Despina Stasi

Strong Hanani-Tutte on the Projective Plane

If a graph can be drawn in the projective plane so that every two nonadjacent edges cross an even number of times, then the graph can be embedded in the projective plane.

Statistical models for cores decomposition of an undirected random graph

The

Goodness of fit for log-linear network models: dynamic Markov bases using hypergraphs

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Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases

-core decomposition is a widely studied summary statistic that describes a graphs global connectivity structure. In this paper, we move beyond using

Hydras: Directed hypergraphs and Horn formulas

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Hydras: directed hypergraphs and horn formulas

-core decomposition as a tool to summarize a graph and propose using

A Socio-Aware Decentralized Topology Construction Protocol

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Toric algebra of hypergraphs

-core decomposition as a tool to model random graphs. We propose using the shell distribution vector, a way of summarizing the decomposition, as a sufficient statistic for a family of exponential random graph models. We study the properties and behavior of the model family, implement a Markov chain Monte Carlo algorithm for simulating graphs from the model, implement a direct sampler from the set of graphs with a given shell distribution, and explore the sampling distributions of some of the commonly used complementary statistics as good candidates for heuristic model fitting. These algorithms provide first fundamental steps necessary for solving the following problems: parameter estimation in this ERGM, extending the model to its Bayesian relative, and developing a rigorous methodology for testing goodness of fit of the model and model selection. The methods are applied to a synthetic network as well as the well-known Sampson monks dataset.

Random Horn formulas and propagation connectivity for directed hypergraphs

Social networks and other sparse data sets pose significant challenges for statistical inference, since many standard statistical methods for testing model/data fit are not applicable in such settings. Algebraic statistics offers a theoretically justified approach to goodness-of-fit testing that relies on the theory of Markov bases. Most current practices require the computation of the entire basis, which is infeasible in many practical settings. We present a dynamic approach to explore the fiber of a model, which bypasses this issue, and is based on the combinatorics of hypergraphs arising from the toric algebra structure of log-linear models. We demonstrate the approach on the Holland–Leinhardt