SimRank is an attractive link-based similarity measure used in fertile fields of Web search and sociometry. However, the existing deterministic method by Kusumoto et al. [
24
] for retrieving SimRank does not always produce high-quality similarity results, as it fails to accurately obtain diagonal correction matrix
D
. Moreover, SimRank has a “connectivity trait” problem: increasing the number of paths between a pair of nodes would decrease its similarity score. The best-known remedy, SimRank++ [
1
], cannot completely fix this problem, since its score would still be zero if there are no common in-neighbors between two nodes.
In this article, we study fast high-quality link-based similarity search on billion-scale graphs. (1) We first devise a “varied-
D
” method to accurately compute SimRank in linear memory. We also aggregate duplicate computations, which reduces the time of [
24
] from quadratic to linear in the number of iterations. (2) We propose a novel “cosine-based” SimRank model to circumvent the “connectivity trait” problem. (3) To substantially speed up the partial-pairs “cosine-based” SimRank search on large graphs, we devise an efficient dimensionality reduction algorithm,
PSR
#
, with guaranteed accuracy. (4) We give mathematical insights to the semantic difference between SimRank and its variant, and correct an argument in [
24
] that “if
D
is replaced by a scaled identity matrix (1-Ɣ)I, their top-K rankings will not be affected much”. (5) We propose a novel method that can accurately convert from Li et al. SimRank ~{S} to Jeh and Widom’s SimRank
S
. (6) We propose
GSR
#
, a generalisation of our “cosine-based” SimRank model, to quantify pairwise similarities across two distinct graphs, unlike SimRank that would assess nodes across two graphs as completely dissimilar. Extensive experiments on various datasets demonstrate the superiority of our proposed approaches in terms of high search quality, computational efficiency, accuracy, and scalability on billion-edge graphs.
Conjectures on Stably Newton Degenerate Singularities
