Symmetry-based coarse-graining of evolved dynamical networks

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.

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.

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end.In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p + 4.

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

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.

Using methods from algebraic graph theory and convex optimization, we study the relationship between local structural features of a network and the eigenvalues of its Laplacian matrix. In particular, we propose a series of semidefinite programs to find new bounds on the spectral radius and the spectral gap of the Laplacian matrix in terms of a collection of local structural features of the network. Our analysis shows that the Laplacian spectral radius is strongly constrained by local structural features. On the other hand, we illustrate how local structural features are usually insufficient to accurately estimate the Laplacian spectral gap. As a consequence, random graph models in which only local structural features are prescribed are, in general, inadequate to faithfully model Laplacian spectral properties of a network.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

The first imaging devices of optical interferometry are likely to be of weak phase, typically: a set of three- element arrays, coherent and stable, independently observing the same object. The study of their imaging capabilities essentially addresses the self-calibration problem and its stability. Like in VLBI, the principle of our self- calibration methods consists in preforming a series of alternate phase calibration operations and Fourier synthesis processes. Algebraic graph theory and algebraic number theory prove to be the key topics involved in the phase calibration operation. The latter can often be written in closed form. As expected, the relative expressions explicitly refer to a set of independent closure phases. To illustrate this essential point, we consider the special case of three-element arrays. The corresponding phase calibration formula, which is then particularly simple, provides all the elements for coping with the possible global instabilities. The Fourier synthesis process, which is also involved in the self-calibration cycles, is performed via WIPE, a methodology recently introduced in radio imaging and optical interferometry. The robustness of the image reconstruction process can then be well controlled.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Partitioning into ExpandersShayan Oveis Gharan and Luca TrevisanShayan Oveis Gharan and Luca Trevisanpp.1256 - 1266Chapter DOI:https://doi.org/10.1137/1.9781611973402.93PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Let G = (V, E) be an undirected graph, λk be the kth smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λk > 0 if and only if G has at most k – 1 connected components. We prove a robust version of this fact. If λk > 0, then for some 1 ≤ ℓ ≤ k – 1, V can be partitioned into ℓ sets P1, …, Pℓ such that each Pi is a low-conductance set in G and induces a high conductance induced subgraph. In particular, and G[Pi] ≳ λk/k2-expander. We make our results algorithmic by designing a simple polynomial time spectral algorithm to find such partitioning of G with a quadratic loss in the inside conductance of Pi's. Unlike the recent results on higher order Cheeger's inequality [6, 9], our results does not use higher order eigenfunctions of G. If there is a sufficiently large gap between λk and λk+1, more precisely if then our algorithm finds a k partitioning of V into sets P1, …, Pk such that the induced subgraph G[Pi] has a singnificantly larger conductance than the conductance of Pi in G. Such a partitioning may represent the best k clusterings of G. Our algorithm is a simple local search that only uses the Spectral Partitioning algorithm as a subroutine. We expect to see further applications of this simple algorithm in clustering applications. Let ρ(k) = mindisjoint max1≤i≤k φ(Ai) be the order k conductance constant of G, in words, ρ(k) is the smallest value of the maximum conductance of any k disjoint subsets of V. Our main technical lemma shows that if (1 + ∊)ρ(k) < ρ(k+1), then V can be partitioned into k sets P1, …, Pk such that for each 1 ≤ i ≤ k, φ(G[Pi]) ≳ ∊ · ρ(k + 1)/k and φ(Pi) ≤ k · ρ(k). This significantly improves a recent result of Tanaka [13] who assumed an exponential (in k) gap between ρ(k) and ρ(k + 1). Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885

A novel method for identifying dense communities of classes (clusters) within an Object-Oriented system has been developed. Such communities might possibly imply relevance of functionality and thus be used as Components. The term Reusable Components is used to refer to a set of interrelated classes that represent a module that can be used in different software systems. To accomplish this identification we employ Algebraic Graph Theory (Spectral Graph Partitioning Techniques). From a modified class diagram of the software system we create a graph in which classes stand for the nodes and the discrete messages exchanged between the classes stand for the edges. Our approach is based on an iterative method for partitioning the graph in order to identify possible reusable components within the system. The eigenvectors of the Laplacian matrix derived from the graph are used for the partitioning.

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.

Graph structures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low-dimensional space using a number of alternative strategies, including principal components analysis (PCA), multidimensional scaling (MDS), and locality preserving projection (LPP). Experimentally, we demonstrate that the embeddings result in well-defined graph clusters. Our experiments with the spectral representation involve both synthetic and real-world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real-world experiments show that the method can be used to locate clusters of graphs.

This paper investigates the periodic event-triggered asymmetric bipartite consensus problem of multi-agent systems (MASs) under structurally balanced undirected signed graphs. In order to reduce the consumption of communication resources and continuous detection in the control process of MASs, a novel periodic detection event-triggered asymmetric bipartite consensus control protocol is proposed. In particular, each agent samples only in a fixed period. For each agent, two event conditions are given, and only when state error satisfies this event condition can the event be triggered. A new type of Laplacian matrix is constructed to achieve asymmetric bipartite consensus. The control method can guarantee the performance of MASs and have less communication resources. Based on the Lyapunov stability theorem and algebraic graph theory, the proposed periodic event-triggered control strategy is able to implement the asymmetric bipartite consensus when the topology is structurally balanced undirected signed graphs. The proposed method in this paper can reduce the restriction on the sampling period and automatically exclude Zeno behavior. Finally, the validity of the conclusion is verified by numerical simulation.

Algebraic Graph Theory with Linear Algebra

The consensus problem of multi-agent nonlinear systems with or without uncertain parameters is studied in this paper. Based on the theory of impulsive differential equations, Lyapunov stability theory and algebraic graph theory, some impulsive consensus conditions are given to realize the consensus of a class of multi-agent nonlinear systems. Compared with the existing investigations of impulsive consensus of multi-agent systems, the proposed impulsive control protocol with uncertain parameters is more rigorous and effective in practical systems. Two numerical simulations are verified to confirm the effectiveness of the proposed methods.