Marked Graphs Research Articles

2-Knot homology and Yoshikawa move

Ng constructed an invariant of knots in [Formula: see text], a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in [Formula: see text] using marked graph diagrams.

ОБОБЩЕННАЯ ФУНКЦИОНАЛЬНАЯ МОДЕЛЬ ХИМИЧЕСКОГО ПРОИЗВОДСТВА И ЕЕ ТЕОРЕТИКО-МНОЖЕСТВЕННОЕ ПРЕДСТАВЛЕНИЕ

The mechanism of constructing functional models in the IDEF0 notation is considered. The process of creating a marked graph is described, it includes adding service vertices and arcs by converting the graph from a separate diagram. A generalized functional model of one-stage chemical production has been developed. A settheoretic description of a graph describing a top-level functional model is presented.

Tosha-Degree Equivalence Signed Graphs

The Tosha-degree of an edge $\alpha $ in a graph $\Gamma$ without multiple edges, denoted by $T(\alpha)$, is the number of edges adjacent to $\alpha$ in $\Gamma$, with self-loops counted twice. A signed graph (marked graph) is an ordered pair $\Sigma=(\Gamma,\sigma)$ ($\Sigma =(\Gamma, \mu)$), where $\Gamma=(V,E)$ is a graph called the underlying graph of $\Sigma$ and $\sigma : E \rightarrow \{+,-\}$ ($\mu : V \rightarrow \{+,-\}$) is a function. In this paper, we define the Tosha-degree equivalence signed graph of a given signed graph and offer a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and $ k^{th}$ iterated Tosha-degree equivalence signed graphs. It is shown that for any signed graph $\Sigma$, its Tosha-degree equivalence signed graph $T(\Sigma)$ is balanced and we offer a structural characterization of Tosha-degree equivalence signed graphs

Complexes of marked graphs in gauge theory

We review the gauge and ghost cyle graph complexes as defined by Kreimer, Sars and van Suijlekom in “Quantization of gauge fields, graph polynomials and graph homology” and compute their cohomology. These complexes are generated by labelings on the edges or cycles of graphs and the differentials act by exchanging these labels. We show that both cases are instances of a more general construction of double complexes associated with graphs. Furthermore, we describe a universal model for these kinds of complexes which allows to treat all of them in a unified way.

Graph-based STA for asynchronous controllers

We present a Graph-based Asynchronous Static Timing Analysis (ASTA) methodology for Asynchronous Control Circuits, which pessimistically computes Critical Cycle(s), instead of Critical Paths, without cycle cutting. Its additional requirement over STA is a graph-based Event Model, Marked Graph or Petri Net. We contrast STA, ASTA results for 23 asynchronous circuit benchmarks, and demonstrate significant timing differences between the ASTA critical cycle and STA critical path, with cut cycles. We also demonstrate our correlation to SPICE level simulations, for 20 of the 23 circuits. Our ASTA flow effectively upper bounds critical cycle delay over SPICE, and is orders of magnitude faster.

Complex of Models for Network Security Assessment of Industrial Automated Control Systems

The modern enterprises apply network technologies to their automated industrial control systems. Along with advantages of the above approach the risk of network attacks on automated control systems increases significantly. Hence there is an urgent need to develop automated monitoring means being capable of unauthorized access detection and of an adequate response to it. The enterprise security system should take into account components interaction and involve the ability of self-renewal throughout the entire life cycle.
 The partial models of functioning of automated control systems of an enterprise under information threats are offered taking into account parameters of states of the enterprise at its different levels, realization of network threats, counteraction measures, etc. For each model it is possible to form the state space of a part of an enterprise and on the basis of the series of tests to define state transition parameters thus enabling model representation in the form of a marked graph. The sequences of states possess the properties of semi-Markov processes so semi-Markov apparatus is applicable. Probabilities of state transitions could be computed as a result of numerical solution of the corresponding system of integral equations by Lagrange-Stieltjes technique.
 Application of Semi-Markov apparatus for the detection of non-authorized activities during data transfer under network scanning attack proved the validity of the above methods. In addition its application results in creation of a set of security assurance measures to be undertaken. Having obtained state transition probabilities the development of integral security indicator becomes possible thus contributing to the enterprise performance enhancement.

An enumeration of immersed surface-links in 4-space

Immersed surface-links can be described by ch-graphs derived from marked graph diagrams, analogously to the case of surface-links. By using the ch-graph description, we enumerate immersed surface-links in 4-space, up to ch-index 9.

Universal Lossless Compression of Graphical Data

Graphical data is comprised of a graph with marks on its edges and vertices. The mark indicates the value of some attribute associated to the respective edge or vertex. Examples of such data arise in social networks, molecular and systems biology, and web graphs, as well as in several other application areas. Our goal is to design schemes that can efficiently compress such graphical data without making assumptions about its stochastic properties. Namely, we wish to develop a universal compression algorithm for graphical data sources. To formalize this goal, we employ the framework of local weak convergence, also called the objective method, which provides a technique to think of a marked graph as a kind of stationary stochastic processes, stationary with respect to movement between vertices of the graph. In recent work, we have generalized a notion of entropy for unmarked graphs in this framework, due to Bordenave and Caputo, to the case of marked graphs. We use this notion to evaluate the efficiency of a compression scheme. The lossless compression scheme we propose in this paper is then proved to be universally optimal in a precise technical sense. It is also capable of performing local data queries in the compressed form.

Biquandle module invariants of oriented surface-links

We define invariants of oriented surface-links by enhancing the biquandle counting invariant using biquandle modules, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials.

A Prediction Method of Node Attack Path based on Optimized-AG

An optimization algorithm for reducing the scale of attack graphs and an attack path prediction method are presented herein to cover the difficulty of predicting the attack path in traditional attack graphs. Firstly, a new distance cost attack graph model is generated and converted into a marked attack graph via the algorithm. Secondly, the original attack graph is optimized by introducing an attack distance weight value and a path redundancy coefficient in order to determine the possible attack paths and calculate the cost of path attack behavior. Finally, the preferred attack path is determined based on an evaluation function. According to the experimental results, the above scheme eliminated the path redundancy of attack graphs and achieved comprehensive prediction of attack paths.

Independence of the Yoshikawa eighth move and a minimal generating set of band moves

Yoshikawa moves were introduced at least quarter-century ago and are still actively used by researchers. For any marked graph diagram we will define its twisted diagram and its mirror cut surface. By using a surface-link group of a mirror cut surface of a twisted diagram we will prove the independence of Yoshikawa eighth move. As a consequence we will establish a minimal generating set of band moves for links with bands.

Analysis and Synthesis of Weighted Marked Graph Petri Nets: Exact and Approximate Methods

Numerous real-world systems can be modeled with Petri nets, which allow a combination of concurrency with synchronizations and conflicts. To alleviate the difficulty of checking their behaviour, a common approach consists in studying specific subclasses. In the converse problem of Petri net synthesis, a Petri net of some subclass has to be constructed efficiently from a given specification, typically from a labelled transition system (lts) describing the behaviour of the desired net. In this paper, we focus on a notorious subclass of persistent Petri nets, the weighted marked graphs (WMGs), also called generalised (or weighted) event (or marked) graphs or weighted T-nets. In such nets, edges have multiplicities (weights) and each place has at most one ingoing and one outgoing transition. Although extensively studied in previous works and benefiting from strong results, both their analysis and synthesis can be further investigated. We provide new behavioural properties of WMGs expressed on their reachability graph, notably backward persistence and strong similarities between any two sequences sharing the same starting state and the same destination state. Besides, we design a general synthesis procedure aiming at the WMG class. Finally, when no solution to the synthesis problem exists, i.e., when the given lts is not WMG-solvable, we show how to construct a WMG whose reachability graph is a minimal over-approximation of the given lts.

On the Design of Time-Constrained and Buffer-Optimal Self-Timed Pipelines

Pipelining is a powerful technique to achieve high performance in computing systems. However, as computing platforms become large-scale and integrate with heterogeneous processing elements (PEs) (CPUs, GPUs, field-programmable gate arrays, etc.), it is difficult to employ a global clock to achieve synchronous pipelines. Therefore, self-timed (or asynchronous) pipelines are usually adopted. Nevertheless, due to their complex running behavior, the performance modeling and systematic optimizations for self-timed pipeline (STP) systems are more complicated than those for synchronous ones. This paper employs marked graph theory to model STPs and presents algorithms to detect performance bottlenecks. Based on the proposed model, we observe that the system performance can be improved by inserting buffers. Due to the limited memory resources on the PEs, it is critical to minimize the number of buffers for STPs while satisfying the required timing constraints. In this paper, we propose integer linear programming formulations to obtain the optimal solutions and devise efficient algorithms to obtain the near-optimal solutions. Experimental results show that the proposed algorithms can achieve 53.10% improvement in the maximum performance and 54.04% reduction in the number of buffers, compared with the technique for the slack matching problem.

An improved approach for marking optimization of timed weighted marked graphs

Timed weighted marked graphs are a mathematical formalism suitable to model automated manufacturing systems in which synchronization and bulk services and arrivals appear, such as assembly lines and kanban systems. In this paper, we aim to develop practically efficient methods for the marking optimization of timed weighted marked graphs, a problem which consists in finding an initial resource assignment to minimize the cost of resources under a given requirement on the cycle time. Starting with a live initial marking, we first compute the critical places of a timed weighted marked graph by exploring an equivalent model called timed marked graph. Then, we develop an analytical method to identify the critical circuit of the system to which tokens will be iteratively added. Application to a real manufacturing system is finally provided, which shows that the developed approach is significantly more efficient than the existing ones.

Estimation of Least-Cost Transition Firing Sequences in Labeled Petri Nets by Using Basis Reachability Graph

This paper develops an approach for solving the problem of least-cost transition firing sequences estimation in labeled Petri nets with unobservable transitions. Each transition in the net is labeled as either observable or unobservable, and has a nonnegative cost. Additionally, we assume that the net system is bounded, the unobservable subnet is acyclic, and the cost of each unobservable transition is strictly positive. We propose the method mainly on the basis of the notions of basis marking and basis reachability graph (BRG), which is a compact representation of the reachability graph of the net system. By sacrificing extra storage space to keep the BRG and other necessary information in memory, some time-consuming computations are moved off-line. This makes our proposed approach more feasible for on-line applications. Finally, we present a brief survey and comparison of some representative methods that use labeled Petri nets as a formalism in the literature.

Generalizations of a Conway algebra for oriented surface-links via marked graph diagrams

In 1987, Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the HOMFLY-PT polynomial invariant. In 2018, Kim generalized a Conway algebra, which is an algebraic structure with two skein relations, which is called a generalized Conway algebra. In 2017, Joung, Kamada, Kawauchi and Lee constructed a polynomial invariant of oriented surface-links by using marked graph diagrams. In this paper, we will introduce generalizations [Formula: see text] and [Formula: see text] of a Conway algebra and a generalized Conway algebra, which are called a marked Conway algebra and a generalized marked Conway algebra, respectively. We will construct invariants valued in [Formula: see text] and [Formula: see text] for oriented marked graphs and oriented surface-links by applying binary operations to classical crossings and marked vertices via marked graph diagrams. The polynomial invariant of oriented surface-links is obtained from the invariant valued in the marked Conway algebra with additional conditions.

Biquandle cohomology and state-sum invariants of links and surface-links

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links.

Presentation of immersed surface-links by marked graph diagrams

It is well known that surface-links in [Formula: see text]-space can be presented by diagrams on the plane of [Formula: see text]-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper, we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some moves on marked graph diagrams that preserve the ambient isotopy classes of their presenting immersed surface-links.

A Driving-Gear Dynamic Analysis

A new approach of mechanical driving-gear dynamic analysis, which includes several modelling stages is observed in the article. On the first stage driving-gear is represented in the form of consistently connected rotation bodies. The driving-gear is represented in a graphic kind by means of the marked graph. On the second stage mathematical model of driving-gear performance with using of mnemonic rule is created. Mathematical model of mechanical driving-gear is a system of second-order regular differential equations (RDEs). The system of second-order regular differential equations is transformed into a system of first-order regular differential equations. There is a standard method for writing a higher-order RDE as a system of the first-order RDEs. On the third stage computer model of driving-gear performance using system Mathcad is created and initial data is defined. On the fourth stage the mechanical driving-gear modelling is performed and calculation data in numerical and graphical forms is obtained. This approach provides high level of the driving-gear dynamic analysis, including the received results presentation, which is especially important on the earliest stages of mechanical driving-gear design. The proposed procedure of mechanical driving-gear dynamic analysis using Mathcad software significantly decreases time and working costs on execution of such computations and helps to execute investigations related with changing of driving-gear elements parameters efficiently.

Performance Optimization for Timed Weighted Marked Graphs Under Infinite Server Semantics

This paper deals with the performance optimization of resource allocation systems with the aim of maximizing the system's throughput under a given budget for acquiring resources. Resources are assumed to be renewable, i.e., they are not consumed by the operations and become available again after they have been released. The systems under consideration are modeled by a subclass of timed Petri nets called deterministic timed weighted marked graphs. In addition, we take into account infinite server semantics, i.e., the degree of self-concurrency of each transition is infinite. We propose an approach that provides an optimal solution, but has a high computational cost. For this reason, we also present two different approaches that can find suboptimal solutions with a reduced computational cost. The performances of the proposed approaches are compared by means of numerical simulations.