Essential Graphs Research Articles

Genus two nilpotent graphs of finite commutative rings

Let [Formula: see text] be a finite commutative ring and [Formula: see text] is nilpotent for some [Formula: see text]. The nilpotent graph [Formula: see text] of [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is nilpotent. In this paper, we observe a relationship between zerodivisor graphs, essential graphs and nilpotent graphs of [Formula: see text]. In continuation of genus characterizations in [T. Asir K. Mano and T. Tamizh Chelvam, Correction to: Classification of non-local rings with genus two zerodivisor graphs, Soft Comput. 25 (2021) 3355–3356; K. Selvakumar and M. Subajini, Finite commutative ring with genus two essential 10 graph, J. Algebra Appl. 16(2) (2018) 1850121], we classify all finite commutative rings (up to isomorphism) whose nilpotent graphs are of genus two.

A simple interpretation of undirected edges in essential graphs is wrong.

Artificial intelligence for causal discovery frequently uses Markov equivalence classes of directed acyclic graphs, graphically represented as essential graphs, as a way of representing uncertainty in causal directionality. There has been confusion regarding how to interpret undirected edges in essential graphs, however. In particular, experts and non-experts both have difficulty quantifying the likelihood of uncertain causal arrows being pointed in one direction or another. A simple interpretation of undirected edges treats them as having equal odds of being oriented in either direction, but I show in this paper that any agent interpreting undirected edges in this simple way can be Dutch booked. In other words, I can construct a set of bets that appears rational for the users of the simple interpretation to accept, but for which in all possible outcomes they lose money. I put forward another interpretation, prove this interpretation leads to a bet-taking strategy that is sufficient to avoid all Dutch books of this kind, and conjecture that this strategy is also necessary for avoiding such Dutch books. Finally, I demonstrate that undirected edges that are more likely to be oriented in one direction than the other are common in graphs with 4 nodes and 3 edges.

Objective Bayes model selection of Gaussian interventional essential graphs for the identification of signaling pathways

A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.

Characterizing the Minimal Essential Graphs of Maximal Ancestral Graphs

Learning ancestor graph is a typical NP-hard problem. We consider the problem to represent a Markov equivalence class of ancestral graphs with a compact representation. Firstly, the minimal essential graph is defined to represent the equivalent class of maximal ancestral graphs with the minimum number of invariant arrowheads. Then, an algorithm is proposed to learn the minimal essential graph of ancestral graphs based on the detection of minimal collider paths. It is the first algorithm to use necessary and sufficient conditions for Markov equivalence as a base to seek essential graphs. Finally, a set of orientation rules is presented to orient edge marks of a minimal essential graph. Theory analysis shows our algorithm is sound, and complete in the sense of recognizing all minimal collider paths in a given ancestral graph. And the experiment results show we can discover all invariant marks by these orientation rules.

On the essential graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if ann[Formula: see text] is an essential ideal. It is proved that [Formula: see text] is connected with diameter at most three and with girth at most four, if [Formula: see text] contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

Approximate Counting of Graphical Models via MCMC Revisited

We apply Markov chain Monte Carlo (MCMC) sampling to approximately calculate some quantities, and discuss their implications for learning directed and acyclic graphs (DAGs) from data. Specifically, we calculate the approximate ratio of essential graphs (EGs) to DAGs for up to 31 nodes. Our ratios suggest that the average Markov equivalence class is small. We show that a large majority of the classes seem to have a size that is close to the average size. This suggests that one should not expect more than a moderate gain in efficiency when searching the space of EGs instead of the space of DAGs. We also calculate the approximate ratio of connected EGs to connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs. These new ratios are interesting because, as we will see, the DAG or EG learnt from some given data is likely to be connected. Furthermore, we prove that the latter ratio is asymptotically 1. Finally, we calculate the approximate ratio of EGs to largest chain graphs for up to 25 nodes. Our ratios suggest that Lauritzen–Wermuth–Frydenberg chain graphs are considerably more expressive than DAGs. We also report similar approximate ratios and conclusions for multivariate regression chain graphs.

Polynomial Time Recognition of Essential Graphs Having Stability Number Equal to Matching Number

For a graph $$G$$G let $$\alpha (G)$$?(G) and $$\mu (G)$$μ(G) denote respectively the cardinality of a maximum stable set and of a maximum matching of $$G$$G. It is well-known that computing $$\alpha (G)$$?(G) is NP-hard and that computing $$\mu (G)$$μ(G) can be done in polynomial time. In particular checking if $$\alpha (G)=\mu (G)$$?(G)=μ(G) is NP-complete and relies on the fact that computing $$\alpha (G)$$?(G) is NP-hard (Mosca, Graphs Combinat 18:367---379, 2002). A well known result of Hammer et al. (SIAM J Alg Disc Math 3(4):511---522, 1982). states that the vertex-set of a graph $$G$$G can be efficiently and uniquely partitioned in two subsets (possibly empty) $$A$$A and $$B$$B, such that $$G[A]$$G[A] has the Konig-Egervary property while $$G[B]$$G[B] can be covered by pairwise disjoint edges and odd cycles: furthermore, one has $$\alpha (G) = \alpha (G[A]) + \alpha (G[B])$$?(G)=?(G[A])+?(G[B]), where computing $$\alpha (G[A])$$?(G[A]) can be done in polynomial time. For that let us call $$essential$$essential those graphs which can be covered by pairwise disjoint edges and odd cycles (in particular computing $$\alpha (G)$$?(G) remains NP-hard for such graphs). This paper shows that: (i) for every essential graph $$G$$G, checking if $$\alpha (G)=\mu (G)$$?(G)=μ(G) can be done in polynomial time; (ii) essential graphs for which $$\alpha (G)=\mu (G)$$?(G)=μ(G) can be recognized in polynomial time and for such graph a maximum stable set can be computed in polynomial time; (iii) a new characterization of graphs which have the Konig-Egervary property can be derived in that context.

Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras

AbstractIn Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.

Chain graph models: topological sorting of meta-arrows and efficient construction of $${\mathcal{B}}$$ -essential graphs

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively, especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices\({\mathcal{B}}\) was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of \({\mathcal{B}}\) . The family of chain graphs identified by a labelled block ordering of vertices \({\mathcal{B}}\) is partitioned into equivalence classes each represented by means of a \({\mathcal{B}}\) -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of \({\mathcal{B}}\) -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.

Characterizing Markov equivalence classes for AMP chain graph models

Chain graphs (CG) (= adicyclic graphs) use undirected and directed edges to represent both structural and associative dependences. Like acyclic directed graphs (ADGs), the CG associated with a statistical Markov model may not be unique, so CGs fall into Markov equivalence classes, which may be superexponentially large, leading to unidentifiability and computational inefficiency in model search and selection. It is shown here that, under the Andersson-Madigan-Perlman (AMP) interpretation of a CG, each Markov-equivalence class can be uniquely represented by a single distinguished CG, the AMP essential graph, that is itself simultaneously Markov equivalent to all CGs in the AMP Markov equivalence class. A complete characterization of AMP essential graphs is obtained. Like the essential graph previously introduced for ADGs, the AMP essential graph will play a fundamental role for inference and model search and selection for AMP CG models.

Controlling a bipedal walking robot actuated by pleated pneumatic artificial muscles

This paper reports on the control structure of the pneumatic biped Lucy. The robot is actuated with pleated pneumatic artificial muscles, which have interesting characteristics that can be exploited for legged locomotion. They have a high power to weight ratio, an adaptable compliance and they can absorb impact effects.The discussion of the control architecture focuses on the joint trajectory generator and the joint trajectory tracking controller. The trajectory generator calculates trajectories represented by polynomials based on objective locomotion parameters, which are average forward speed, step length, step height and intermediate foot lift. The joint trajectory tracking controller is divided in three parts: a computed torque module, a delta-p unit and a bang-bang pressure controller. The control design is formulated for the single support and double support phase, where specifically the trajectory generator and the computed torque differs for these two phases.The first results of the incorporation of this control architecture in the real biped Lucy are given. Several essential graphs, such as pressure courses, are discussed and the effectiveness of the proposed algorithm is shown by the small deviations between desired and actual attained objective locomotion parameters.

On Block Ordering of Variables in Graphical Modelling

In graphical modelling, the existence of substantive background knowledge on block ordering of variables is used to perform structural learning within the family of chain graphs in which every block corresponds to an undirected graph and edges joining vertices in different blocks are directed in accordance with the ordering. We show that this practice may lead to an inappropriate restriction of the search space and introduce the concept of labelled block ordering B corresponding to a family of B-consistent chain graphs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a chain graph. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of chain graphs considered in the literature are families of B-consistent chain graphs for the appropriate choice of B. Structural learning within a family of B-consistent chain graphs requires to deal with Markov equivalence. We provide a graphical characterisation of equivalence classes of B-consistent chain graphs, namely the B-essential graphs, as well as a procedure to construct the B-essential graph for any given equivalence class of B-consistent chain graphs. Both largest chain graphs and essential graphs turn out to be special cases of B-essential graphs.

ON BIALGEBRAS ASSOCIATED WITH PATHS AND ESSENTIAL PATHS ON ADE GRAPHS

We define a graded multiplication on the vector space of essential paths on a graph G (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length-preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product. It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

. A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags. Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs, whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs. In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.

CHARACTERIZATION OF ESSENTIAL GRAPHS BY MEANS OF THE OPERATION OF LEGAL MERGING OF COMPONENTS

One of the most common ways of representing classes of equivalent Bayesian networks is the use of essential graphs which are also known in the literature as completed patterns or completed pdags. The name essential graph was proposed by Andersson, Madigan and Perlman who also gave a graphical characterization of essential graphs. In this paper an alternative characterization of essential graphs is presented. The main observation is that every essential graph is the largest chain graph within a special class of chain graphs. More precisely, every equivalence class of Bayesian networks is contained in an equivalence class of chain graphs without flags (= certain induced subgraphs). A special operation of legal merging of (connectivity) components for a chain graph without flags is introduced. This operation leads to an algorithm for finding the essential graph on the basis of any graph in that equivalence class of chain graphs without flags which contains the equivalence class of a Bayesian network. In particular, the algorithm may start with any Bayesian network.

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.

Left Cells in the Affine Weyl Group of TypeC4

We find a representative set of left cells of the affine Weyl groupWaof typeC4as well as its left cell graphs by applying an algorithm. This algorithm was designed in my previous paper (Tôhoku J. Math.46,1994, 105–124). It is reformulated and improved in a more efficient form here. These representatives of left cells are presented as the vertices of so-called essential graphs so that the generalized τ-invariants of left cells ofWaare actually described explicitly, the latter almost characterize the left cells ofWa. Some comments and conjectures are proposed on cells of affine Weyl groups, mostly for the case of typeCl,l≥2.

Left Cells in the Affine Weyl Group of TypeF̃4

By applying an algorithm designed before, we complete the description for all the left cells of the affine Weyl groupWaof typeF̃4by finding a representative set of its left cells together with all its left cell graphs (or with all the associated essential graphs) in each of its two-sided cells. The generalized τ-invariants of left cells ofWaare exhibited graphically. A group-theoretical interpretation is given on the numbers of left cells ofWain some two-sided cells. Thus so far the left cells of all the affine Weyl groups of ranks less than or equal to 4 have been known explicitly. Some techniques are developed in applying the algorithm. As a consequence, we complete the verification of a conjecture concerning the characterization of left cells of Weyl groups and affine Weyl groups.

A characterization of Markov equivalence classes for acyclic digraphs

Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.

Turbulent flow in annular pipes

AbstractAn extension to annular conduits of the well‐known method of calculating the resistance coefficient for turbulent flow in circular pipes is presented. The method is new in that it does not borrow any information from laminar flow theory; it gives, therefore, a location for the maximum velocity based only on turbulent flow characteristics. Expressions and a few essential graphs are given for the average as well as inner‐ and outer‐wall resistance coefficients for both hydraulically smooth and rough regimes and also for the transitional regime. A comparison with available experimental information for fully developed turbulent flow is presented.