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1
Il’ev, A. V., and V. P. Il’ev. "ALGORITHMS FOR SOLVING SYSTEMS OF EQUATIONS OVER VARIOUS CLASSES OF FINITE GRAPHS." Prikladnaya Diskretnaya Matematika, no. 53 (2021): 89–102. http://dx.doi.org/10.17223/20710410/53/6.
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The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n > 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.
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Asif, Muhammad, Hamad Almohamedh, Muhammad Hussain, Khalid M. Alhamed, Abdulrazaq A. Almutairi, and Sultan Almotairi. "An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths." Complexity 2021 (June 24, 2021): 1–13. http://dx.doi.org/10.1155/2021/3745862.
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Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γ n k , l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ . Let A α Γ n k , l and A α β Γ n k , l be the transformed graphs under the fact of transformations A α and A α β , 0 ≤ α ≤ l , 0 ≤ β ≤ k − 1 , respectively. In this work, we obtained new inequalities for the graph family Γ n k , l and transformed graphs A α Γ n k , l and A α β Γ n k , l which involve GA Γ . The presence of GA Γ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.
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Ustimenko, Vasyl O., and Oleksandr S. Pustovit. "On security of GIS systems with N-tier architecture and family of graph based ciphers." Environmental safety and natural resources 47, no. 3 (September 29, 2023): 113–32. http://dx.doi.org/10.32347/2411-4049.2023.3.113-132.
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Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.
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RANAWAKE, U. A., P. M. LENDERS, and S. M. GOODNICK. "ON LOWER BOUNDS FOR THE COMMUNICATION VOLUME IN DISTRIBUTED SYSTEMS." Parallel Processing Letters 01, no. 02 (December 1991): 125–33. http://dx.doi.org/10.1142/s0129626491000070.
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In this paper we derive a lower bound for the total communication volume when mapping arbitrary task graphs onto a distributed processor system. For a K processor system this lower bound can be computed with only the K (possibly) largest eigen values of the adjacency matrix of the task graph and the eigen values of the adjacency matrix of the processor graph. We also derive the eigen values of the adjacency matrix of the processor graph for a hypercube multiprocessor and illustrate the concept with a simple example for the two processor case.
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Lobov, Alexander A., and Mikhail B. Abrosimov. "About uniqueness of the minimal 1-edge extension of hypercube Q4." Prikladnaya Diskretnaya Matematika, no. 58 (2023): 84–93. http://dx.doi.org/10.17223/20710410/58/8.
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One of the important properties of reliable computing systems is their fault tolerance. To study fault tolerance, you can use the apparatus of graph theory. Minimal edge extensions of a graph are considered, which are a model for studying the failure of links in a computing system. A graph G* = (V*,α*) with n vertices is called a minimal k-edge extension of an n-vertex graph G = (V, α) if the graph G is embedded in every graph obtained from G* by deleting any of its k edges and has the minimum possible number of edges. The hypercube Qn is a regular 2n-vertex graph of order n, which is the Cartesian product of n complete 2-vertex graphs K2. The hypercube is a common topology for building computing systems. Previously, a family of graphs Q*n was described, whose representatives for n>1 are minimal edge 1-extensions of the corresponding hypercubes. In this paper, we obtain an analytical proof of the uniqueness of minimal edge 1-extensions of hypercubes for n≤4 and establish a general property of an arbitrary minimal edge 1-extension of a hypercube Qn for n>2: it does not contain edges connecting vertices, the distance between which in the hypercube is equal to 2.
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Chen, Haiyan, and Fuji Zhang. "Spectral Dynamics of Graph Sequences Generated by Subdivision and Triangle Extension." Electronic Journal of Linear Algebra 32 (February 6, 2017): 454–63. http://dx.doi.org/10.13001/1081-3810.3583.
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For a graph G and a unary graph operation X, there is a graph sequence \G_k generated by G_0=G and G_{k+1}=X(G_k). Let Sp({G_k}) denote the set of normalized Laplacian eigenvalues of G_k. The set of limit points of \bigcup_{k=0}^\infty Sp(G_k)$, $\liminf_{k\rightarrow\infty}Sp(G_k) and $\limsup_{k\rightarrow \infty}Sp(G_k)$ are considered in this paper for graph sequences generated by two operations: subdivision and triangle extension. It is obtained that the spectral dynamic of graph sequence generated by subdivision is determined by a quadratic function, which is closely related to the the well-known logistic map; while that generated by triangle extension is determined by a linear function. By using the knowledge of dynamic system, the spectral dynamics of graph sequences generated by these two operations are characterized. For example, it is found that, for any initial non-trivial graph $G$, chaos takes place in the spectral dynamics of iterated subdivision graphs, and the set of limit points is the entire closed interval [0,2].
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Razumovsky, P. V., and M. B. Abrosimov. "THE MINIMAL VERTEX EXTENSIONS FOR COLORED COMPLETE GRAPHS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 13, no. 4 (2021): 77–89. http://dx.doi.org/10.14529/mmph210409.
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The article proposes the results of the search for minimal vertex extensions of undirected colored complete graphs. The research topic is related to the modelling of full fault tolerant technical systems with a different type of their objects in the terminology of graph theory. Let a technical system be Σ, then there is a graph G(Σ), which vertices reflects system’s objects and edges reflects connections between these objects. Type of each object reflected in a mapping of some color from F = {1,2…,i} to the corresponding vertex. System’s Σ vertex extension is a graph G(Σ) which contains additional vertices. System reflected by graph G(Σ) can work even if there are k faults of its objects. Complete graph is a graph where each two vertices have an edge between them. Complete graphs have no edge extensions because there is no way to add additional edge to the graph with a maximum number of edges. In other words, the system reflected by some complete graph cannot be able to resist connection faults. Therefore the article research is focused on vertex extensions only. There is a description of vertex extensions existence condition for those colored complete graphs. This paper considers generating schemes for such minimal vertex extensions along with formulas, which allows to calculate number of additional edges to have an ability to construct minimal vertex extension.
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Aslam, Adnan, Safyan Ahmad, Muhammad Ahsan Binyamin, and Wei Gao. "Calculating topological indices of certain OTIS interconnection networks." Open Chemistry 17, no. 1 (April 10, 2019): 220–28. http://dx.doi.org/10.1515/chem-2019-0029.
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AbstractRecently, increasing attention has been paid to The Optical Transpose Interconnection System (OTIS) network because of its prospective applications in architectures for parallel as well as distributed systems [27, 28]. Different interconnection networks in the context of topological indices are researched recently in [25, 26]. This article includes the computions of the general Randi´c, first and second Zagreb, general sum connectivity, first and second multiple zagreb, hyper zagreb, ABC and GA indices for OTIS (swapped and biswapped) networks by taking path and k-regular graph on n vertices as a base graphs. In addition, some delicated formulas are also obtained for the ABC4 and GA5 indices for the OTIS biswapped networks by considering basis graph as a path and k-regular graph of order n.
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Shirdel, G. H., M. Ghanbari, and M. Ramezani. "Generalized 3-Rainbow Domination in Graphs and Honeycomb System (HC(n))." Utilitas Mathematica 119, no. 1 (March 31, 2024): 3–7. http://dx.doi.org/10.61091/um119-01.
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In this paper, we introduce the concept of the generalized 3 -rainbow dominating function of a graph G . This function assigns an arbitrary subset of three colors to each vertex of the graph with the condition that every vertex (including its neighbors) must have access to all three colors within its closed neighborhood. The minimum sum of assigned colors over all vertices of G is defined as the g 3 -rainbow domination number, denoted by γ g 3 r . We present a linear-time algorithm to determine a minimum generalized 3-rainbow dominating set for several graph classes: trees, paths ( P n ) , cycles ( C n ) , stars ( K 1 , n ) , generalized Petersen graphs ( G P ( n , 2 ) , GP ( n , 3 ) ) , and honeycomb networks ( H C ( n ) ) .
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Pavlopoulou, Niki. "Dynamic Diverse Summarisation in Heterogeneous Graph Streams: a Comparison between Thesaurus/Ontology-based and Embeddings-based Approaches." International Journal of Graph Computing 1, no. 1 (March 1, 2020): 70–94. http://dx.doi.org/10.35708/gc1868-126724.
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Nowadays, there is a lot of attention drawn in smart environments, like Smart Cities and Internet of Things. These environments generate data streams that could be represented as graphs, which can be analysed in real-time to satisfy user or application needs. The challenges involved in these environments, ranging from the dynamism, heterogeneity, continuity, and high-volume of these real-world graph streams create new requirements for graph processing algorithms. We propose a dynamic graph stream summarisation system with the use of embeddings that provides expressive graphs while ensuring high usability and limited resource usage. In this paper, we examine the performance comparison between our embeddings-based approach and an existing thesaurus/ontology-based approach (FACES) that we adapted in a dynamic environment with the use of windows and data fusion. Both approaches use conceptual clustering and top-k scoring that can result in expressive, dynamic graph summaries with limited resources. Evaluations show that sending top-k fused diverse summaries, results in 34% to 92% reduction of forwarded messages and redundancy-awareness with an F-score ranging from 0.80 to 0.95 depending on the k compared to sending all the available information without top-k scoring. Also, the summaries' quality follows the agreement of ideal summaries determined by human judges. The summarisation approaches come with the expense of reduced system performance. The thesaurus/ontology-based approach proved 6 times more latency-heavy and 3 times more memory-heavy compared to the most expensive embeddings-based approach while having lower throughput but provided slightly better quality summaries.
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SOBOLEVA, E. "VASSILIEV KNOT INVARIANTS COMING FROM LIE ALGEBRAS AND 4-INVARIANTS." Journal of Knot Theory and Its Ramifications 10, no. 01 (February 2001): 161–69. http://dx.doi.org/10.1142/s0218216501000809.
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We study the 4-bialgebra of graphs and the bialgebra of 4-invariants introduced by S. K. Lando. Our main goal is the investigation of the relationship between 4-invariants of graphs and weight systems arising in the theory of finite order invariants of knots. In particular, we show that the corank of the adjacency matrix of a graph leads to the weight system coming from the defining representation of the Lie algebra gl(N).
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Ionescu, Marius, and Yasuo Watatani. "C*-Algebras Associated with Mauldin–Williams Graphs." Canadian Mathematical Bulletin 51, no. 4 (December 1, 2008): 545–60. http://dx.doi.org/10.4153/cmb-2008-054-0.
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AbstractA Mauldin–Williams graph is a generalization of an iterated function system by a directed graph. Its invariant set K plays the role of the self-similar set. We associate a C*-algebra (K) with a Mauldin–Williams graph and the invariant set K, laying emphasis on the singular points. We assume that the underlying graph G has no sinks and no sources. If satisfies the open set condition in K, and G is irreducible and is not a cyclic permutation, then the associated C*-algebra (K) is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.
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Huang, Da, Jibin Yang, Xing Chen, and Xiaolin Fan. "On Consensus Indices of Triplex Multiagent Networks Based on Complete k-Partite Graph." Symmetry 14, no. 8 (August 2, 2022): 1586. http://dx.doi.org/10.3390/sym14081586.
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In this article, the performance indices on consensus problems for three-layered, multiagent systems are studied from the perspective of algebraic graph theory, where the indices can be used as a measurement of the system performance and refer to the network coherence and algebraic connectivity. Specifically, some operations of two graphs are applied to established the three-layered networks based on k-partite structure, and the mathematical expression of the coherence is derived by the methods of algebraic graph theory. We found that the operations of adding star-shaped copies or fan-graph copies will make the coherence increase by some scalars under the computations of limitation. Then, the indices of the three-layered systems with non-isomorphic topologies but the same number of nodes were compared and simulated; it is found that, when the number of nodes in the counterpart node classes tend to infinity, their difference in coherence are only relevant with the number of peripheral nodes in the sense of limitation.
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Mirakyan, Martin. "ABCDE: Approximating Betweenness-Centrality ranking with progressive-DropEdge." PeerJ Computer Science 7 (September 6, 2021): e699. http://dx.doi.org/10.7717/peerj-cs.699.
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Betweenness-centrality is a popular measure in network analysis that aims to describe the importance of nodes in a graph. It accounts for the fraction of shortest paths passing through that node and is a key measure in many applications including community detection and network dismantling. The computation of betweenness-centrality for each node in a graph requires an excessive amount of computing power, especially for large graphs. On the other hand, in many applications, the main interest lies in finding the top-k most important nodes in the graph. Therefore, several approximation algorithms were proposed to solve the problem faster. Some recent approaches propose to use shallow graph convolutional networks to approximate the top-k nodes with the highest betweenness-centrality scores. This work presents a deep graph convolutional neural network that outputs a rank score for each node in a given graph. With careful optimization and regularization tricks, including an extended version of DropEdge which is named Progressive-DropEdge, the system achieves better results than the current approaches. Experiments on both real-world and synthetic datasets show that the presented algorithm is an order of magnitude faster in inference and requires several times fewer resources and time to train.
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Vilakone, Phonexay, and Doo-Soon Park. "The Efficiency of a DoParallel Algorithm and an FCA Network Graph Applied to Recommendation System." Applied Sciences 10, no. 8 (April 23, 2020): 2939. http://dx.doi.org/10.3390/app10082939.
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This article investigates the efficiency of a doParallel algorithm and a formal concept analysis (FCA) network graph applied to recommendation systems. It is the first article using the FCA method to create a network graph and apply this graph to improve the accuracy of a recommendation system. According to the fundamental knowledge about users who have similar feature information, they may like the same items. This idea was used to create an FCA network graph. In the proposed process, the k-clique method was used to divide this network graph into various communities. A combination of the k-nearest neighbor and the betweenness centrality methods was used to find a suitable community for a new user based on the feature information similarities between the new user and an existing user in each community. Finally, a data mining method created a list of items from suitable communities and recommended them to the new user. In essence, the execution in this article uses a doParallel algorithm as a mechanism in parallel processing technology. The result of the implementation is satisfactory. It proved that the proposed method could resolve the cold-start problem in a recommendation system and may overcome the vast time consumption when a huge dataset is involved.
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LOERA, J. A., J. LEE, S. MARGULIES, and S. ONN. "Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz." Combinatorics, Probability and Computing 18, no. 4 (July 2009): 551–82. http://dx.doi.org/10.1017/s0963548309009894.
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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.
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Panić, Biljana, Nataša Kontrec, Mirko Vujošević, and Stefan Panić. "A Novel Approach for Determination of Reliability of Covering a Node from K Nodes." Symmetry 12, no. 9 (September 5, 2020): 1461. http://dx.doi.org/10.3390/sym12091461.
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In this paper, a stochastic problem of multicenter location on a graph was formulated through the modification of the existing p-center problem to determine the location of a given number of facilities, to maximize the reliability of supplying the system. The system is represented by a graph whose nodes are the locations of demand and the potential facilities, while the weights of the arcs represent the reliability, i.e., the probability that an appropriate branch is available. First, k locations of facilities are randomly determined. Using a modified Dijkstra’s algorithm, the elementary path of maximal reliability for every demand node is determined. Then, a graph of all of elementary paths for demand node is formed. Finally, a new algorithm for calculating the reliability of covering a node from k nodes (k—covering reliability) was formulated.
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Wang, Shiying, and Yunxia Ren. "g-Good-Neighbor Diagnosability of Arrangement Graphs under the PMC Model and MM* Model." Information 9, no. 11 (November 7, 2018): 275. http://dx.doi.org/10.3390/info9110275.
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Diagnosability of a multiprocessor system is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph G = ( V , E ) . In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. Under the PMC model, to diagnose the system, two adjacent nodes in G are can perform tests on each other. Under the MM model, to diagnose the system, a node sends the same task to two of its neighbors, and then compares their responses. The MM* is a special case of the MM model and each node must test its any pair of adjacent nodes of the system. As a famous topology structure, the ( n , k ) -arrangement graph A n , k , has many good properties. In this paper, we give the g-good-neighbor diagnosability of A n , k under the PMC model and MM* model.
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Chang, G. J., and F. K. Hwang. "Optimal Consecutive-k−Out−of−n Systems under a Fixed Budget." Probability in the Engineering and Informational Sciences 2, no. 1 (January 1988): 63–73. http://dx.doi.org/10.1017/s0269964800000632.
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A consecutive-k−out−of−n system is a graph with n vertices where the system fails if and only if some path of k consecutive vertices all fail. The assumption is made that qi, is the failing probability of the ith vertex and the states of the vertices are statistically independent. The problem is to find the best system, i.e., a set {q1, …,qn}, and its assignment to the vertices of the system graph that maximizes the reliability of the system, under the constraint that the product of qi's is a constant. We solve the problem for all lines and some cycles. We then extend our results to trees and show that for given n and k the best tree is a star and the worst is a line.
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Adinayev, Arthur, and Itamar Stein. "Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System." Fundamenta Informaticae 178, no. 3 (January 15, 2021): 173–85. http://dx.doi.org/10.3233/fi-2021-2002.
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In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.
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Yao, Y. C. "On Optimal Consecutive k-out-of-n: F Systems Subject to a Fixed Product of Failure Probabilities." Probability in the Engineering and Informational Sciences 3, no. 2 (April 1989): 165–73. http://dx.doi.org/10.1017/s0269964800001078.
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A consecutive−k−out−of−n:F system is an n−vertex graph where the system fails if and only if some k consecutive vertices all fail. Assuming that the n vertices have, independently, the respective failure probabilities q1,…, qn, the problem is to find, subject to πqi = Q (a constant), a set of q1,…, qn so as to maximize the system reliability. In the case that the graph is a circle (or cycle), Chang and Hwang [1] conjectured that q1 =… = qn, = Q1/n is optimal if n and k are relatively prime. In this paper, it is shown that the conjecture is true for sufficiently small Q. It is also shown by counterexample that the conjecture is not true in general.
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JENAB, K., and B. S. DHILLON. "k-OUT-OF-n SYSTEM WITH SELF-LOOP UNITS." International Journal of Reliability, Quality and Safety Engineering 12, no. 01 (February 2005): 61–73. http://dx.doi.org/10.1142/s0218539305001665.
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This paper presents an analytical approach for a k-out-of-n system with units having the failure detection, isolation, and recovery mechanisms. The approach uses the flow-graph concept and moment generating function (MGF) to analyze the reliability of the k-out-of-n system with self-loop units. This newly developed analytical approach provides the probability of system failure, system mean, and standard deviation time to failure. A solved example is presented to demonstrate the application of the approach.
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Pramanik, Tarasankar, G. Muhiuddin, Abdulaziz M. Alanazi, and Madhumangal Pal. "An Extension of Fuzzy Competition Graph and Its Uses in Manufacturing Industries." Mathematics 8, no. 6 (June 19, 2020): 1008. http://dx.doi.org/10.3390/math8061008.
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Competition graph is a graph which constitutes from a directed graph (digraph) with an edge between two vertices if they have some common preys in the digraph. Moreover, Fuzzy competition graph (briefly, FCG) is the higher extension of the crisp competition graph by assigning fuzzy value to each vertex and edge. Also, Interval-valued FCG (briefly, IVFCG) is another higher extension of fuzzy competition graph by taking each fuzzy value as a sub-interval of the interval [ 0 , 1 ] . This graph arises in many real world systems; one of them is discussed as follows: Each and every species in nature basically needs ecological balance to survive. The existing species depends on one another for food. If there happens any extinction of any species, there must be a crisis of food among those species which depend on that extinct species. The height of food crisis among those species varies according to their ecological status, environment and encompassing atmosphere. So, the prey to prey relationship among the species cannot be assessed exactly. Therefore, the assessment of competition of species is vague or shadowy. Motivated from this idea, in this paper IVFCG is introduced and several properties of IVFCG and its two variants interval-valued fuzzy k-competition graphs (briefly, IVFKCG) and interval-valued fuzzy m-step competition graphs (briefly, IVFMCG) are presented. The work is helpful to assess the strength of competition among competitors in the field of competitive network system. Furthermore, homomorphic and isomorphic properties of IVFCG are also discussed. Finally, an appropriate application of IVFCG in the competition among the production companies in market is presented to highlight the relevance of IVFCG.
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Zhang, Zhanqi, and Yingqing Xiao. "Graph Concatenations to Derive Weighted Fractal Networks." Complexity 2020 (July 18, 2020): 1–9. http://dx.doi.org/10.1155/2020/4906878.
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Given an initial weighted graph G0, an integer m>1, and m scaling factors f1,…,fm∈0,1, we define a sequence of weighted graphs Gkk=0∞ iteratively. Provided that Gk−1 is given for k≥1, we let Gk−11,…,Gk−1m be m copies of Gk−1, whose weighted edges have been scaled by f1,…,fm, respectively. Then, Gk is constructed by concatenating G0 with all the m copies. The proposed framework shares several properties with fractal sets, and the similarity dimension dfract has a great impact on the topology of the graphs Gk (e.g., node strength distribution). Moreover, the average geodesic distance of Gk increases logarithmically with the system size; thus, this framework also generates the small-world property.
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LIU, WENJUN. "The Relationship Between Extra Connectivity and t/k-Diagnosability of Regular Networks." Journal of Interconnection Networks 20, no. 01 (March 2020): 2050003. http://dx.doi.org/10.1142/s0219265920500036.
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The g-extra connectivity of a multiprocessor system modeled by a graph G, denoted by [Formula: see text] (G), is the minimum number of removed vertices such that the network is disconnected and each residual component has no less than g + 1 vertices. The t/k-diagnosis strategy can detect up to t faulty processors which might include at most k misdiagnosed processors. These two parameters are important to measure the fault tolerant ability of a multiprocessor system. The extra connectivity and t/k-diagnosability of many well-known networks have been investigated extensively and independently. However, the general relationship between the extra connectivity and the t/k-diagnosability of general regular networks has not been established. In this paper, we explore the relationship between the k-extra connectivity and t/k-diagnosability for regular networks under the classic PMC diagnostic model. More specifically, we derive the relationship between 1-extra connectivity and pessimistic diagnosability for regular networks. Furthermore, the t/k-diagnosability and pessimistic diagnosability of some networks, including star network, BC networks, Cayley graphs generated by transposition trees etc., are determined.
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Krieger, Wolfgang, and Kengo Matsumoto. "A lambda-graph system for the Dyck shift and its $K$-groups." Documenta Mathematica 8 (2003): 79–96. http://dx.doi.org/10.4171/dm/139.
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Jiang, Yaozhi. "Dialectical Logic K-Model: A Mathematical Model for Machine." Journal of Mathematics Research 9, no. 6 (October 27, 2017): 82. http://dx.doi.org/10.5539/jmr.v9n6p82.
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An axiom system for dialectical logic K-model which based on Kirchhoff energy-method is established by author in the paper. The author describes that subjective-laws is the mirror imagine reflected from objective-laws and defines that the three-step which named by sensation, abstraction and thinking in artificial intelligence. At same time, describes that axiom system for dialectical logic K-model, in which contains such as logic-variable energy conservation law, Mozi’s principle( mini-max principle) and forbidden law, etc. In the axiom system also contain such as a continuous true-value-function system valued on interval and the K-graph for logic-variable. And describes the giving value method by matrix based on K-graph satisfied Kirchhoff laws to the logic variable. The author describes simply the linear and nonlinear logic variable system. And describes simply the logic variable involved three-dimension Euclidean space and topology networks space separately. Dialectical logic K-model would supply an computation algorithm idea for machine so that the machine is able to think by dialectical logic method, thus an important information-treated method maybe the dialectical logic.
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Wu, Zhihao, and Fang Jia. "Construction and Application of a Major-Specific Knowledge Graph Based on Big Data in Education." International Journal of Emerging Technologies in Learning (iJET) 17, no. 07 (April 12, 2022): 64–79. http://dx.doi.org/10.3991/ijet.v17i07.30405.
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The main problems lying in the learning process of learners in different majors are that they study blindly and do not have a complex knowledge structure, which are seriously affecting their learning effects. But if a knowledge graph can be modeled for the major-specific curriculum system using the quantitative method from the perspective of knowledge network, it may be able to improve the existing teaching problems and optimize the teaching quality. The existing major-specific knowledge graphs were all constructed in an abstract form, ignoring the inherent prior learning relationship between teaching units and curriculum knowledge. To this end, taking English major as an example, this paper studied the construction and application of a major-specific knowledge graph based on the big data in education. Firstly, the English major-specific knowledge graph was modeled, the calculation process of node importance was shown, and a localized graph of the knowledge network of English major courses was given. Then, a multi-node feature selection framework for the English major-specific knowledge graph was constructed based on the context of nodes, and the importance of the top k nodes in the constructed knowledge graph was extracted using the multi-node feature extraction technology. After that, the experimental results verified the stability and connectivity of the nodes in the constructed knowledge graph.
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Oz, Mert Sinan, Roberto Cruz, and Juan Rada. "Computation method of the Hosoya index of primitive coronoid systems." Mathematical Biosciences and Engineering 19, no. 10 (2022): 9842–52. http://dx.doi.org/10.3934/mbe.2022458.
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<abstract><p>Coronoid systems are natural graph representations of coronoid hydrocarbons associated with benzenoid systems, but they differ in that they contain a hole. The Hosoya index of a graph $ G $ is defined as the total number of independent edge sets, that are called $ k $-matchings in $ G $.</p> <p>The Hosoya index is a significant molecular descriptor that has an important position in QSAR and QSPR studies. Therefore, the computation of the Hosoya index of various molecular graphs is needed for making progress on investigations. In this paper, a method based on the transfer matrix technique and the Hosoya vector for computing the Hosoya index of arbitrary primitive coronoid systems is presented. Moreover, the presented method is customized for hollow hexagons by using six parameters. As a result, the Hosoya indices of both each arbitrary primitive coronoid system and also each hollow hexagon can be computed by means of a summation of four selected multiplications consisting of presented transfer matrices and two vectors.</p></abstract>
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Meghanathan, Natarajan. "Unit Disk Graph-Based Node Similarity Index for Complex Network Analysis." Complexity 2019 (March 14, 2019): 1–22. http://dx.doi.org/10.1155/2019/6871874.
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We seek to quantify the extent of similarity among nodes in a complex network with respect to two or more node-level metrics (like centrality metrics). In this pursuit, we propose the following unit disk graph-based approach: we first normalize the values for the node-level metrics (using the sum of the squares approach) and construct a unit disk graph of the network in a coordinate system based on the normalized values of the node-level metrics. There exists an edge between two vertices in the unit disk graph if the Euclidean distance between the two vertices in the normalized coordinate system is within a threshold value (ranging from 0 tok, where k is the number of node-level metrics considered). We run a binary search algorithm to determine the minimum value for the threshold distance that would yield a connected unit disk graph of the vertices. We refer to “1 − (minimum threshold distance/k)” as the node similarity index (NSI; ranging from 0 to 1) for the complex network with respect to the k node-level metrics considered. We evaluate the NSI values for a suite of 60 real-world networks with respect to both neighborhood-based centrality metrics (degree centrality and eigenvector centrality) and shortest path-based centrality metrics (betweenness centrality and closeness centrality).
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Jardine, J. F. "Stable Components and Layers." Canadian Mathematical Bulletin 63, no. 3 (October 23, 2019): 562–76. http://dx.doi.org/10.4153/s000843951900064x.
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AbstractComponent graphs $\unicode[STIX]{x1D6E4}_{0}(F)$ are defined for arrays of sets $F$, and, in particular, for arrays of path components for Vietoris–Rips complexes and Lesnick complexes. The path components of $\unicode[STIX]{x1D6E4}_{0}(F)$ are the stable components of the array $F$. The stable components for the system of Lesnick complexes $\{L_{s,k}(X)\}$ for a finite data set $X$ decompose into layers, which are themselves path components of a graph. Combinatorial scoring functions are defined for layers and stable components.
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Liu, Jiafei, Shuming Zhou, Zhendong Gu, Yihong Wang, and Qianru Zhou. "A Note of Independent Number and Domination Number of Qn,k,m-Graph." Parallel Processing Letters 29, no. 03 (September 2019): 1950011. http://dx.doi.org/10.1142/s0129626419500117.
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The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.
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Suparta, I. Nengah, Mathiyazhagan Venkatachalam, I. Gede Aris Gunadi, and Putu Andi Cipta Pratama. "GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS." Ural Mathematical Journal 9, no. 2 (December 27, 2023): 193. http://dx.doi.org/10.15826/umj.2023.2.016.
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A graph \(G(V,E)\) is a system consisting of a finite non empty set of vertices \(V(G)\) and a set of edges \(E(G)\). A (proper) vertex colouring of \(G\) is a function \(f:V(G)\rightarrow \{1,2,\ldots,k\},\) for some positive integer \(k\) such that \(f(u)\neq f(v)\) for every edge \(uv\in E(G)\). Moreover, if \(|f(u)-f(v)|\neq |f(v)-f(w)|\) for every adjacent edges \(uv,vw\in E(G)\), then the function \(f\) is called graceful colouring for \(G\). The minimum number \(k\) such that \(f\) is a graceful colouring for \(G\) is called the graceful chromatic number of \(G\). The purpose of this research is to determine graceful chromatic number of Cartesian product graphs \(C_m \times P_n\) for integers \(m\geq 3\) and \(n\geq 2\), and \(C_m \times C_n\) for integers \(m,n\geq 3\). Here, \(C_m\) and \(P_m\) are cycle and path with \(m\) vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs.
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BUDKO, NIKITA P. "THE METHOD OF SYNTHESIS OF THE SUBSYSTEM OF INTELLIGENT MONITORING OF THE INFORMATION AND TELECOMMUNICATIONS NETWORK OF THE SITUATION CENTER." H&ES Research 13, no. 5 (2021): 38–56. http://dx.doi.org/10.36724/2409-5419-2021-13-5-38-56.
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Introduction: in accordance with the task of forming a system of distributed situational centers of public authorities, it is necessary to interface heterogeneous segments of the department's information and telecommunications networks into a geographically distributed infrastructure with the construction of a subsystem for monitoring the state of its elements on it. The purpose of the study: based on the use of basic concepts of structural synthesis methods, structural analysis of network infrastructures, as well as parametric synthesis procedures, to develop a methodology for the synthesis of a new generation monitoring subsystem based on intelligent methods for identifying the state of the network. Results: the structuring of the controlled space by the terms "monitoring zone"," critical element"," state classes "form the basis of new methods of intelligent monitoring that are" insensitive " to the properties of the constant evolution of network infrastructures. The proposed method of synthesis of the intelligent monitoring subsystem consists of sequentially performed stages of structural synthesis, parametric synthesis and structural analysis of the network. Practical significance: the successive stages of the methodology using graph distance measurement procedures and the modified k-means algorithm allow not only to identify the type of network state, but also to reasonably, using instrumental calculation methods, present in the interests of the decision support system sets of acceptable values of the main parameters and probabilistic-temporal characteristics of the monitoring subsystem for subsequent reconfiguration of the network and preventing its transition to an inoperable state. Discussion: the modification of the k-means algorithm proposed in the study differs in that in the classical algorithm, work is carried out on points of Euclidean space, and in the proposed method we are talking about a graph space with metrics in the form of graph distances, while data clouds are used as initial data for classification in the k-means algorithm as disordered data sets that are not tied to any of the measurement scales, and in the proposed algorithm, the data cloud is represented by a set of graphs in a given topological space of graph metrics, describing the state of the network in time.
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Nhi, Nguyen Thi Uyen, Thanh Manh Le, and Thanh The Van. "A Model of Semantic-Based Image Retrieval Using C-Tree and Neighbor Graph." International Journal on Semantic Web and Information Systems 18, no. 1 (January 2022): 1–23. http://dx.doi.org/10.4018/ijswis.295551.
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The problems of image mining and semantic image retrieval play an important role in many areas of life. In this paper, a semantic-based image retrieval system is proposed that relies on the combination of C-Tree, which was built in our previous work, and a neighbor graph (called Graph-CTree) to improve accuracy. The k-Nearest Neighbor (k-NN) algorithm is used to classify a set of similar images that are retrieved on Graph-CTree to create a set of visual words. An ontology framework for images is created semi-automatically. SPARQL query is automatically generated from visual words and retrieve on ontology for semantics image. The experiment was performed on image datasets, such as COREL, WANG, ImageCLEF, and Stanford Dogs, with precision values of 0.888473, 0.766473, 0.839814, and 0.826416, respectively. These results are compared with related works on the same image dataset, showing the effectiveness of the methods proposed here.
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Salihoglu, Semih. "Kùzu: A Database Management System For "Beyond Relational" Workloads." ACM SIGMOD Record 52, no. 3 (October 30, 2023): 39–40. http://dx.doi.org/10.1145/3631504.3631514.
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I would like to share my opinions on the following question: how should a modern graph DBMS (GDBMS) be architected? This is the motivating research question we are addressing in the K`uzu project at University of Waterloo [4, 5].1 I will argue that a modern GDBMS should optimize for a set of what I will call, for lack of a better term, "beyond relational" workloads. As a background, let me start with a brief overview of GDBMSs.
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NI, TIAN-JIA, and ZHI-YING WEN. "SELF-SIMILARITY OF SATISFIABLE BOOLEAN EXPRESSIONS DECIPHERED IN TERMS OF GRAPH DIRECTED ITERATED FUNCTION SYSTEMS." Fractals 16, no. 04 (December 2008): 305–15. http://dx.doi.org/10.1142/s0218348x0800406x.
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The decision problem of satisfiability of Boolean expression in k-conjunctive normal form (kSAT) is a typical NP-complete problem. In this paper, by mapping the whole Boolean expressions in k-conjunctive normal form onto a unit hypercube, we visualize the problem space of kSAT. The pattern of kSAT is shown to have self-similarity which can be deciphered in terms of graph directed iterated function system. We provide that the Hausdorff dimension of the pattern of kSAT is equal to the box-counting dimension and increases with k. This suggests that the time complexity of kSAT increases with k.
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Usha, Subramaniam, Murugesan Karthik, Marappan Jothibasu, and Vijayaragavan Gowtham. "An integrated exploration of heat kernel invariant feature and manifolding technique for 3D object recognition system." Acta Scientiarum. Technology 46, no. 1 (November 6, 2023): e62608. http://dx.doi.org/10.4025/actascitechnol.v46i1.62608.
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Spectral Graph theory has been utilized frequently in the field of Computer Vision and Pattern Recognition to address challenges in the field of Image Segmentation and Image Classification. In the proposed method, for classification techniques, the associated graph's Eigen values and Eigen vectors of the adjacency matrix or Laplacian matrix created from the images are employed. The Laplacian spectrum and a graph's heat kernel are inextricably linked. Exponentiating the Laplacian eigensystem over time yields the heat kernel, which is the solution to the heat equations. In the proposed technique K-Nearest neighborhood and Delaunay triangulation techniques are used to generate a graph from the 3D model. The graph is then represented into Normalized Laplacian (NL) and Laplacian matrix (L). From each Normalized Laplacian and Laplacian matrix, the feature vectors like Heat Content Invariant and Laplacian Eigen values are extracted. Then, using all of the available clustering algorithms on datasets, the optimum feature vector for clustering is determined. For clustering various manifolding techniques are employed. In the suggested method, the graph heat kernel is constructed using industry-standard objects which are taken from the Engineering bench mark Data set.
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Zalkow, Frank, Julian Brandner, and Meinard Müller. "Efficient Retrieval of Music Recordings Using Graph-Based Index Structures." Signals 2, no. 2 (May 17, 2021): 336–52. http://dx.doi.org/10.3390/signals2020021.
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Flexible retrieval systems are required for conveniently browsing through large music collections. In a particular content-based music retrieval scenario, the user provides a query audio snippet, and the retrieval system returns music recordings from the collection that are similar to the query. In this scenario, a fast response from the system is essential for a positive user experience. For realizing low response times, one requires index structures that facilitate efficient search operations. One such index structure is the K-d tree, which has already been used in music retrieval systems. As an alternative, we propose to use a modern graph-based index, denoted as Hierarchical Navigable Small World (HNSW) graph. As our main contribution, we explore its potential in the context of a cross-version music retrieval application. In particular, we report on systematic experiments comparing graph- and tree-based index structures in terms of the retrieval quality, disk space requirements, and runtimes. Despite the fact that the HNSW index provides only an approximate solution to the nearest neighbor search problem, we demonstrate that it has almost no negative impact on the retrieval quality in our application. As our main result, we show that the HNSW-based retrieval is several orders of magnitude faster. Furthermore, the graph structure also works well with high-dimensional index items, unlike the tree-based structure. Given these merits, we highlight the practical relevance of the HNSW graph for music information retrieval (MIR) applications.
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Jiang, Yaozhi. "Dialectical Logic K-Model: the Discrete Time Dynamical Sampling System, Multidimensional Logic Variable and Associate Database(ADB)." Journal of Mathematics Research 10, no. 2 (March 5, 2018): 88. http://dx.doi.org/10.5539/jmr.v10n2p88.
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Following the earlier works about dialectical logic K-model by the author, in this succeed paper author described the three problems : the first is that discrete time dynamical sampling system to solve which the true-value function is unknown and need discrete time dynamical sampling system to obtain a series of sampled discrete time true-value function points to predict the continuous true-value function or we need some properties of true value function in the frequency domain, a several formulas for true-value function of single-dimensional logic variable via discrete Fourier transformation are explained; the second is the graph expression and matrices expression for the multidimensional logic variables in dialectical logic K-model. Multidimensional logic variable is important that can be used in multidimensional contradictions and in multiple-person games. In fact, author also described the graph $G_K^p$ and corresponding matrices of the multidimensional logic variables; the third is the machine oriented database, associated database i.e. ADB, this is a new database for artificial intelligence, in present paper author describes theoretical properties and some features of ADB.
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Lin, Yiming, Yeye He, and Surajit Chaudhuri. "Auto-BI: Automatically Build BI-Models Leveraging Local Join Prediction and Global Schema Graph." Proceedings of the VLDB Endowment 16, no. 10 (June 2023): 2578–90. http://dx.doi.org/10.14778/3603581.3603596.
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Business Intelligence (BI) is crucial in modern enterprises and billion-dollar business. Traditionally, technical experts like database administrators would manually prepare BI-models (e.g., in star or snowflake schemas) that join tables in data warehouses, before less-technical business users can run analytics using end-user dashboarding tools. However, the popularity of self-service BI (e.g., Tableau and Power-BI) in recent years creates a strong demand for less technical end-users to build BI-models themselves. We develop an Auto-BI system that can accurately predict BI models given a set of input tables, using a principled graph-based optimization problem we propose called k-Min-Cost-Arborescence (k-MCA), which holistically considers both local join prediction and global schema-graph structures, leveraging a graph-theoretical structure called arborescence. While we prove k-MCA is intractable and inapproximate in general, we develop novel algorithms that can solve k-MCA optimally, which is shown to be efficient in practice with sub-second latency and can scale to the largest BI-models we encounter (with close to 100 tables). Auto-BI is rigorously evaluated on a unique dataset with over 100K real BI models we harvested, as well as on 4 popular TPC benchmarks. It is shown to be both efficient and accurate, achieving over 0.9 F1-score on both real and synthetic benchmarks.
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Mugeni, John Bosco, and Toshiyuki Amagasa. "A Graph-Based Blocking Approach for Entity Matching Using Contrastively Learned Embeddings." ACM SIGAPP Applied Computing Review 22, no. 4 (December 2022): 37–46. http://dx.doi.org/10.1145/3584014.3584017.
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Data integration is considered a crucial task in the entity matching process. In this process, redundant and cunning entries must be identified and eliminated to improve the data quality. To archive this, a comparison between all entities is performed. However, this has quadratic computational complexity. To avoid this, `blocking' limits comparisons to probable matches. This paper presents a k-nearest neighbor graph-based blocking approach utilizing state-of-the-art context-aware sentence embeddings from pre-trained transformers. Our approach maps each database tuple to a node and generates a graph where nodes are connected by edges if they are related. We then invoke unsupervised community detection techniques over this graph and treat blocking as a graph clustering problem. Our work is motivated by the scarcity of training data for entity matching in real-world scenarios and the limited scalability of blocking schemes in the presence of proliferating data. Additionally, we investigate the impact of contrastively trained embeddings on the above system and test its capabilities on four data sets exhibiting more than 6 million comparisons. We show that our block processing times on the target benchmarks vary owing to the efficient data structure of the k-nearest neighbor graph. Our results also show that our method achieves better performance in terms of F1 score when compared to current deep learning-based blocking solutions.
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Zhang, Xinyi, Qichen Wang, Cheng Xu, Yun Peng, and Jianliang Xu. "FedKNN: Secure Federated k-Nearest Neighbor Search." Proceedings of the ACM on Management of Data 2, no. 1 (March 12, 2024): 1–26. http://dx.doi.org/10.1145/3639266.
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Nearest neighbor search is a fundamental task in various domains, such as federated learning, data mining, information retrieval, and biomedicine. With the increasing need to utilize data from different organizations while respecting privacy regulations, private data federation has emerged as a promising solution. However, it is costly to directly apply existing approaches to federated k-nearest neighbor (kNN) search with difficult-to-compute distance functions, like graph or sequence similarity. To address this challenge, we propose FedKNN, a system that supports secure federated kNN search queries with a wide range of similarity measurements. Our system is equipped with a new Distribution-Aware kNN (DANN) algorithm to minimize unnecessary local computations while protecting data privacy. We further develop DANN*, a secure version of DANN that satisfies differential obliviousness. Extensive evaluations show that FedKNN outperforms state-of-the-art solutions, achieving up to 4.8× improvement on federated graph kNN search and up to 2.7× improvement on federated sequence kNN search. Additionally, our approach offers a trade-off between privacy and efficiency, providing strong privacy guarantees with minimal overhead.
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Gao, Yan, Haojun Xu, Jie Li, Nannan Wang, and Xinbo Gao. "Multi-Scene Generalized Trajectory Global Graph Solver with Composite Nodes for Multiple Object Tracking." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 3 (March 24, 2024): 1842–50. http://dx.doi.org/10.1609/aaai.v38i3.27953.
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The global multi-object tracking (MOT) system can consider interaction, occlusion, and other ``visual blur'' scenarios to ensure effective object tracking in long videos. Among them, graph-based tracking-by-detection paradigms achieve surprising performance. However, their fully-connected nature poses storage space requirements that challenge algorithm handling long videos. Currently, commonly used methods are still generated trajectories by building one-forward associations across frames. Such matches produced under the guidance of first-order similarity information may not be optimal from a longer-time perspective. Moreover, they often lack an end-to-end scheme for correcting mismatches. This paper proposes the Composite Node Message Passing Network (CoNo-Link), a multi-scene generalized framework for modeling ultra-long frames information for association. CoNo-Link's solution is a low-storage overhead method for building constrained connected graphs. In addition to the previous method of treating objects as nodes, the network innovatively treats object trajectories as nodes for information interaction, improving the graph neural network's feature representation capability. Specifically, we formulate the graph-building problem as a top-k selection task for some reliable objects or trajectories. Our model can learn better predictions on longer-time scales by adding composite nodes. As a result, our method outperforms the state-of-the-art in several commonly used datasets.
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Ma, Weigang, Jing Wang, Chaohui Zhang, Qiao Jia, Lei Zhu, Wenjiang Ji, and Zhoukai Wang. "Application of Variational Graph Autoencoder in Traction Control of Energy-Saving Driving for High-Speed Train." Applied Sciences 14, no. 5 (February 29, 2024): 2037. http://dx.doi.org/10.3390/app14052037.
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In a high-speed rail system, the driver repeatedly adjusts the train’s speed and traction while driving, causing a high level of energy consumption. This also leads to the instability of the train’s operation, affecting passengers’ experiences and the operational efficiency of the system. To solve this problem, we propose a variational graph auto-encoder (VGAE) model using a neural network to learn the posterior distribution. This model can effectively capture the correlation between the components of a high-speed rail system and simulate drivers’ operating state accurately. The specific traction control is divided into two parts. The first part employs an algorithm based on the K-Nearest Neighbors (KNN) algorithm and undersampling to address the negative impact of imbalanced quantities in the training dataset. The second part utilizes a variational graph autoencoder to derive the initial traction control of drivers, thereby predicting the energy performance of the drivers’ operation. An 83,786 m long high-speed train driving section is used as an example for verification. By using a confusion matrix for our comparative analysis, it was concluded that the energy consumption is approximately 18.78% less than that of manual traction control. This shows the potential and effect of the variational graph autoencoder model for optimizing energy consumption in high-speed rail systems.
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Chen, Gang, Jinbo Ni, and Xinyu Fu. "Existence, and Ulam's types stability of higher-order fractional Langevin equations on a star graph." AIMS Mathematics 9, no. 5 (2024): 11877–909. http://dx.doi.org/10.3934/math.2024581.
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<abstract><p>A study was conducted on the existence of solutions for a class of nonlinear Caputo type higher-order fractional Langevin equations with mixed boundary conditions on a star graph with $ k+1 $ nodes and $ k $ edges. By applying a variable transformation, a system of fractional differential equations with mixed boundary conditions and different domains was converted into an equivalent system with identical boundary conditions and domains. Subsequently, the existence and uniqueness of solutions were verified using Krasnoselskii's fixed point theorem and Banach's contraction principle. In addition, the stability results of different types of solutions for the system were further discussed. Finally, two examples are illustrated to reinforce the main study outcomes.</p></abstract>
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Ghani, Muhammad Usman, Francis Joseph H. Campena, Muhammad Kashif Maqbool, Jia-Bao Liu, Sanaullah Dehraj, Murat Cancan, and Fahad M. Alharbi. "Entropy Related to K-Banhatti Indices via Valency Based on the Presence of C6H6 in Various Molecules." Molecules 28, no. 1 (January 3, 2023): 452. http://dx.doi.org/10.3390/molecules28010452.
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Entropy is a measure of a system’s molecular disorder or unpredictability since work is produced by organized molecular motion. Shannon’s entropy metric is applied to represent a random graph’s variability. Entropy is a thermodynamic function in physics that, based on the variety of possible configurations for molecules to take, describes the randomness and disorder of molecules in a given system or process. Numerous issues in the fields of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines are resolved using distance-based entropy. These applications cover quantifying molecules’ chemical and electrical structures, signal processing, structural investigations on crystals, and molecular ensembles. In this paper, we look at K-Banhatti entropies using K-Banhatti indices for C6H6 embedded in different chemical networks. Our goal is to investigate the valency-based molecular invariants and K-Banhatti entropies for three chemical networks: the circumnaphthalene (CNBn), the honeycomb (HBn), and the pyrene (PYn). In order to reach conclusions, we apply the method of atom-bond partitioning based on valences, which is an application of spectral graph theory. We obtain the precise values of the first K-Banhatti entropy, the second K-Banhatti entropy, the first hyper K-Banhatti entropy, and the second hyper K-Banhatti entropy for the three chemical networks in the main results and conclusion.
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FARKAS, ÁBEL. "Dimension approximation of attractors of graph directed IFSs by self-similar sets." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 01 (August 31, 2018): 193–207. http://dx.doi.org/10.1017/s0305004118000282.
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AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.
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Madeo, Dario, Chiara Mocenni, Giulia Palma, and Simone Rinaldi. "Optimal colorings of Max k-Cut game." Pure Mathematics and Applications 30, no. 1 (June 1, 2022): 82–89. http://dx.doi.org/10.2478/puma-2022-0013.
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Abstract We investigate strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors, where vertices represent players and the edges indicate their relations. Each player v chooses one of the available colors as its own strategy, and its payoff (or utility) is the number of neighbors of v that has chosen a different color. Such games are significant since they model loads of real-worlds scenario with selfish agents and, moreover, they are related to fundamental classes of games. Few results are known concerning the existence of strong equilibria in max k-cut games in this direction. In this paper we make some progress in the understanding of the properties of strong equilibria. In particular, our main result is to show that optimal solutions are 7-strong equilibria. This means that in order a coalition of nodes is able to deviate and drive the system towards a different configuration, i.e. a different coloring, a number of nodes of the coalition strictly larger than 7 is necessary. We also conjecture that, in a generic graph with n nodes, any optimal coloring is also an n-strong equilibrium.
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Sugimoto, Takuya, Tetsuya Toyota, and Hajime Nobuhara. "A Recommendation System with the Use of Comprehensive Trend Indication Based on Weighted Complete Graph." Journal of Advanced Computational Intelligence and Intelligent Informatics 16, no. 2 (March 20, 2012): 266–72. http://dx.doi.org/10.20965/jaciii.2012.p0266.
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Recently, Internet shopping has become widespread, websites of which are equipped with a recommendation system to help users easily find their target items from among vast product information. As a typical method to create recommendation information, collaborative filtering is used but it has a problem that recommendation results tend to be biased toward the same category. Since this study intends recommendation with a high discoverability from a large point of view of category, we define dissimilarity between products based on information on Browse Node ID held by some products in Amazon and use k-medoids to newly categorize the products. Moreover, we create a weighted complete graph with those categories as nodes and indicate the trend across different categories. The proposed system estimates and recommends a category strongly related to a category that is thought to be unknown to the user but the user will like based on information of the weighted complete graph. We evaluate the effectiveness of the proposed system through experiments with 9 undergraduate students, 12 graduate students, and 2 office workers as subjects and show that the proposed system is better in recommending unknown products to the user than existing recommendation systems.