Cluster consensus of opinion dynamics with stubborn individuals

This paper investigates the cluster consensus problem for second-order multi-agent systems by applying the pinning control method to a small collection of the agents. Consensus is attained independently for different agent clusters according to the community structure generated by the group partition of the underlying graph and sufficient conditions for both cluster and general consensus are obtained by using results from algebraic graph theory and the LaSalle Invariance Principle. Finally, some simple simulations are presented to illustrate the technique.

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

We investigate the cluster consensus problem in directed networks of nonlinearly coupled multi-agent systems by using pinning control. Depending on the community structure generated by the group partition of the underlying digraph, various clusters can be made coherently independent by applying feedback injections to a fraction of the agents. Sufficient conditions for cluster consensus are obtained using algebraic graph theory and matrix theory and some simulations results are included to illustrate the method.

In this paper, we use nonlinear continuous opinion dynamics models to investigate how the opinions in social networks are affected by the existence of stubborn individuals who remain attached to their initial opinions. Specifically, we are interested in the case where trust-mistrust relations also exist in the network. We extend the existing models, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn positives</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn extremists</i> , by using signed graphs and allowing negative edges to model mistrust interactions. The effect of stubborn individuals is studied for both structurally balanced and unbalanced graphs. We show that, due to the stubborn individuals, a bipartite consensus (polarization) might not occur even if the associated graph is structurally balanced. Moreover, for structurally unbalanced graphs, the opinion vector does not always converge to 0 as compared to the well-known Altafini's model because of stubborn individuals. We present theorems that provide the above convergence guarantees.

The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.

Scaled consensus control of heterogeneous multi-agent systems with switching topologies

Unsupervised consensus analysis for on-line review and questionnaire data

Spectral Graph Theory is the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of systems has received much interest in the last 15 years. Both algorithm design theory and practical implementations have made substantial progress. There is also a growing number of problems that can be efficiently solved using SDD solvers including: image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics. All these examples can be viewed as special case of convex optimization problems.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsDarren GlassDarren Glass is a professor of mathematics at Gettysburg College. His talk at the 1996 Joint Mathematics Meetings was about group theory and Rubik's puzzles, but now his mathematical interests tend toward algebraic geometry and graph theory.

Character theory has diverse applications in the applied sciences. In particular, it plays a crucial role in understanding symmetries and features of quantum systems in quantum mechanics, constructing error-correcting codes in coding theory, and analyzing and predicting molecular structures in computational chemistry. In addition, the characters of finite groups are utilized in algebraic geometry, graph theory, cryptography, number theory, network analysis, and cohomology theory. This article studies the derived Chevalley group $G_2'(2)$, which is isomorphic to the group of all $3 \times 3$ invertible matrices preserving a non-singular Hermitian form over the Galois field $\mathbb{F}_9$ of order 9. The computation of p-decomposition matrices relative to all prime divisors of the group order enables the investigation of the decomposition matrices of its subgroups and supergroups. Absolutely irreducible p-characters provide valuable insights into the structure of the group.

In this paper we describe an iterative method for indefinite saddle-point systems arising from mixed finite element discretizations of second-order elliptic boundary value problems subject to mixed boundary conditions and posed over polyhedral three-dimensional domains. The method is based on a decoupling of the vector of velocities in the saddle-point system from the vector of pressures, resulting in a symmetric positive definite velocity system and a triangular pressure system. The crucial step in this approach is the construction of the divergence-free Raviart--Thomas--Nedelec elements from the curls of Nedelec's edge elements. Because of the large kernel of the curl-operator, this representation is not unique. To find a basis we consider the graph made up of the nodes and edges of the mesh and eliminate the edge elements associated with a spanning tree in this graph. To prove that this technique works in the general case considered here, we employ fundamental results from algebraic topology and graph theory. We also include some numerical experiments, where we solve the (decoupled) velocity system by ILU-preconditioned conjugate gradients and the pressure system by simple back substitutions. We compare our method with a standard ILU-based block preconditioner for the original saddle-point system, and we find that our method is faster by a factor of at least 4.5 in all cases, with the greatest improvement occurring in the nonuniform mesh case.

Over the last several years there have been major breakthroughs in the design of approximation algorithms for such classic problems as finding the maximum flow in a graph. Maximum flow for undirected graphs can now be approximately solved in almost linear time. This result by researchers at Berkeley and MIT, I claim, is only the beginning of a new era in efficient algorithm design.Graph theoretic optimization problems, that have been dormant for fifty years are now seeing new and exciting algorithms. These advances have been made possible by Spectral Graph Theory, the interplay between linear algebra and combinatorial graph theory. One application of this interplay is a nearly linear time solver for Symmetric Diagonally Dominate systems (SDD). This seemingly restrictive class of linear systems has received substantial interest in the last 15 years. Both algorithm design theory and practical implementations have made major progress. Surprisingly, there is an ever growing list of problems that can be efficiently solved using SDD solvers including image segmentation, image denoising, finding solutions to elliptic equations, computing maximum flow in a graph, graph sparsification, and graphics. All these examples can be viewed as special cases of convex optimization problems that arise from graph problems. I can imagine a world where such optimization problems that seem to require at least quadratic work will all be solvable by practical algorithms guaranteed to run in near linear work and are very parallel.

A multi-level finite element nodal ordering using algebraic graph theory

Flocking of a team of Lagrangian agents is investigated in this paper. The motion equation of agents with Lagrangian dynamics and its properties are reviewed. The centrifugal and coriolis effects and gravitational terms in the dynamics of agents have been neglected in previous works on flocking problem. However the dynamics of many dynamical systems which would be the agents in practical applications can be assumed as Lagrangian with the above mentioned terms. The proposed flocking scheme is based on a gradient algorithm in which the collective potential is systematically constructed to cause the velocity of the agents to reach a consensus. It is explained that the suggested algorithm satisfies three rules of flocking demonstrated by Reynolds: cohesion, separation, and alignment. For this purpose the Hamiltonian of a team of agents with Lagrangian dynamics is considered and is treated as the Lyapunov like for the Multi-Agent system. Then considering the properties of the Lagrangian systems and also using the algebraic graph theory, it is shown that the derivative of the Hamiltonian is in order that the Reynolds' rules are satisfied and the conditions under which flocking is achieved are derived. The analytical tools provided here are relayed on algebraic graph theory, matrix theory, and control theory. In simulation results, a group of wheeled mobile robots are used to show the effectiveness of the suggested algorithm.

There are numerous applications of graph theory and algebraic graph theory in combinatorial optimization and optimal structural analysis. In this paper, a new canonical form as well as its relation with four structural models often encountered in practice and their corresponding graphs are presented. Furthermore, the block diagonalization of this form, which is performed using three Kronecker products and unsymmetric matrices, is studied. This block diagonalization leads to an efficient method for the eigensolution of adjacency and Laplacian matrices of special graphs. The eigenvalues and eigenvectors are used for efficient nodal ordering and partitioning of large structural models. The present method is far more simple than any existing general approach.