Uncovering periodic group behaviors in temporal bipartite social graphs

<jats:p> ( <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> </jats:inline-formula> , <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> </jats:inline-formula> )-bicliques serve to identify cohesive node groups in bipartite graphs, where each node is required to maintain at least <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> </jats:inline-formula> or <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> </jats:inline-formula> connections within the biclique. Although ( <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> </jats:inline-formula> , <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> </jats:inline-formula> )-biclique enumeration has been extensively studied, its temporal periodicity remains largely unexplored. To capture recurrent group behaviors in temporal bipartite graphs, we introduce the novel problem of periodic ( <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>α</mml:mi> </mml:math> </jats:inline-formula> , <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> </jats:inline-formula> )-biclique enumeration. Our approach presents three core innovations: First, a <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>θ</mml:mi> </mml:math> </jats:inline-formula> -periodic biclique model, which enforces that all edges within a biclique exhibit <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>θ</mml:mi> </mml:math> </jats:inline-formula> -periodic temporal support. Second, an edge-based reduction framework that avoids the combinatorial explosion typical in node-based enumeration. This is achieved by first constructing condensed subgraphs via <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>θ</mml:mi> </mml:math> </jats:inline-formula> -temporal edge filtering, followed by biclique enumeration within the reduced space. Third, two optimized data structures: (i) a linked edge list (LE-list)–based periodic edge aggregator that enables constant-time subgraph access, and (ii) a Trie-based periodicity detector that accelerates the validation of temporal recurrence. Extensive experiments on six real-world temporal networks demonstrate up to a 10 <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo>×</mml:mo> </mml:math> </jats:inline-formula> speedup over baseline methods, highlighting substantial gains in efficiency, accuracy, and scalability. </jats:p>