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We study maximum likelihood estimation for the statistical model for both directed and undirected random graph models in which the degree sequences are minimal sufficient statistics. In the undirected case, the model is known as the beta model. We derive necessary and sufficient conditions for the existence of the MLE that are based on the polytope of… (More)

We address the problem of studying the toric ideals of phylo-genetic invariants for a general group-based model on an arbitrary claw tree. We focus on the group Z2 and choose a natural recursive approach that extends to other groups. The study of the lattice associated with each phylogenetic ideal produces a list of circuits that generate the corresponding… (More)

- SONJA PETROVIĆ
- 2008

A universal Gröbner basis of an ideal is the union of all its reduced Gröbner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp degree bound for their elements, is a nontrivial task. In their '95 paper, Graham, Diaconis and Sturmfels give a nice… (More)

The p 1 model is a directed random graph model used to describe dyadic interactions in a social network in terms of effects due to differential attraction (popularity) and expansiveness, as well as an additional effect due to reciprocation. In this article we carry out an algebraic statistics analysis of this model. We show that the p 1 model is a toric… (More)

We develop a rigorous and interpretable statistical model for networks using their shell structure to construct minimal sufficient statistics. The model provides the formalism necessary for using k-cores in statistical considerations of random graphs, and in particular social networks. It cannot be specialized to any known degree-centric model and does not… (More)

A reference set, or a fiber, of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and… (More)

The structure of the space of possible contingency table real-izations under some constraints such as marginal totals has been studied in the algebraic statistics literature. The properties of the space of tables, i.e., fibers, are important for conducting exact conditional inferences, calculating cell bounds, imputing missing cell values, and assessing the… (More)

- SONJA PETROVIC
- 2011

Algebraic statistics has flourished in recent years as a branch of applied algebraic geometry. This field is fundamentally connected to and driven by methods from computational algebraic geometry and combinatorics. The algebraic methods have applications to statistical models where standard computational tools do not scale well, for example, phylogenetics,… (More)

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is… (More)