The @math -core, defined as the largest subgraph of minimum degree @math , of the random graph @math has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [JCTB 67 (1996) 111--151] determined the threshold @math for the appearance of an extensive @math -core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the @math -core is embedded into the random graph for any @math and any fixed average degree @math . This generalises prior results on, e.g., the internal structure of the @math -core.
Some results on the local structure of the core and the mantle follow directly from the aforementioned analyses. For instance, the Poisson cloning model @cite_11 immediately implies that the internal local structure of the @math -core can be described by a simple (single-type) Galton-Watson process. Riordan also pointed out that this local description follows from his analysis @cite_23 . Furthermore, Cooper @cite_22 derived the asymptotic distribution of the internal and the external degree sequences of the vertices in the mantle, i.e., of the number of vertices with a given number of neighbours in the core and a given number of neighbours outside. In addition, Sato @cite_6 studied the robustness of the @math -core (i.e., the impact of deleting random edges).
@cite_6 @cite_22 @cite_23 @cite_11
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The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores (with fixed k>=3). Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seen as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c->k, then the new k-core is empty with probability 1-o(1). We define a constant c"k^' such that when [email protected]c"k^'[email protected](n) with @j(n)[email protected](n^-^1^ ^2logn), @j(n)>0 and c is bounded, then the probability that the new k-core has less than n-h(n) vertices goes to zero, for any function h(n)-> . We study random r-uniform n vertex hypergraphs with fixed degree sequence d = (d1…,dn), maximum degree D = o(n1 24) and total degree tn, where t is bounded. We give the size, number of edges and degree sequence of the k ≥ 2) up to a whp error of O(n2 3 D4 3 log n). In the case of graphs (r = 2) we give further structural details such as the number of tree components and, for the case of smooth degree sequences, the size of the mantle. We give various examples, such as the cores of r-uniform hypergraphs with a near Poisson degree sequence, and an improved upper bound for the first linear dependence among the columns in the independent column model of random Boolean matrices. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 25, 2004 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 example, we study the k-core in a certain power-law or 'scale-free' graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is e above the threshold. In contrast to G(n, λ n), this fraction tends to 0 as e➝0. Arrangement for converting a differential code into an absolute code which comprises a non-recursive interpolating digital filter, to which input code words are applied, with a sampling frequency fs, for supplying auxiliary code words occurring with an increased sampling frequency mfs, and a digital intergrator for converting the auxiliary code words into output code words, also with a sampling frequency mfs.