Efficient Computation of Cohesive Subgraphs in Uncertain Bipartite Graphs

Bipartite graphs are extensively used to model relationships between two different types of entities. In many real-world bipartite graphs, relationships are naturally uncertain due to various reasons such as data noise, measurement error and imprecision of data, leading to uncertain bipartite graphs. In this paper, we propose the ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha,\beta,\eta$</tex> )-core model, which is the first cohesive subgraph model on uncertain bipartite graphs. To capture the uncertainty of relationships/edges, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> -degree is adopted to measure the vertex engagement level, which is the largest integer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> such that the probability of a vertex having at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> neighbors is not less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> . Given degree constraints <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> , and a probability threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> , the ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha, \beta, \eta$</tex> )-core requires that each vertex on the upper or lower level have <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> -degree no less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta$</tex> , respectively. An ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha, \beta, \eta$</tex> )-core can be derived by iteratively removing a vertex with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> -degree below the degree constraint and updating the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> -degrees of its neighbors. This incurs prohibitively high cost due to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> -degree computation and updating, and is not scalable to large bipartite graphs. This motivates us to develop index-based approaches. We propose a basic full index that stores ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha, \beta, \eta$</tex> )-core for all possible <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha, \beta$</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eta$</tex> combinations, thus supporting optimal retrieval of the vertices in any ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha, \beta, \eta$</tex> )-core. Due to its long construction time and high space complexity, we further propose a probability-aware index to achieve a balance between time and space costs. To efficiently build the probability-aware index, we design a bottom-up index construction algorithm and a top-down index construction algorithm. Extensive experiments are conducted on real-world datasets with generated edge probabilities under different distributions, which show that (1) ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha,\beta,\eta$</tex> )-core is an effective model; (2) index construction and query processing are significantly sped up by the proposed techniques.