29,021 research outputs found
Equivalence of the random intersection graph and G(n,p)
We solve the conjecture posed by Fill, Scheinerman and Singer-Cohen and show
the equivalence of the sharp threshold functions of the random intersection
graph G(n,m,p) with and a graph in which each edge appears
independently. Moreover we prove sharper equivalence results under some
additional assumptions
Duality in Graphical Models
Graphical models have proven to be powerful tools for representing
high-dimensional systems of random variables. One example of such a model is
the undirected graph, in which lack of an edge represents conditional
independence between two random variables given the rest. Another example is
the bidirected graph, in which absence of edges encodes pairwise marginal
independence. Both of these classes of graphical models have been extensively
studied, and while they are considered to be dual to one another, except in a
few instances this duality has not been thoroughly investigated. In this paper,
we demonstrate how duality between undirected and bidirected models can be used
to transport results for one class of graphical models to the dual model in a
transparent manner. We proceed to apply this technique to extend previously
existing results as well as to prove new ones, in three important domains.
First, we discuss the pairwise and global Markov properties for undirected and
bidirected models, using the pseudographoid and reverse-pseudographoid rules
which are weaker conditions than the typically used intersection and
composition rules. Second, we investigate these pseudographoid and reverse
pseudographoid rules in the context of probability distributions, using the
concept of duality in the process. Duality allows us to quickly relate them to
the more familiar intersection and composition properties. Third and finally,
we apply the dualization method to understand the implications of faithfulness,
which in turn leads to a more general form of an existing result
On the Intersection Property of Conditional Independence and its Application to Causal Discovery
This work investigates the intersection property of conditional independence.
It states that for random variables and we have that
independent of given and independent of given implies
independent of given . Under the assumption that the joint
distribution has a continuous density, we provide necessary and sufficient
conditions under which the intersection property holds. The result has direct
applications to causal inference: it leads to strictly weaker conditions under
which the graphical structure becomes identifiable from the joint distribution
of an additive noise model
On String Graph Limits and the Structure of a Typical String Graph
We study limits of convergent sequences of string graphs, that is, graphs
with an intersection representation consisting of curves in the plane. We use
these results to study the limiting behavior of a sequence of random string
graphs. We also prove similar results for several related graph classes.Comment: 18 page
Branching Processes, and Random-Cluster Measures on Trees
Random-cluster measures on infinite regular trees are studied in conjunction
with a general type of `boundary condition', namely an equivalence relation on
the set of infinite paths of the tree. The uniqueness and non-uniqueness of
random-cluster measures are explored for certain classes of equivalence
relations. In proving uniqueness, the following problem concerning branching
processes is encountered and answered. Consider bond percolation on the
family-tree of a branching process. What is the probability that every
infinite path of , beginning at its root, contains some vertex which is
itself the root of an infinite open sub-tree
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