new proof of the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ/n) is given using a local coupling of the graph to a suitable branching process and extends to a general model of inhomogeneous random graphs with independence between the edges.
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.
Asymptotic normality of the k-core in random graphs
TLDR
It is shown that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold.
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 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
Size and connectivity of the k-core of a random graph
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
Poisson Cloning Model for Random Graphs
TLDR
The Poisson cloning model GP C(n, p) equipped with the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the t-core of G( n, p), and yields not only elegant proofs but also improved bounds that are essentially best possible.
...
...