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Bayesian networks are widely used to learn and reason about the dependence structure of discrete variables. However, they are only capable of formally encoding symmetric conditional independence, which in practice is often too strict to…
In many applications involving binary variables, only pairwise dependence measures, such as correlations, are available. However, for multi-way tables involving more than two variables, these quantities do not uniquely determine the joint…
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from…
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs,…
Bayesian networks, with structure given by a directed acyclic graph (DAG), are a popular class of graphical models. However, learning Bayesian networks from discrete or categorical data is particularly challenging, due to the large…
This paper explores the conditions for determining fixed nodes in structured networks, specifically focusing on directed acyclic graphs (DAGs). We introduce several necessary and sufficient conditions for determining fixed nodes in…
It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
An intense activity is nowadays devoted to the definition of models capturing the properties of complex networks. Among the most promising approaches, it has been proposed to model these graphs via their clique incidence bipartite graphs.…
The displayed tree phylogenetic network model is shown to sit as a natural submodel of the graphical model associated to a directed acyclic graph (DAG). This representation allows to derive a number of results about the displayed tree…
Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a dominating induced matching (d.i.m.) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The Dominating Induced Matching (DIM) problem asks…
Every directed acyclic graph (DAG) over a finite non-empty set of variables (= nodes) N induces an independence model over N, which is a list of conditional independence statements over N.The inclusion problem is how to characterize (in…
If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…
Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a…
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…
A graphical model is a statistical model that is associated to a graph whose nodes correspond to variables of interest. The edges of the graph reflect allowed conditional dependencies among the variables. Graphical models admit…
A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be…
Given two subsets A and B of nodes in a directed graph, the conduciveness of the graph from A to B is the ratio representing how many of the edges outgoing from nodes in A are incoming to nodes in B. When the graph's nodes stand for the…
We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex…
We present a simple characterization of equivalent Bayesian network structures based on local transformations. The significance of the characterization is twofold. First, we are able to easily prove several new invariant properties of…