Perron向量最优化在搜索引擎中的应用
Perron vector optimization applied to search engines
摘要：
在过去的几年里，Google的PageRank最优化问题得到了广泛的研究。在这种情况下，排名由随机阵的不变测度给出。在本文中，我们考虑更一般的情况下，排名是由Perron特征向量的非负的，但不一定是随机的，矩阵，以涵盖Kleinberg的HITS算法。我们也给出了Tomlin的HOTS算法的一些结果。问题在于，然后在找到一个最佳的外链策略的设计约束和给定的搜索引擎。我们研究这些问题的放松版本，这意味着我们应该接受加权超链接。我们提供了一个有效的算法计算矩阵的偏导数的标准，使用这个矩阵的低秩属性。我们给出了一个耦合梯度和幂迭代的可扩展算法，并给出了Perron向量最优化问题的局部极小值。我们证明了收敛性考虑它作为一个近似的梯度法。然后，我们表明，最佳的链接策略的HITS和HOTS最优化问题验证阈值属性。我们报告这些搜索引擎最优化问题的真实网络图的片段的数值结果。
英文摘要：
In the last years, Google's PageRank optimization problems have been extensively studied. In that case, the ranking is given by the invariant measure of a stochastic matrix. In this paper, we consider the more general situation in which the ranking is determined by the Perron eigenvector of a nonnegative, but not necessarily stochastic, matrix, in order to cover Kleinberg's HITS algorithm. We also give some results for Tomlin's HOTS algorithm. The problem consists then in finding an optimal outlink strategy subject to design constraints and for a given search engine. We study the relaxed versions of these problems, which means that we should accept weighted hyperlinks. We provide an efficient algorithm for the computation of the matrix of partial derivatives of the criterion, that uses the low rank property of this matrix. We give a scalable algorithm that couples gradient and power iterations and gives a local minimum of the Perron vector optimization problem. We prove convergence by considering it as an approximate gradient method. We then show that optimal linkage stategies of HITS and HOTS optimization problems verify a threshold property. We report numerical results on fragments of the real web graph for these search engine optimization problems.
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