2,883 research outputs found
β models for random hypergraphs with a given degree sequence
We introduce the beta model for random hypergraphs in order to represent
the occurrence of multi-way interactions among agents in a social network. This model
builds upon and generalizes the well-studied beta model for random graphs, which instead only considers pairwise interactions. We provide two algorithms for fitting the
model parameters, IPS (iterative proportional scaling) and fixed point algorithm, prove
that both algorithms converge if maximum likelihood estimator (MLE) exists, and provide algorithmic and geometric ways of dealing the issue of MLE existence
Degree sequences of sufficiently dense random uniform hypergraphs
We find an asymptotic enumeration formula for the number of simple
-uniform hypergraphs with a given degree sequence, when the number of edges
is sufficiently large. The formula is given in terms of the solution of a
system of equations. We give sufficient conditions on the degree sequence which
guarantee existence of a solution to this system. Furthermore, we solve the
system and give an explicit asymptotic formula when the degree sequence is
close to regular. This allows us to establish several properties of the degree
sequence of a random -uniform hypergraph with a given number of edges. More
specifically, we compare the degree sequence of a random -uniform hypergraph
with a given number edges to certain models involving sequences of binomial or
hypergeometric random variables conditioned on their sum
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
Structure of large random hypergraphs
The theme of this paper is the derivation of analytic formulae for certain
large combinatorial structures. The formulae are obtained via fluid limits of
pure jump type Markov processes, established under simple conditions on the
Laplace transforms of their Levy kernels. Furthermore, a related Gaussian
approximation allows us to describe the randomness which may persist in the
limit when certain parameters take critical values. Our method is quite
general, but is applied here to vertex identifiability in random hypergraphs. A
vertex v is identifiable in n steps if there is a hyperedge containing v all of
whose other vertices are identifiable in fewer than n steps. We say that a
hyperedge is identifiable if every one of its vertices is identifiable. Our
analytic formulae describe the asymptotics of the number of identifiable
vertices and the number of identifiable hyperedges for a Poisson random
hypergraph on a set of N vertices, in the limit as N goes to infinity.Comment: Revised version with minor conceptual improvements and additional
discussion. 32 pages, 5 figure
Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
We generalize the belief-propagation algorithm to sparse random networks with
arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in
these networks belongs to a given set of motifs (generalization of the
configuration model). These networks can be treated as sparse uncorrelated
hypergraphs in which hyperedges represent motifs. Here a hypergraph is a
generalization of a graph, where a hyperedge can connect any number of
vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which
crucially simplify the problem and allow us to apply the belief-propagation
algorithm to these loopy networks with arbitrary motifs. As natural examples,
we consider motifs in the form of finite loops and cliques. We apply the
belief-propagation algorithm to the ferromagnetic Ising model on the resulting
random networks. We obtain an exact solution of this model on networks with
finite loops or cliques as motifs. We find an exact critical temperature of the
ferromagnetic phase transition and demonstrate that with increasing the
clustering coefficient and the loop size, the critical temperature increases
compared to ordinary tree-like complex networks. Our solution also gives the
birth point of the giant connected component in these loopy networks.Comment: 9 pages, 4 figure
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