1. How does the core sit inside the mantle?
- Author
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Kathrin Skubch, Mihyun Kang, Oliver Cooley, and Amin Coja-Oghlan
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,General Mathematics ,05C80 ,Structure (category theory) ,0102 computer and information sciences ,Characterization (mathematics) ,01 natural sciences ,Mantle (geology) ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,0101 mathematics ,Mathematics ,Branching process ,Discrete mathematics ,Random graph ,Series (mathematics) ,Degree (graph theory) ,Applied Mathematics ,Probability (math.PR) ,Computer Graphics and Computer-Aided Design ,Tree (graph theory) ,Vertex (geometry) ,010201 computation theory & mathematics ,Bounded function ,Core (graph theory) ,Combinatorics (math.CO) ,Combinatorial theory ,Mathematics - Probability ,Software ,Computer Science - Discrete Mathematics - Abstract
The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017
- Published
- 2017