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We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear…
We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, $\mathrm{MLD}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that $\mathrm{MLD}_F(X)$ is equal to the…
In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the…
We study the neighborhood polynomial and the complexity of its computation for chordal graphs. The neighborhood polynomial of a graph is the generating function of subsets of its vertices that have a common neighbor. We introduce a…
Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the…
In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical…
We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials…
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum…
In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree…
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…
Bollob\'as-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\{W_j\}_{j\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first…
A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In…
The matching polynomial of a graph encodes rich combinatorial information through its roots. We determine the maximum multiplicity of a non-zero matching polynomial root and characterize all graphs attaining the bound. We also generalize…
An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for…
We construct classes of graphs that are variants of the so-called layered wheel. One of their key properties is that while the treewidth is bounded by a function of the clique number, the construction can be adjusted to make the dependance…
The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition…
Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that…
Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is…
Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and…
Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations…