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Getting Social: Graph Theory and Social Network Analysis
Published in Jesús Rogel-Salazar, Advanced Data Science and Analytics with Python, 2020
We can calculate the PageRank centrality of a node as follows: ()xi=α∑kak,ioutdeg(k)xk+β, where α and β are constants, outdeg(k) is the outdegree of node k, and ak, i is the (k, i) entry in the adjacency matrix for the graph. We can write this equation in matrix form as x = αxD–1A + β, with D–1 being a diagonal matrix with k-th diagonal element equal to 1/outdeg(k). We can solve for x as x = β(I – αD–1A)–1. We call α the damping factor, and its value should be chosen between 0 and 1/ρ(D–1A), where ρ(D–1A) is the largest eigenvalue of D–1A. It must be said that for large networks, it is more efficient to compute the PageRank via power iteration, as it does not have to deal with matrix decomposition and it works very well with sparse matrices. Note that if the outdegree of k is null, we require that outdeg(k) = 1 in this calculation.See more about power iteration in Appendix B.
Subspace Algorithms
Published in Philipos C. Loizou, Speech Enhancement, 2013
where u1 is the estimated principal eigenvector. Applying the power iteration method (Equation 8.230) to the new matrix R′ will yield the second largest eigenvalue and corresponding eigenvector. The deflation approach is simple; however, it introduces additional complexity. Other deflation techniques can be found in [52].
Diffusion-Based Finite Element Method to Estimate the Reactivity Changes due to Core Deformation in an SFR
Published in Nuclear Science and Engineering, 2018
The implemented code generates the problem matrices and solves the system of a linear equation using the conjugate gradient (CG) method as an iterative solver. The fundamental eigenvalue of the matrix problem is calculated using the standard power iteration. The 2-D GFEM code takes any absolute position for each node in a problem domain so that any type of triangular element is applicable. This is for accommodating to any irregular deformation of assemblies.
The RAPID Fission Matrix Approach to Reactor Core Criticality Calculations
Published in Nuclear Science and Engineering, 2018
William J. Walters, Nathan J. Roskoff, Alireza Haghighat
In order to solve the resulting linear eigenvalue problem [Eq. (2)], the power iteration scheme is used in this work, due to ease of implementation and low memory requirements. A tolerance of was used for both eigenvector and eigenvalue convergence. In the future, more advanced linear methods will be explored to further reduce the convergence time.
Application of Krylov Acceleration Technique in Method of Characteristics–Based Neutron Transport Code
Published in Nuclear Science and Engineering, 2018
In linear algebra, the Krylov subspace method is quite popular for solving a large matrix equation in an iterative manner. Convergence of the Krylov subspace method depends on the condition number, which is the ratio of the maximum eigenvalue to the minimum eigenvalue of the matrix. When this number is close to unity, the matrix is called well conditioned and, hence, converges faster. In the case of an ill-conditioned matrix, preconditioning can be adopted to improve the convergence. The convergence of the power iteration method depends on the dominance ratio, which is the ratio of the second largest eigenvalue to the largest eigenvalue of the matrix. If these two eigenvalues are closely spaced, then the convergence of the power iteration will become slow. The power iteration method is implemented using the conventional inner-outer iteration scheme, shown in Fig. 2. In the proposed iteration scheme, shown in Fig. 5, the outer one is the power iteration while the inner one is the BICG iteration. The number of inner iterations (or BICG) per outer iterations in both schemes is optimized to achieve the lowest computation time. Since BICG is more efficient in finding the solution of the matrix equation than the conventional power iteration, the former offers a better estimate of the solution in a limited number of inner iterations per outer iterations. Improvement in estimating the solution during the inner iteration, especially in the initial rounds, drastically reduces the total number of outer iterations for the BICG iteration. Hence, the Krylov iteration takes less time than the inner-outer iteration to obtain a solution of comparable accuracy. However, the extent up to which the computation time is reduced by the Krylov iteration mainly depends on the scattering of neutrons from one group to the other, which again depends on the number of energy groups involved in the problem. Neutron scattering from one group to the other group is predominant in the four-energy-group problems of MZA (see Sec. III.A.1), KNK-II (see Sec. III.A.2), and the six-energy-group problem of HTTR (see Sec. III.B.6), discussed in Sec. III. Therefore, while solving these problems, the computation time is reduced by a factor of 3, 4, and 5 or even more. The rest of the benchmark problems involve one or two energy groups, and the computation time is reduced by a factor of only 2 or less. However, solving a many-energy-group problem with simple geometry (e.g., a square filled with single homogeneous material) hardly reflects any advantage of the Krylov iteration over the inner-outer iteration.