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Brain-Organization Paradigm
Published in Villalba-Diez Javier, The Lean Brain Theory, 2017
The eigenvalues and eigenvectors of the Laplacian have a significant physical meaning: The second smallest non-zero eigenvalue of Lij represents the Fiedler Value or algebraic connectivity (Figure 1.3F). The magnitude of this value reflects how well connected the overall graph is and has been related to the value stream performance in attaining consensual states (Olfati-Saber and Murray, 2004).The algebraic connectivity has also been used in analyzing the robustness and synchronizability of networks (Mohar, 1991) and can therefore serve to compare different complex networks. The higher the value, the more robust and synchronizable a value stream network will be.The eigenvector associated to the algebraic connectivity is the Fiedler vector . The sign structure of this vector can be used to partition a value stream graph into different classes and identify which organizational nodes are critical bottlenecks from a network perspective (Mohar, 1991).
Research on missing value processing methods based on advanced community detection
Published in Jimmy C.M. Kao, Wen-Pei Sung, Civil, Architecture and Environmental Engineering, 2017
Fuqiang Zhao, Guijun Yang, Xue Xu
The second smallest eigenvalue λ2(L) is called the algebraic connectivity of the graph G, and the corresponding normalized eigenvector is called the Fiedler vector. The algebraic connectivity is considered a measure how well-connected a graph is. That is, the more connected graph has the greater algebraic connectivity on the same vertex set. The magnitude of this value reflects how well connected the overall graph is. λ2(L) > 0 if and only if G is connected. λ2(L) is monotone increasing in the edge set. The algebraic connectivity function of complex networks is a monotone convex function. If G1 = (V, E1) and G2 = (V, E2), E1 ⊂ E2, then λ2(L1) ≤ λ2(L2). The Courant–Fischer Minimax Principle implies (Mohar, 1991):
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 control for bi-connectivity of multi-robot network
Published in SICE Journal of Control, Measurement, and System Integration, 2023
To achieve the team objectives through communication or information exchange, it is important that a network modelling the communication topology is connected. The task accomplishment will be hard once the connectedness is lost since the network topology will be divided into two or more components. For this challenge, some autonomous distributed control laws to preserve the connectedness have been proposed [2,3]. Since it is known that the connectedness is equivalent to the second smallest eigenvalue of the graph Laplacian matrix is positive, a lot of the control laws for the connectedness preservation employ the gradient of the second smallest eigenvalue to decide the direction of the robot motion. The second smallest eigenvalue of the graph Laplacian is called algebraic connectivity.
The Law of Scale Independence
Published in Annals of GIS, 2022
Two key measures from algebraic graph theory will be used. The spectral radius is the largest eigenvalue of the graph adjacency matrix A, which has N eigenvalues λ, such that λ1 ≥ λ2 ≥ … ≥ λN. Spectral radius (λ1) is a standard measure of graph complexity, and is directly related to graph entropy (Mowshowitz and Detmer, 2012). The second is algebraic connectivity, which measures synchronization or synchronizability of the network. It is calculated from the Laplacian matrix L of A, where L = D – A and D is the degree matrix, where the diagonal represents the degree of each node and all other entries are zero. Eigenvalues of L are all positive except for the smallest, λ(L)N = 0. Algebraic connectivity is given by the smallest nonzero eigenvalue, α = λ(L)N-1. Literal synchronization may or may not be applicable to scale hierarchies, but α also represents inferential synchronization, or the extent to which inferences or observations at one point in the network can be applied to other components (Phillips 2013). High algebraic connectivity and synchronization in a hierarchical network indicates relatively seamless scale linkage, and vice versa. These measures are discussed in more detail in texts on algebraic or spectral graph theory (e.g. Biggs 1994), and elsewhere in a geographical and geoscience context by Phillips (2013, 2016).
Computation of the target state and feedback controls for time optimal consensus in multi-agent systems
Published in International Journal of Control, 2018
Ameer K. Mulla, Deepak U. Patil, Debraj Chakraborty
Most of the recent literature studies consensus achieved asymptotically only through local communication among neighbouring agents (Jadbabaie et al., 2003; Olfati-Saber et al., 2007). However, the speed of achieving consensus is also an important consideration. In almost all available consensus algorithms, the speed of approximately reaching consensus can be characterised by the algebraic connectivity of the communication graph (Ren & Beard, 2007). As a result, various properties of the communication graph have been designed (Olfati-Saber, 2005; Xiao & Boyd, 2004) to maximise the algebraic connectivity so as to speed up the convergence to consensus. On the other hand, Cortés (2006) and Zheng and Wang (2012) tackle the problem of computing control laws to achieve consensus in finite time. However, the problem of achieving consensus in the minimum time possible is not addressed anywhere except recently in Mulla et al. (2014) and Mulla and Chakraborty (2015). In spite of the unavailability of a solution, it is clear from the above extensive efforts that time-optimal consensus is an important problem. As applications of such algorithms, one might consider multiple unmanned aerial vehicles (UAVs) operating in a hostile environment, when they may want toteam up as soon as possible to counter any perceived threats (Richards et al., 2005). Similarly, minimum time consensus is important for networked spacecrafts in deep space which may want to align their attitudes in minimum time (Kapila, Sparks, Buffington, & Yan, 2000; Scrivener & Thompson, 1994).