In the binomial random graph @math , when @math changes from @math (subcritical case) to @math and then to @math (supercritical case) for @math , with high probability the order of the largest component increases smoothly from @math to @math and then to @math . As a natural generalisation of random graphs and connectedness, we consider the binomial random @math -uniform hypergraph @math (where each @math -tuple of vertices is present as a hyperedge with probability @math independently) and the following notion of high-order connectedness. Given an integer @math , two sets of @math vertices are called if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least @math vertices. A @math -connected component is a maximal collection of pairwise @math -connected @math -tuples of vertices. Recently, the threshold for the appearance of the giant @math -connected component in @math and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure, order, and size of the largest @math -connected components, with the help of a certain class of hypertrees' and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant @math -connected component.
The case @math of @math -tuple-connectedness corresponds to vertex-connectedness of @math and is the most studied among the higher-dimensional analogues of the phase transition in @math . Enumeration results for the asymptotic number of @math -connected @math -uniform hypergraphs were obtained by Karo 'nski and uczak @cite_11 , later improved by Andriamampianina and Ravelomanana @cite_1 via enumerative techniques. The threshold for the emergence of the giant @math -component, @math , was first determined by Schmidt-Pruzan and Shamir @cite_25 . Subsequently, Karo 'nski and uczak @cite_3 studied the distribution of the order of the largest component in the early supercritical regime. The studied range was extended by Ravelomanana and Rijamamy @cite_15 , although they only computed the expected order of the largest component and not its distribution. Behrisch, Coja-Oghlan, and Kang @cite_10 @cite_5 provided central and local limit theorems for @math , with @math arbitrarily small but fixed, while more recently Bollob 'as and Riordan @cite_2 showed that the distribution of the order of the largest component tends to a normal distribution for every @math .
@cite_1 @cite_3 @cite_2 @cite_5 @cite_15 @cite_10 @cite_25 @cite_11
2093252122 2080289118 2079728415 2953042812 2161355281 1977759320
We show that in the evolution of the random d-uniform hypergraph Gd(n,M) the phase transition occurs when M = n d(d - 1) + O(n2 3). We also prove local limit theorems for the distribution of the size of the largest component of Gd(n,M) in the subcritical and in the early supercritical phase. Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-Lof, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph G(n,p) above the phase transition. Here we show that the same method applies to the analogous model of random k -uniform hypergraphs, establishing asymptotic normality throughout the (sparse) supercritical regime. Previously, asymptotic normality was known only towards the two ends of this regime. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc. Let H d (n,p) signify a random d -uniform hypergraph with n vertices in which each of the @math possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d -uniform hypergraph with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of H d (n,p) and H d (n,m) in the regime @math , resp. d(d−1)m n >1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach. Denote by an @math -component a connected @math -uniform hypergraph with @math edges and @math vertices. We prove that the expected number of creations of @math -component during a random hypergraph process tends to 1 as @math and @math tend to @math with the total number of vertices @math such that @math . Under the same conditions, we also show that the expected number of vertices that ever belong to an @math -component is approximately @math . As an immediate consequence, it follows that with high probability the largest @math -component during the process is of size @math . Our results give insight about the size of giant components inside the phase transition of random hypergraphs. We establish central and local limit theorems for the number of vertices in the largest component of a random d-uniform hypergraph Hd(n,p) with edge probability p = c- @math , where c > (d - 1)-1 is a constant. The proof relies on a new, purely probabilistic approach. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 The component structure of the most general random hypergraphs, with edges of differen sizes, is analyzed. We show that, as this is the case for random graphs, there is a “double jump” in the probable and almost sure size of the greatest component of hypergraphs, when the average vertex degree passes the value 1.