Truncated and Sparse Power Methods with Partially Updating for Large and Sparse Higher-Order PageRank Problems

Abstract

Higher-order Markov chain plays an important role in analyzing a variety of stochastic processes over time in multi-dimensional spaces. One of the important applications is the study of higher-order PageRank problem. The truncated power method proposed recently provides us with another idea to solve this problem, however, its accuracy and efficiency can be poor in practical computations. In this work, we revisit the higher-order PageRank problem and consider how to solve it efficiently. The contribution of this work is as follows. First, we accelerate the truncated power method for higher-order PageRank. In the improved version, it is neither to form and store the vectors arising from the dangling states, nor to store an auxiliary matrix. Second, we propose a truncated power method with partially updating to further release the overhead, in which one only needs to update some important columns of the approximation in each iteration. However, the truncated power method solves a modified higher-order PageRank model for convenience, which is not mathematically equivalent to the original one. Thus, the third contribution of this work is to propose a sparse power method with partially updating for the original higher-order PageRank problem. The convergence of all the proposed methods are discussed. Numerical experiments on large and sparse real-world and synthetic data sets are performed. Numerical results demonstrate the superiority of our new algorithms over some state-of-the-art ones for large and sparse higher-order PageRank problems.

Appendices

Appendix A: Numerical Results of Subsection 6.2 on Real-World Data Set

See Tables 3, 4, 5 and 6.

Table 3 A comparison of the six algorithms for high-order PageRank on .GOV Web collection, $\alpha =0.85$, and the parameter $\beta$ for sparsing is chosen as $\frac{1}{n^{2}},\frac{1}{n^{3}}$ and $\frac{1}{n^{4}}$, respectively. Notice that the approximation obtained from the power method is generally dense

Table 4 A comparison of the six algorithms for high-order PageRank on .GOV Web collection, $\alpha =0.90$, and the parameter $\beta$ for sparsing is chosen as $\frac{1}{n^{2}},\frac{1}{n^{3}}$ and $\frac{1}{n^{4}}$, respectively. Notice that the approximation obtained from the power method is generally dense

Table 5 A comparison of the six algorithms for high-order PageRank on .GOV Web collection, $\alpha =0.95$, and the parameter $\beta$ for sparsing is chosen as $\frac{1}{n^{2}},\frac{1}{n^{3}}$ and $\frac{1}{n^{4}}$, respectively.Notice that the approximation obtained from the power method is generally dense

Table 6 A comparison of the six algorithms for high-order PageRank on .GOV Web collection, $\alpha =0.99$, and the parameter $\beta$ for sparsing is chosen as $\frac{1}{n^{2}},\frac{1}{n^{3}}$ and $\frac{1}{n^{4}}$, respectively. Notice that the approximation obtained from the power method is generally dense

Table 7 The number of top-ten ranking results of the .GOV Web collection obtained from the five algorithms, $\alpha =0.85$

Appendix B: Numerical Results of Subsection 6.3 on Synthetic Data Set

See Tables 8, 9, 10 and 11.

Table 8 A comparison of the algorithms on synthetic data, $n=10000,\alpha = 0.85$. Notice that the approximations obtained from the power method to EIOM are generally dense

Table 9 A comparison of the algorithms on synthetic data, $n=10000,\alpha = 0.90$. Notice that the approximations obtained from the power method to EIOM are generally dense

Table 10 A comparison of the algorithms on synthetic data, $n=10000,\alpha = 0.95$. Notice that the approximations obtained from the power method to EIOM are generally dense

Table 11 A comparison of the algorithms on synthetic data, $n=10000,\alpha = 0.99$. Notice that the approximations obtained from the power method to EIOM are generally dense