Dynamical SimRank search on time-varying networks

SimRank is an appealing pair-wise similarity measure based on graph structure. It iteratively follows the intuition that two nodes are assessed as similar if they are pointed to by similar nodes. Many real graphs are large, and links are constantly subject to minor changes. In this article, we study the efficient dynamical computation of all-pairs SimRanks on time-varying graphs. Existing methods for the dynamical SimRank computation [e.g., LTSF (Shao et al. in PVLDB 8(8):838–849, 2015) and READS (Zhang et al. in PVLDB 10(5):601–612, 2017)] mainly focus on top-k search with respect to a given query. For all-pairs dynamical SimRank search, Li et al.’s approach (Li et al. in EDBT, 2010) was proposed for this problem. It first factorizes the graph via a singular value decomposition (SVD) and then incrementally maintains such a factorization in response to link updates at the expense of exactness. As a result, all pairs of SimRanks are updated approximately, yielding (Formula presented.) time and (Formula presented.) memory in a graph with n nodes, where r is the target rank of the low-rank SVD. Our solution to the dynamical computation of SimRank comprises of five ingredients: (1) We first consider edge update that does not accompany new node insertions. We show that the SimRank update (Formula presented.) in response to every link update is expressible as a rank-one Sylvester matrix equation. This provides an incremental method requiring (Formula presented.) time and (Formula presented.) memory in the worst case to update (Formula presented.) pairs of similarities for K iterations. (2) To speed up the computation further, we propose a lossless pruning strategy that captures the “affected areas” of (Formula presented.) to eliminate unnecessary retrieval. This reduces the time of the incremental SimRank to (Formula presented.), where m is the number of edges in the old graph, and (Formula presented.) is the size of “affected areas” in (Formula presented.), and in practice, (Formula presented.). (3) We also consider edge updates that accompany node insertions, and categorize them into three cases, according to which end of the inserted edge is a new node. For each case, we devise an efficient incremental algorithm that can support new node insertions and accurately update the affected SimRanks. (4) We next study batch updates for dynamical SimRank computation, and design an efficient batch incremental method that handles “similar sink edges” simultaneously and eliminates redundant edge updates. (5) To achieve linear memory, we devise a memory-efficient strategy that dynamically updates all pairs of SimRanks column by column in just (Formula presented.) memory, without the need to store all (Formula presented.) pairs of old SimRank scores. Experimental studies on various datasets demonstrate that our solution substantially outperforms the existing incremental SimRank methods and is faster and more memory-efficient than its competitors on million-scale graphs

Fast Exact CoSimRank Search on Evolving and Static Graphs

In real Web applications, CoSimRank has been proposed as a powerful measure of node-pair similarity based on graph topologies. However, existing work on CoSimRank is restricted to static graphs. When the graph is updated with new edges arriving over time, it is cost-inhibitive to recompute all CoSimRank scores from scratch, which is impractical. In this study, we propose a fast dynamic scheme, \DCoSim for accurate CoSimRank search over evolving graphs. Based on \DCoSim, we also propose a fast scheme, \FCoSim, that greatly accelerates CoSimRank search over static graphs. Our theoretical analysis shows that \DCoSim and \FCoSim guarantee the exactness of CoSimRank scores. On the static graph G, to efficiently retrieve CoSimRank scores \mathbfS , \FCoSim is based on three ideas: (i) It first finds a "spanning polytree»» T over G. (ii) On T, a fast algorithm is designed to compute the CoSimRank scores \mathbfS (T) over the "spanning polytree»» T. (iii) On G, \DCoSim is employed to compute the changes of \mathbfS (T) in response to the delta graph $(G øminus T)$. Experimental evaluations verify the superiority of \DCoSim over evolving graphs, and the fast speedup of \FCoSim on large-scale static graphs against its competitors, without any loss of accuracy

RoleSim* : scaling axiomatic role-based similarity ranking on large graphs

RoleSim and SimRank are among the popular graph-theoretic similarity measures with many applications in, e.g., web search, collaborative filtering, and sociometry. While RoleSim addresses the automorphic (role) equivalence of pairwise similarity which SimRank lacks, it ignores the neighboring similarity information out of the automorphically equivalent set. Consequently, two pairs of nodes, which are not automorphically equivalent by nature, cannot be well distinguished by RoleSim if the averages of their neighboring similarities over the automorphically equivalent set are the same. To alleviate this problem: 1) We propose a novel similarity model, namely RoleSim*, which accurately evaluates pairwise role similarities in a more comprehensive manner. RoleSim* not only guarantees the automorphic equivalence that SimRank lacks, but also takes into account the neighboring similarity information outside the automorphically equivalent sets that are overlooked by RoleSim. 2) We prove the existence and uniqueness of the RoleSim* solution, and show its three axiomatic properties (i.e., symmetry, boundedness, and non-increasing monotonicity). 3) We provide a concise bound for iteratively computing RoleSim* formula, and estimate the number of iterations required to attain a desired accuracy. 4) We induce a distance metric based on RoleSim* similarity, and show that the RoleSim* metric fulfills the triangular inequality, which implies the sum-transitivity of its similarity scores. 5) We present a threshold-based RoleSim* model that reduces the computational time further with provable accuracy guarantee. 6) We propose a single-source RoleSim* model, which scales well for sizable graphs. 7) We also devise methods to scale RoleSim* based search by incorporating its triangular inequality property with partitioning techniques. Our experimental results on real datasets demonstrate that RoleSim* achieves higher accuracy than its competitors while scaling well on sizable graphs with billions of edges

Similarity Search over Network Structure

With the advent of the Internet, graph-structured data are ubiquitous. An essential task for graph-structured data management is similarity search based on graph topology, with a wide spectrum of applications, e.g., web search, outlier detection, co-citation analysis, and collaborative filtering. These graph topology data arrive from multiple sources at an astounding velocity, volume and veracity. While the scale of network structured data is increasing, existing similarity search algorithms on large graphs are impractical due to their expensive costs in terms of computational time and memory space. Moreover, dynamic changes (e.g., noise and abnormality) exists in network data, and it arises from many factors, such as data loss in transfer, data incompleteness, and dirty reading. Thus, the dynamic changes have become the main barrier to gaining accurate results for efficient network analysis. In real Web applications, CoSimRank has been proposed as a robust measure of node-pair similarity based on graph topology. It follows a SimRank-like notion that “two nodes are considered as similar if their in-neighbours are similar”, but the similarity of each node with itself is not constantly 1, which is different from SimRank. However, existing work on CoSimRank is restricted to static graphs. Each node pair CoSimRank score is retrieved from the sum of dot products of two Personalised PageRank vectors. When the graph is updated with edges (nodes) addition and deletion over time, it is cost-inhibitive to recompute all CoSimRank scores from scratch, which is impractical. RoleSim is a popular graph-structural role similarity search measure with many applications (e.g., sociometry), it can get the automorphic equivalence of nodes pair similarity, which SimRank and CoSimRank lack. But the accuracy of RoleSim algorithm can be improved. In this study, (1) we propose fast dynamic scheme, D-CoSim and D-deCoSim, for accurate CoSimRank search over large-scale evolving graphs. (2) Based on D-CoSim, we also propose fast scheme, F-CoSim and Opt_F-CoSim, which greatly accelerates CoSimRank search over static graphs. Our theoretical analysis shows that D-CoSim, D-deCoSim F-CoSim and Opt_F-CoSim guarantee the exactness of CoSimRank scores. Experimental evaluations verify the superiority of D-CoSim and D-deCoSim over evolving graphs, and the fast speedupof F-CoSim and Opt_F-CoSim on large-scale static graphs against its competitors, without any loss of accuracy. (3) We propose a novel role similarity search algorithm FaRS, and a speedup algorithm Opt_FaRS, which guarantees the automorphic equivalence capture, and captures the information from the neighbour’s class. The experimental results of FaRS and Opt_FaRS show that our algorithms achieves higher accuracy than baseline algorithms