A graphical characterization of the largest chain graphs

The paper presents a graphical characterization of the largest chain graphs which serve as unique representatives of classes of Markov equivalent chain graphs. The characterization is a basis for an algorithm constructing, for a given chain graph, the largest chain graph equivalent to it. The algorithm was used to generate a catalog of the largest chain graphs with at most five vertices. Every item of the catalog contains the largest chain graph of a class of Markov equivalent chain graphs and an economical record of the induced independency model.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

Learning marginal AMP chain graphs under faithfulness revisited

Marginal AMP chain graphs are a recently introduced family of models that is based on graphs that may have undirected, directed and bidirected edges. They unify and generalize the AMP and the multivariate regression interpretations of chain graphs. In this paper, we present a constraint based algorithm for learning a marginal AMP chain graph from a probability distribution which is faithful to it. We show that the marginal AMP chain graph returned by our algorithm is a distinguished member of its Markov equivalence class. We also show that our algorithm performs well in practice. Finally, we show that the extension of Meek's conjecture to marginal AMP chain graphs does not hold, which compromises the development of efficient and correct score+search learning algorithms under assumptions weaker than faithfulness.     

An efficient algorithm for finding the largest chain graph according to a given chain graph

Chain graph (CG) is a general model of graphical Markov models. Some different chain graphs may describe the same conditional independence structure, then we say that these CGs are Markov equivalent. In 1990 Frydenberg showed that every class of Markov equivalent CGs has a CG which is called the largest chain graph with the greatest number of lines. This paper presents an efficient algorithm for finding the largest chain graph of the corresponding Markov equivalent class of a given CG. The computational complexity of the algorithm is O(n3). It is more efficient than the complexity O(n!) of the present algorithms. Also a more intuitive graphical characterization of the largest chain graph is provided based on the algorithm in this paper.     

The product representation of a locally dependent random graph

Directed graphs with random black and white colourings of edges such that the colours of edges from different vertices are mutually independent are called locally dependent random graphs. Two random graphs are equivalent if they cannot be distinguished from percolation processes on them if only the vertices are seen. A necessary and sufficient condition is given for when a locally dependent random graph is equivalent to a product random graph; that is one in which the edges can be grouped in such a way that within each group the colours of the edges are equivalent and between groups they are independent. As an application the random graph corresponding to a spatial general epidemic model is considered.     

Chain graph interpretations and their relations revisited

In this paper we study how different theoretical concepts of Bayesian networks have been extended to chain graphs. Today there exist mainly three different interpretations of chain graphs in the literature. These are the Lauritzen–Wermuth–Frydenberg, the Andersson–Madigan–Perlman and the multivariate regression interpretations. The different chain graph interpretations have been studied independently and over time different theoretical concepts have been extended from Bayesian networks to also work for the different chain graph interpretations. This has however led to confusion regarding what concepts exist for what interpretation.In this article we do therefore study some of these concepts and how they have been extended to chain graphs as well as what results have been achieved so far. More importantly we do also identify when the concepts have not been extended and contribute within these areas. Specifically we study the following theoretical concepts: Unique representations of independence models, the split and merging operators, the conditions for when an independence model representable by one chain graph interpretation can be represented by another chain graph interpretation and finally the extension of Meek's conjecture to chain graphs. With our new results we give a coherent overview of how each of these concepts is extended for each of the different chain graph interpretations.     

Learning AMP chain graphs and some marginal models thereof under faithfulness

This paper deals with chain graphs under the Andersson–Madigan–Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. Moreover, we show that the extension of Meek's conjecture to AMP chain graphs does not hold, which compromises the development of efficient and correct score + search learning algorithms under assumptions weaker than faithfulness.We also study the problem of how to represent the result of marginalizing out some nodes in an AMP CG. We introduce a new family of graphical models that solves this problem partially. We name this new family maximal covariance–concentration graphs because it includes both covariance and concentration graphs as subfamilies.     

Conditions on Ramsey Nonequivalence

Given a graph H , a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G , H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. (J Combin Theory Ser B 109 (2014), 120–133) asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Ne?et?il et al. showed that any two graphs with different clique number (Combinatorica 1(2) (1981), 199–202) or different odd girth (Comment Math Univ Carolin 20(3) (1979), 565–582) are not Ramsey equivalent. These are the only structural graph parameters we know that “distinguish” two graphs in the above sense. This article provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, the chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most five vertices are not Ramsey equivalent to any other connected graph. Moreover, two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique‐reduction properties.     

Pairs of Chromatically Equivalent Graphs

Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. In this paper we give the means to construct infinitely many pairs of chromatically equivalent graphs where one graph in the pair is clique-separable, that is, can be obtained by identifying an r-clique in some graph H ₁ with an r-clique in some graph H ₂, and the other graph is non-clique-separable. There are known methods for finding pairs of chromatically equivalent graphs where both graphs are clique-separable or both graphs are non-clique-separable. Although examples of pairs of chromatically equivalent graphs where only one of the graphs is clique-separable are known, a method for the construction of infinitely many such pairs was not known. Our method constructs such pairs of graphs with odd order n ≥ 9.     

Enumeration of matchings in families of self-similar graphs

The number of matchings of a graph G is an important graph parameter in various contexts, notably in statistical physics (dimer-monomer model). Following recent research on graph parameters of this type in connection with self-similar, fractal-like graphs, we study the asymptotic behavior of the number of matchings in families of self-similar graphs that are constructed by a very general replacement procedure. Under certain conditions on the geometry of the graphs, we are able to prove that the number of matchings generally follows a doubly exponential growth. The proof depends on an independence theorem for the number of matchings that has been used earlier to treat the special case of Sierpiński graphs. We also further extend the technique to the matching-generating polynomial (equivalent to the partition function for the dimer-monomer model) and provide a variety of examples.     

Two characterizations of chain partitioned probe graphs

Chain graphs are exactly bipartite graphs without induced 2K ₂ (a graph with four vertices and two disjoint edges). A graph G=(V,E) with a given independent set S⊆V (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if G can be extended to a chain graph by adding edges between certain vertices in S. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: G is a chain partitioned probe graph if and only if the enhanced graph G ^* is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an O(m ²)-time recognition algorithm for chain partitioned probe graphs.     

The edge intersection graphs of paths in a tree

The class of edge intersection graphs of a collection of paths in a tree (EPT graphs) is investigated, where two paths edge intersect if they share an edge. The cliques of an EPT graph are characterized and shown to have strong Helly number 4. From this it is demonstrated that the problem of finding a maximum clique of an EPT graph can be solved in polynomial time. It is shown that the strong perfect graph conjecture holds for EPT graphs. Further complexity results follow from the observation that every line graph is an EPT graph. The class of EPT graphs is equivalent to the class of fundamental cycle graphs.     

Simple Markov‐chain algorithms for generating bipartite graphs and tournaments

We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the near‐regular case, i.e., when all the degrees are almost equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs, which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near‐regular. The proof techniques we use for the two problems are similar. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 293–308, 1999     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

On expressiveness of the chain graph interpretations

In this article we study the expressiveness of the different chain graph interpretations. Chain graphs is a class of probabilistic graphical models that can contain two types of edges, representing different types of relationships between the variables in question. Chain graphs is also a superclass of directed acyclic graphs, i.e. Bayesian networks, and can thereby represent systems more accurately than this less expressive class of models. Today there do however exist several different ways of interpreting chain graphs and what conditional independences they encode, giving rise to different so-called chain graph interpretations. Previous research has approximated the number of representable independence models for the Lauritzen–Wermuth–Frydenberg and the multivariate regression chain graph interpretations using an MCMC based approach. In this article we use a similar approach to approximate the number of models representable by the latest chain graph interpretation in research, the Andersson–Madigan–Perlman interpretation. Moreover we summarize and compare the different chain graph interpretations with each other. Our results confirm previous results that directed acyclic graphs only can represent a small fraction of the models representable by chain graphs, even for a low number of nodes. The results also show that the Andersson–Madigan–Perlman and multivariate regression interpretations can represent about the same amount of models and twice the amount of models compared to the Lauritzen–Wermuth–Frydenberg interpretation. However, at the same time almost all models representable by the latter interpretation can only be represented by that interpretation while the former two have a large intersection in terms of representable models.     

Random graphons and a weak Positivstellensatz for graphs

In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2‐variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a “sum of squares” in the algebra of partially labeled graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory     

On a new reformulation of Hadwiger's conjecture

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwiger's conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let F be a finite set of 2-connected graphs, and let C be the class of graphs with no minor from F. We prove that if C has a maximum, then any maximum of C must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nešet?il and Ossona de Mendez.     

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A two-dimensional framework (G,p) is a graph G = (V,E) together with a map p: V → ℝ². We view (G,p) as a straight line realization of G in ℝ². Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G. The graph G is globally rigid if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs. As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3].     

Chip-firing via open covers

We investigate chip-firing with respect to open covers of discrete graphs and metric graphs. For the case of metric graphs we show that given an open cover and a sink q, stabilization of a divisor D is unique and that there is a distinguished configuration equivalent to D, which we call the critical configuration. Also, we show that given a double cover of the metric graph by stars, which is the continuous analogue of the sandpile model, the critical configurations are in bijection with reduced divisors. Passing to the discrete case, we interpret open covers of a graph as simplicial complexes on the vertex and observe that chip-firing with respect to a simplicial complex is equivalent to the model introduced by Paoletti [G. Paoletti. July 11 2007: Master in Physics at University of Milan, defending thesis “Abelian sandpile models and sampling of trees and forests”; supervisor: Prof. S. Caracciolo. http://pcteserver.mi.infn.it/caraccio/index.html]. We generalize this setup for directed graphs using weighted simplicial complexes on the vertex set and show that the fundamental results extend. In the undirected case we present a generalization of the Cori-Le Borgne algorithm for chip-firing models via open covers, giving an explicit bijection between the critical configurations and the spanning trees of a graph.(http://www.elsevier.com/locate/endm)