Given a graph G with n nodes, and two nodes u,v in G, the CoSim-Rank value s(u,v) quantifies the similarity between u and v based on graph topology. Compared to SimRank, CoSimRank has been shown to be more accurate and effective in many real-world applications including synonym expansion, lexicon extraction, and entity relatedness in knowledge graphs. The computation of all-pair CoSimRank values in G is highly expensive and challenging. Existing methods all focus on devising approximate algorithms for the computation of all-pair CoSimRanks. To attain the desired absolute error delta, the state-of-the-art approximate algorithm for computing all-pair CoSimRank values requires O(n^3log2(ln(1/delta))) time. In this paper, we propose RP-CoSim, a randomized algorithm for computing all-pair CoSimRank values. The basic idea of RP-CoSim is to reduce the n*n matrix multiplications into a k-dimensional(k<<n) subspace via a random projection such that the pairwise inner product is preserved within a certain error, and then iteratively approximate CoSimRank values in the k-dimensional subspace in O(n^2k) time. Theoretically, RP-CoSimruns in O(n^2*ln(n)*ln(1/delta)/delta^2) time, and meanwhile ensures an absolute error of at most delta in the CoSimRank value of every two nodes in G with a high probability.

翻译：以图形 G 和 n 节点和 G 中的 2 节点 u, v, CoSim- Rank 值 s (u,v) 来测量 u 和 v 之间的相似性。 与 SimRank 相比, CoSimRank 显示在许多真实世界应用中更加准确和有效, 包括同义扩展、 立字提取和 知识图形中的实体关系。 在本文中, 计算 G 中的所有 pair coSim1/ Rank 值非常昂贵且具有挑战性。 现有的方法都侧重于为计算全部 pair coSir 的绝对值设计近似算法 。 要达到理想的绝对错误 del- sank, CoS imR% 的状态近似性算法需要 O( n3log) (1/ delta) 时间图中的 O( CoS) 。 我们建议 RP- CoSim- pal- pal- co- mark 数值。 RBR- RV 最基本的想法是用来在 ir- bir- silental- broal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- 和 mal- cal- sal- cal- sal- sal- sal- sal- sir- cal- sal- sal- sir- sir- sir- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- sal- mal- sal- sal- sal- sal- sal- sal- sal- sal- sal- salsalsalsalsalsalsals)。