Sudden emergence of a giant k-core in a random graph
Journal of Combinatorial Theory Series B
A critical point for random graphs with a given degree sequence
Random Graphs 93 Proceedings of the sixth international seminar on Random graphs and probabilistic methods in combinatorics and computer science
Cores in random hypergraphs and Boolean formulas
Random Structures & Algorithms
The phase transition in the uniformly grown random graph has infinite order
Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
The cores of random hypergraphs with a given degree sequence
Random Structures & Algorithms
The Small Giant Component in Scale-Free Random Graphs
Combinatorics, Probability and Computing
A simple solution to the k-core problem
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
Efficient control of epidemics over random networks
Proceedings of the eleventh international joint conference on Measurement and modeling of computer systems
Random graphs with forbidden vertex degrees
Random Structures & Algorithms
New graph polynomials from the bethe approximation of the ising partition function
Combinatorics, Probability and Computing
External-memory network analysis algorithms for naturally sparse graphs
ESA'11 Proceedings of the 19th European conference on Algorithms
Exact thresholds for DPLL on random XOR-SAT and NP-complete extensions of XOR-SAT
Theoretical Computer Science
Survey: The cook-book approach to the differential equation method
Computer Science Review
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The k-core of a graph G is the maximal subgraph ofG having minimum degree at least k. In 1996, Pittel,Spencer and Wormald found the threshold λc forthe emergence of a non-trivial k-core in the random graphG(n, λ/n), and the asymptotic size ofthe k-core above the threshold. We give a new proof of thisresult using a local coupling of the graph to a suitable branchingprocess. This proof extends to a general model of inhomogeneousrandom graphs with independence between the edges. As an example,we study the k-core in a certain power-law or 'scale-free'graph with a parameter c controlling the overall density ofedges. For each k ≥ 3, we find the threshold value ofc at which the k-core emerges, and the fraction ofvertices in the k-core when c is ε above thethreshold. In contrast to G(n, λ/n),this fraction tends to 0 as ε➝0.