The graph model for conflict resolution provides a convenient and effective means to model and analyze a strategic conflict. Standard practice is to carry out a stability analysis of a graph model, ...and then to follow up with a post-stability analysis, an important component of which is status quo analysis. A graph model can be viewed as an edge-colored graph, but the fundamental problem of status quo analysis – to find a shortest colored path from the status quo node to a desired equilibrium – is different from the well-known network analysis problem of finding the shortest path between two nodes. The only matrix method that has been proposed cannot track all aspects of the evolution of a conflict from the status quo state. Our explicit algebraic approach is convenient for computer implementation and, as demonstrated with a real world case study, easy to use. It provides new insights into a graph model, not only identifying all equilibria reachable from the status quo, but also how to reach them. Moreover, this approach bridges the gap between stability analysis and status quo analysis in the graph model for conflict resolution.
Fast pairwise grouping methods have recently shown great promising for large-scale clustering. However, the relationships among objects of the real world are often high-order rather than pairwise. ...Naively applying pairwise grouping methods to high-order grouping tasks will not always obtain the desired clustering results since there is a loss of information. In this paper, we propose a fast hypergraph clustering algorithm via the Nyström Extension. As the hyperedge of the hypergraph allows us to join any number of vertices, high-order relationships among objects can be captured. In contrast to some previous works in the Nyström research focusing on the error bound for the Nyström-approximated kernel matrix, we present an error bound for the approximated eigenvectors associated with the Nyström Extension of hypergraph Laplacian. In addition, a novel formulation for sampling size selection is provided. Our experiments on a series of UCI data sets show that the proposed algorithm achieves significantly speeding and lower storage. Specifically, our algorithm outperforms the pairwise spectral grouping method based on the Nyström method in terms of accuracy, computation time and space occupation.
In this paper, we present equivalent characterizations for the minimal ∞-norm of generalized inverses of the incidence matrix of a tree. We also compute the minimal ∞-norm for a class of trees.
A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, and the opposite orientation is assigned the inverse of this quaternion unit. In this paper, we ...provide a combinatorial description of the determinant of the Laplacian matrix of a quaternion unit gain graph by using row-column noncommutative determinants recently introduced by one of the authors. A numerical example is presented for illustrating our results.
Service is the process of meeting needs through the activities of others directly. Service is usually synonymous with queues, and queues are what many people complain about. Most of the taxpayers ...complained about the queues, indirectly they would blame the poor service because of the queues that had piled up. Queues can be reduced by improving services, while one way to improve services is to analyze services using the Petri Net model. Petri Net is mathematical modeling for discrete event systems. Petri Net can be used to model and analyze algebraic problems of transportation networks, manufacturing systems, telecommunications networks, parallel process systems, and so on. In this study, a Petri Net Model of the annual vehicle tax payment service system was created as many as 16 places, 14 transitions, 2 operators, and 30 arcs using WOPED 3.2.0 software. The length of time for tax payment services for taxpayers who have completed the file is faster with a total time of 27 minutes compared to those who have not completed the file with a total time of 35 minutes. The Petri Net model of the annual type of vehicle tax payment service system can be presented in the form of a backward incidence and forward incidence matrix which is used to see the queuing pattern at Samsat Oku Timur 1 with a mathematical model. Columns in the backward incidence matrix can be used to determine which transitions are enabled.
The Laplacian matrix L of a signed graph G may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of L. This is achieved by finding a combinatorial formula for ...the Moore-Penrose inverse of an incidence matrix of G. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas.
This paper studies the edge controllability of signed networks, where the dynamics taking place on edges and the interaction type over each edge can be cooperative or antagonistic. First, the edge ...controllable subspace is studied quantitatively from the graph-theoretic perspective. And the lower and upper bounds for the dimension of edge controllable subspace are derived, respectively. Second, the relationship between edge controllability and vertex controllability is analyzed. We prove that the controllability of the edge dynamics is equivalent to that of the vertex dynamics when the communication graph is structurally unbalanced. Furthermore, the edge structural controllability of signed networks is investigated, and a graph-theoretic necessary and sufficient condition for edge structural controllability is presented. Finally, we demonstrate possible applications of our theoretical results to the traffic flow control of urban traffic networks.
The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective ...line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of were studied over different finite fields.
The signless Laplacian Q and signless edge-Laplacian S of a given graph may or may not be invertible. The Moore-Penrose inverses of Q and S are studied. In particular, using the incidence matrix, we ...find combinatorial formulas of the Moore-Penrose inverses of Q and S for trees. Also, we present combinatorial formulas of the inverses of Q and S for odd unicyclic graphs.
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