17 research outputs found
Creation and Growth of Components in a Random Hypergraph Process
Denote by an -component a connected -uniform hypergraph with
edges and vertices. We prove that the expected number of
creations of -component during a random hypergraph process tends to 1 as
and tend to with the total number of vertices such that
. Under the same conditions, we also show that
the expected number of vertices that ever belong to an -component is
approximately . As an immediate
consequence, it follows that with high probability the largest -component
during the process is of size . Our results
give insight about the size of giant components inside the phase transition of
random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend
Asymptotic normality of the size of the giant component in a random hypergraph
Recently, we adapted random walk arguments based on work of Nachmias and
Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic
normality of the size of the giant component in the random graph above
the phase transition. Here we show that the same method applies to the
analogous model of random -uniform hypergraphs, establishing asymptotic
normality throughout the (sparse) supercritical regime. Previously, asymptotic
normality was known only towards the two ends of this regime.Comment: 11 page
Counting Connected Graphs Asymptotically
We find the asymptotic number of connected graphs with vertices and
edges when approach infinity, reproving a result of Bender,
Canfield and McKay. We use the {\em probabilistic method}, analyzing
breadth-first search on the random graph for an appropriate edge
probability . Central is analysis of a random walk with fixed beginning and
end which is tilted to the left.Comment: 23 page
Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets
Goldreich suggested candidates of one-way functions and pseudorandom
generators included in . It is known that randomly generated
Goldreich's generator using -wise independent predicates with input
variables and output variables is not pseudorandom generator with
high probability for sufficiently large constant . Most of the previous
works assume that the alphabet is binary and use techniques available only for
the binary alphabet. In this paper, we deal with non-binary generalization of
Goldreich's generator and derives the tight threshold for linear programming
relaxation attack using local marginal polytope for randomly generated
Goldreich's generators. We assume that input
variables are known. In that case, we show that when , there is an
exact threshold
such
that for , the LP relaxation can determine
linearly many input variables of Goldreich's generator if
, and that the LP relaxation cannot determine
input variables of Goldreich's generator if
. This paper uses characterization of LP solutions by
combinatorial structures called stopping sets on a bipartite graph, which is
related to a simple algorithm called peeling algorithm.Comment: 14 pages, 1 figur