240 research outputs found
Connectivity of Random Annulus Graphs and the Geometric Block Model
We provide new connectivity results for {\em vertex-random graphs} or {\em
random annulus graphs} which are significant generalizations of random
geometric graphs. Random geometric graphs (RGG) are one of the most basic
models of random graphs for spatial networks proposed by Gilbert in 1961,
shortly after the introduction of the Erd\H{o}s-R\'{en}yi random graphs. They
resemble social networks in many ways (e.g. by spontaneously creating cluster
of nodes with high modularity). The connectivity properties of RGG have been
studied since its introduction, and analyzing them has been significantly
harder than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge
formation.
Our next contribution is in using the connectivity of random annulus graphs
to provide necessary and sufficient conditions for efficient recovery of
communities for {\em the geometric block model} (GBM). The GBM is a
probabilistic model for community detection defined over an RGG in a similar
spirit as the popular {\em stochastic block model}, which is defined over an
Erd\H{o}s-R\'{en}yi random graph. The geometric block model inherits the
transitivity properties of RGGs and thus models communities better than a
stochastic block model. However, analyzing them requires fresh perspectives as
all prior tools fail due to correlation in edge formation. We provide a simple
and efficient algorithm that can recover communities in GBM exactly with high
probability in the regime of connectivity
High dimensional Hoffman bound and applications in extremal combinatorics
One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on -wise intersecting families
The Lov\'asz-Cherkassky theorem in infinite graphs
Infinite generalizations of theorems in finite combinatorics were initiated
by Erd\H{o}s due to his famous Erd\H{o}s-Menger conjecture (now known as the
Aharoni-Berger theorem) that extends Menger's theorem to infinite graphs in a
structural way. We prove a generalization of this manner of the classical
result about packing edge-disjoint -paths in an ``inner Eulerian'' setting
obtained by Lov\'asz and Cherkassky independently in the '70s
Topological methods in zero-sum Ramsey theory
A cornerstone result of Erd\H os, Ginzburg, and Ziv (EGZ) states that any
sequence of elements in contains a zero-sum subsequence
of length . While algebraic techniques have predominated in deriving many
deep generalizations of this theorem over the past sixty years, here we
introduce topological approaches to zero-sum problems which have proven
fruitful in other combinatorial contexts. Our main result (1) is a topological
criterion for determining when any -coloring of an -uniform
hypergraph contains a zero-sum hyperedge. In addition to applications for
Kneser hypergraphs, for complete hypergraphs our methods recover Olson's
generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we
(2) give a fractional generalization of the EGZ theorem with applications to
balanced set families and (3) provide a constrained EGZ theorem which imposes
combinatorial restrictions on zero-sum sequences in the original result.Comment: 18 page
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