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The local Extrapolated Di usion (EDF) method was studied in [5] for a torus network using communication among 4 adjacent neighbors. In the present paper we study the performance of the same method with the extension that we add four additional edges to the central node in position (i,j) of the torus network. We develop the EDF method for a 8 -regular torus network. The method uses two sets of parameters and for each node in order to increase its rate of convergence. The conventional way to analyze the convergence of the Di usion method is to use matrix analysis. This approach depends heavily upon the property of the Laplacian matrix being circulant. However, the Laplacian matrix of our method does not have this property. To circumvent this problem we use Fourier analysis to determine optimum values for the set of parameters via a closed form formulae resulting in the maximization of its rate of convergence. It is shown that the optimum value of the convergence factor depends only upon the dimensions of the torus. Moreover, by keeping xed the one dimension and increasing the other dimension of the torus the rate of convergence of the EDF method with 8 adjacent neighbors is also increased compared to the EDF method with 4 adjacent neighbors. keywords Laplacian matrix, load balancing, regular graphs, iterative di usion, Fourier analysis.

Optimal Nine Node Diffusion for Special Torus Graphs