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)
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)
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)
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… (More)
We propose a new exponential family of models for random graphs. Starting from the standard exponential random graph model (ERGM) framework, we propose an extension that addresses some of the well-known issues with ERGMs. Specifically, we solve the problem of computational in-tractability and 'degenerate' model behavior by an interpretable support… (More)
This is a comment on the paper arXiv:1410.2840 by Ji and Jin, to appear in the AOAS.
The edge-degeneracy model is an exponential random graph model that uses the graph degeneracy, a measure of the graph's connection density, and number of edges in a graph as its sufficient statistics. We show this model is relatively well-behaved by studying the statistical degeneracy of this model through the geometry of the associated polytope.
Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and… (More)