Given a finite directed acyclic graph, the space of non-negative unit flows is a lattice polytope called the flow polytope of the graph. We consider the volumes of flow polytopes for directed acyclic graphs on $n+1$ vertices with a fixed degree sequence, with a focus on graphs having in- and out-degree two on every internal vertex. When the out-degree of the source is three and the number of vertices is fixed, we prove that there is an interchange operation on the edge set of these graphs that induces a partial order on the graphs isomorphic to a Boolean algebra. Further, we prove that as we move up through this partial order, the volumes of the corresponding flow polytopes weakly decrease. Finally, we show that each such graph is strongly planar and we provide an alternative interpretation of our results in the context of linear extensions for posets that are bipartite non-crossing trees.
The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection.
Given a finite quiver (directed graph) without loops and multiedges, the convex hull of the column vector of the incidence matrix is called the directed edge polytope and is an interesting example of lattice polytopes. In this paper, we give a complete characterization of facets of the directed edge polytope of an arbitrary finite quiver without loops and multiedges in terms of the connectivity and the existence of a rank function. Our result can be regarded as an extension of the result of Higashitani et al. on facets of symmetric edge polytopes to directed edge polytopes. When the quiver in …
It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-polynomials. In particular, we show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F-polynomials, or of the cluster variables, respectively.
Let $H$ be a directed acyclic graph other than a rooted star. It is known that there are constants $c(H)$ and $C(H)$ such that the following holds for the complete directed graph $D_n$. There are at most $C\log n$ directed acyclic subgraphs of $D_n$ that cover every $H$-copy of $D_n$, while fewer than $c\log n$ directed acyclic subgraphs of $D_n$ do not cover all $H$-copies. Here this dichotomy is considerably strengthened. Let ${\vec G}(n,p)$ denote the random directed graph. The {\em fractional arboricity} of $H$ is $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where the maximum is over all no…
We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark-unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark-unmark sequence can also be realized by a mark sequence. We also investigate the quest…
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.
The Kuramoto model is one of the most widely studied model for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform heterogeneous sparse network is an important yet difficult problem due to the complicated nonlinear interactions. In this paper, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks that will form the foundation of a divide-and-conquer approach for studying the frequency sy…
We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree…
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a charact…
In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of directed acyclic graphs and the number of strongly connected directed graphs.
The polytopes $\mathcal{U}_{I,\bar{J}}$ were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of $(I,\bar{J})$-Tamari lattices. They observed a connection between certain $\mathcal{U}_{I,\bar{J}}$ and acyclic root polytopes, and wondered if Mészáros' subdivision algebra can be used to subdivide all $\mathcal{U}_{I,\bar{J}}$. We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that $\mathcal{U}_{I,\bar{J}}$ is integrally equivalent to a flow polytope that can be subdivided using t…
In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit Kähler--Einstein metrics.
This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs".
For a lattice path $ν$ from the origin to a point $(a,b)$ using steps $E=(1,0)$ and $N=(0,1)$, we construct an associated flow polytope $\mathcal{F}_{\hat{G}_B(ν)}$ arising from an acyclic graph where bidirectional edges are permitted. We show that the flow polytope $\mathcal{F}_{\hat{G}_B(ν)}$ admits a subdivision dual to a $w$-simplex, where $w$ is the number of valleys in the path $\barν = EνN$. Refinements of this subdivision can be obtained by reductions of a polynomial $P_ν$ in a generalization of Mészáros' subdivision algebra for acyclic root polytopes where negative roots are allowed. …
Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every non-empty face F of P, the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of its graph. This article is a broader study of ``good orientations'' (of the graphs) on matroid polytopes.
F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation bet…
A descent of a labeled acyclic digraph is a directed edge $x\to y$ with $x>y$. In this paper, we find a recurrence for the number of labeled acyclic digraphs with a given number of descents.
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.
Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and $n$ is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of $P$ terminates after $O(\log^2 n)$ steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.
Boltzmann sampling and optimal exact-size sampling for directed acyclic graphs
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072（卡昂诺曼底大学，ENSICAEN，CNRS，诺曼底大学，GREYC联合研究实验室）
The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak (1960). Fishburn (1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). Apart for a recognition of the facets by Suck (2002), the geometric structure of the MCP was apparently not much investigated. Recently, Chang, Narita and Saito (2022) refer to the adjacency of vertices while Turansick (2022) uses a condition which we show to be equivalent to the non-adjacency of two v…
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
A well-known theorem of Blind and Mani says that every simple polytope is uniquely determined by its graph. Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai's proof can be suitably generalized without much effort. We apply our results to a special class of cubical polytopes.
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
We consider a well known model of random directed acyclic graphs of order $n$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree $d\ge2$ and the endpoints of the $d$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number $X$ of vertices that are descendants of $n$. We show that $X/\sqrt n$ converges in distribution; the limit distribution is, up to a constant factor, given by the $d$th root of a Gamma distributed variable. $Γ(d/(d-1))$. When $d=2$, the limit distribution can also be described as a …
A positive definite completion problem pertains to determining whether the unspecified positions of a partial (or incomplete) matrix can be completed in a desired subclass of positive definite matrices. In this paper we study an important and new class of positive definite completion problems where the desired subclasses are the spaces of covariance and inverse-covariance matrices of probabilistic models corresponding to directed acyclic graph models (also known as Bayesian networks). We provide fast procedures that determine whether a partial matrix can be completed in either of these spaces …
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 constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed and bidirected edges, we g…
We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $η_t…
