11 research outputs found
Operators on random hypergraphs and random simplicial complexes
Random hypergraphs and random simplicial complexes have potential
applications in computer science and engineering. Various models of random
hypergraphs and random simplicial complexes on n-points have been studied. Let
L be a simplicial complex. In this paper, we study random sub-hypergraphs and
random sub-complexes of L. By considering the minimal complex that a
sub-hypergraph can be embedded in and the maximal complex that can be embedded
in a sub-hypergraph, we define some operators on the space of probability
functions on sub-hypergraphs of L. We study the compositions of these operators
as well as their actions on the space of probability functions. As applications
in computer science, we give algorithms generating large sparse random
hypergraphs and large sparse random simplicial complexes.Comment: 22 page
On the structure of random graphs with constant -balls
We continue the study of the properties of graphs in which the ball of radius
around each vertex induces a graph isomorphic to the ball of radius in
some fixed vertex-transitive graph , for various choices of and .
This is a natural extension of the study of regular graphs. More precisely, if
is a vertex-transitive graph and , we say a graph is
{\em -locally } if the ball of radius around each vertex of
induces a graph isomorphic to the graph induced by the ball of radius
around any vertex of . We consider the following random graph model: for
each , we let be a graph chosen uniformly at
random from the set of all unlabelled, -vertex graphs that are -locally
. We investigate the properties possessed by the random graph with
high probability, for various natural choices of and .
We prove that if is a Cayley graph of a torsion-free group of polynomial
growth, and is sufficiently large depending on , then the random graph
has largest component of order at most with high
probability, and has at least automorphisms with high
probability, where depends upon alone. Both properties are in
stark contrast to random -regular graphs, which correspond to the case where
is the infinite -regular tree. We also show that, under the same
hypotheses, the number of unlabelled, -vertex graphs that are -locally
grows like a stretched exponential in , again in contrast with
-regular graphs. In the case where is the standard Cayley graph of
, we obtain a much more precise enumeration result, and more
precise results on the properties of the random graph . Our proofs
use a mixture of results and techniques from geometry, group theory and
combinatorics.Comment: Minor changes. 57 page