Nested Dissection and Kemeny’s Constant: A Comparative Study

In this article, we present a novel approach to efficiently compute Kemeny’s constant, which is a crucial component in various matrix computations. The traditional method for computing Kemeny’s constant is based on LU factorization, but it can be computationally expensive for large matrices. To address this challenge, we propose a low-precision randomized approximation technique that significantly reduces the computational complexity while maintaining the accuracy of the result.
Our approach is built around a clever observation: by applying a suitable permutation to the matrix, we can transform it into a "quasi-block-diagonal" structure, making it easier to compute its LU factorization. We illustrate this idea through an example in Figure 1, which shows how the resulting matrix has a more structured layout after the permutation step.
To apply this technique, we need to solve several intermediate linear systems, each corresponding to a line in Algorithm 1. These systems can be solved using an iterative method, and their solution is essential for computing Kemeny’s constant. We demonstrate the effectiveness of our approach by presenting performance metrics for various test matrices in Table 2.
In summary, our work provides a practical and efficient method for computing Kemeny’s constant, which is crucial in many matrix computations. By leveraging the structural properties of the matrix, we can significantly reduce the computational complexity without sacrificing accuracy. Our approach has broad applicability and can be particularly useful when dealing with large-scale matrices.