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Analyzing Emergence in Biological Neural Networks Using Graph Signal Processing
Published in Larry B. Rainey, O. Thomas Holland, Emergent Behavior in System of Systems Engineering, 2022
Kevin Schultz, Marisel Villafañe-Delgado, Elizabeth P. Reilly, Anshu Saksena, Grace M. Hwang
The adjacency matrix of a graph, indicated by A, represents the edge relationships among the vertices with Aij = 1 if {vi, v_j} ∈ E and Aij = 0 otherwise. For a weighted graph, Aij = wij where wij is the weight of the edge {vi, vj}. Note that if G is simple, then Aii = 0 for all i. Additionally, A is symmetric when G is undirected. For a simple undirected graph, the Laplacian matrix is defined as L = D − A, where D is a diagonal matrix with Dii = deg(vi), the degree of vertex vi and A is the adjacency matrix. The Laplacian matrix is a rich representation of a graph, as it encodes several interesting properties of the graph such as the number of spanning trees, the number of connected components, and the overall strength of connectedness of the graph. For undirected graphs with nonnegative, real-valued weights, both the adjacency and Laplacian matrices are real and symmetric, meaning they have n (not necessarily unique) real eigenvalues and one can find a corresponding set of n orthonormal eigenvectors, L = UΛUT. While the eigenvalues of the adjacency matrix sum to 0, the Laplacian matrix is positive semi-definite and thus has non-negative eigenvalues.
A Self-Organizing Map-Based Spectral Clustering on Shortest Path of a Graph
Published in Siddhartha Bhattacharyya, Anirban Mukherjee, Indrajit Pan, Paramartha Dutta, Arup Kumar Bhaumik, Hybrid Intelligent Techniques for Pattern Analysis and Understanding, 2017
Parthajit Roy, Swati Adhikari, J. K. Mandal
The idea behind the spectral geometry leads towards the deep understanding of the study of graph eigenvalues. By the term “spectra of Laplacian matrix”, we mean the eigenvalues and their corresponding eigenvectors of the Laplacian matrix. There are m eigenvalues of an m×m Laplacian matrix and m eigenvectors corresponding to these eigenvalues. If the assigned weights are all non-negative, then the Laplacian matrix has all real eigenvalues. The eigenvalues are all positive and 0 is the smallest eigenvalue and the corresponding eigenvector is (1,1,1,⋯,1)T. These m eigenvalues are not all distinct.
Graph-Theoretic Algorithms for Energy Saving in IP Networks
Published in F. Richard Yu, Xi Zhang, Victor C. M. Leung, Green Communications and Networking, 2016
Francesca Cuomo, Antonio Cianfrani, Marco Polverini
The algebraic connectivity 𝒜(G) of a graph G(𝒩, ℰ) is evaluated by using the Laplacian matrix L(G) [4]. This matrix is equal to the difference between D(G) and A(G). The Laplacian matrix of a bidirectional graph is symmetric and all its row and column sums are equal to 0. The eigenspectrum of L(G) is defined as the set of its N eigenvalues, denoted as λ(G), that can be ordered from the smallest to the greatest, i.e., λ1(G) ≼ λ2(G) ≼ ... ≼ λ𝒩(G). The eigenvalues of the Laplacian matrix measure the connectivity of the graph and the number of its connected components. The smallest eigenvalue of the Laplacian of a bidirectional graph G is equal to 0 (i.e., λ1(G) = 0) and the number of eigenvalues equal to 0 is the number of connected components of G [7].
Distributed dynamic event-triggered algorithm with minimum inter-event time for multi-agent convex optimisation
Published in International Journal of Systems Science, 2021
Xiasheng Shi, Zhiyun Lin, Tao Yang, Xuesong Wang
The communication network of multi-agent system with n agents can be denoted as a graph . is the node set, is the edge set and A is the adjacency matrix of G. For the definition of matrix A, if , otherwise . represents the communication ability from agent j to agent i. Besides, let be the set of the in-neighbour agents. For graph G, the degree matrix D and Laplacian matrix L are defined as and L = D−A, where . Obviously, is the right eigenvector of matrix L at eigenvalue 0. Besides, is the left eigenvector of matrix L at eigenvalue 0 if the graph G is undirected. The network G is assumed to be connected if there exists at least one path between any two agents.
Distributed robust adaptive flocking for uncertain nonlinear multi-agent systems with time-varying communication delay
Published in International Journal of Systems Science, 2020
Ali Izadipour, Jafar Ghaisari, Javad Askari
For further analysis, there are several quantitative criteria for flocking behaviour evaluation, which has been introduced by Olfati-Saber (2006). Some of them are velocity mismatch that denotes velocity consensus of agents, and Fiedler value that shows the connectivity of the communication network. The good convergence of the agents is shown in Figure 6(a) that velocity mismatch tends to zero ultimately. This behaviour demonstrates the robust behaviour in the presence of the perturbation of communication links and intrinsic nonlinear. In Figure 6(b), the eigenvalues of the Laplacian matrix, as a measure of graph connectivity, are shown. Finally, the four parts of the input signal equation (7), which is applied to the agents, are depicted in Figure 7.
Robust output consensus of networked negative imaginary systems via an integral quadratic constraint approach
Published in International Journal of Control, 2023
Qian Zhang, Liu Liu, Yufeng Lu
Given an undirected and connected graph with n nodes, then the following assertions hold. The Laplacian matrix of is symmetric and positive semi-definite., where is a vector with all elements being 1.