Related papers: Configured spaces and their M\"obius polynomials
We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal…
Subgraph reconfiguration is a family of problems focusing on the reachability of the solution space in which feasible solutions are subgraphs, represented either as sets of vertices or sets of edges, satisfying a prescribed graph structure…
Relating macroscopic observables to microscopic interactions is a central challenge in the study of complex systems. While current approaches often focus on pairwise interactions, a complete understanding requires going beyond these to…
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in…
Given a $k$-uniform hypergraph $\mathcal{H}$ and sufficiently large $m \gg m_0(\mathcal{H})$, we show that an $m$-element set $I \subseteq V(\mathcal{H})$, chosen uniformly at random, with probability $1 - e^{-\omega(m)}$ is either not…
Considering a linearly ordered set, we introduce its symmetric version, and endow it with two operations extending supremum and infimum, so as to obtain an algebraic structure close to a commutative ring. We show that imposing symmetry…
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple…
We consider the challenging problem of statistical inference for exponential-family random graph models based on a single observation of a random graph with complex dependence. To facilitate statistical inference, we consider random graphs…
Graphs and hypergraphs are foundational structures in discrete mathematics. They have many practical applications, including the rapidly developing field of bioinformatics, and more generally, biomathematics. They are also a source of…
We propose and axiomatize preferences on a product state space in light of uncertainty regarding the dependency of different payoff-relevant factors. Dependence structures allow to decompose probabilities and allow to pin down behavior…
A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of…
Higher-order networks, naturally described as hypergraphs, are essential for modeling real-world systems involving interactions among three or more entities. Stochastic block models offer a principled framework for characterizing mesoscale…
Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating…
Decomposable models and Bayesian networks can be defined as sequences of oligo-dimensional probability measures connected with operators of composition. The preliminary results suggest that the probabilistic models allowing for effective…
F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $\Omega$, under what conditions will there exist a total ordering on $\Omega$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are…
Research in transportation frequently involve modelling and predicting attributes of events that occur at regular intervals. The event could be arrival of a bus at a bus stop, the volume of a traffic at a particular point, the demand at a…
Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to $d$-separation and structural causal models.…
Orthogonal polynomials appear naturally in the study of compositions of M\"obius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic M\"obius map.…