536 research outputs found
Positivity for Gaussian graphical models
Gaussian graphical models are parametric statistical models for jointly
normal random variables whose dependence structure is determined by a graph. In
previous work, we introduced trek separation, which gives a necessary and
sufficient condition in terms of the graph for when a subdeterminant is zero
for all covariance matrices that belong to the Gaussian graphical model. Here
we extend this result to give explicit cancellation-free formulas for the
expansions of nonzero subdeterminants.Comment: 16 pages, 3 figure
A Lexicographic Product Cancellation Property for Digraphs
There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted , for some digraphs and . The paper concludes by proving a cancellation property for the lexicographic product of digraphs , , , and : if and , then . It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments
Decomposition and factorisation of transients in Functional Graphs
Functional graphs (FGs) model the graph structures used to analyze the
behavior of functions from a discrete set to itself. In turn, such functions
are used to study real complex phenomena evolving in time. As the systems
involved can be quite large, it is interesting to decompose and factorize them
into several subgraphs acting together. Polynomial equations over functional
graphs provide a formal way to represent this decomposition and factorization
mechanism, and solving them validates or invalidates hypotheses on their
decomposability. The current solution method breaks down a single equation into
a series of \emph{basic} equations of the form (with , ,
and being FGs) to identify the possible solutions. However, it is able to
consider just FGs made of cycles only. This work proposes an algorithm for
solving these basic equations for general connected FGs. By exploiting a
connection with the cancellation problem, we prove that the upper bound to the
number of solutions is closely related to the size of the cycle in the
coefficient of the equation. The cancellation problem is also involved in
the main algorithms provided by the paper. We introduce a polynomial-time
semi-decision algorithm able to provide constraints that a potential solution
will have to satisfy if it exists. Then, exploiting the ideas introduced in the
first algorithm, we introduce a second exponential-time algorithm capable of
finding all solutions by integrating several `hacks' that try to keep the
exponential as tight as possible
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
Equations defining probability tree models
Coloured probability tree models are statistical models coding conditional
independence between events depicted in a tree graph. They are more general
than the very important class of context-specific Bayesian networks. In this
paper, we study the algebraic properties of their ideal of model invariants.
The generators of this ideal can be easily read from the tree graph and have a
straightforward interpretation in terms of the underlying model: they are
differences of odds ratios coming from conditional probabilities. One of the
key findings in this analysis is that the tree is a convenient tool for
understanding the exact algebraic way in which the sum-to-1 conditions on the
parameter space translate into the sum-to-one conditions on the joint
probabilities of the statistical model. This enables us to identify necessary
and sufficient graphical conditions for a staged tree model to be a toric
variety intersected with a probability simplex.Comment: 22 pages, 4 figure
- …