Efficient Temporal Butterfly Counting and Enumeration on Temporal Bipartite Graphs

TL;DR

This work defines temporal butterflies on temporal bipartite graphs and initiates a study of counting and enumerating these motifs under a duration constraint, identifying six non-isomorphic temporal types. It proposes a wedge-set framework and a wedge-priority-guided optimization (TBC, TBC$^+$, TBC$^{++}$) that dramatically reduces time while preserving space efficiency, and extends the approach to enumeration (TBE$^+$). It also introduces streaming adaptations (STBC, STBC$^+$) with parallel batch updates to handle dynamic graphs, plus a robust handling of extreme cases using red-black trees (TBC$^{++}$). Extensive experiments on 11 real-world datasets demonstrate substantial speedups and scalability, validating the practicality of the methods for both static and streaming temporal bipartite graphs.

Abstract

Bipartite graphs characterize relationships between two different sets of entities, like actor-movie, user-item, and author-paper. The butterfly, a 4-vertices 4-edges (2,2)-biclique, is the simplest cohesive motif in a bipartite graph and is the fundamental component of higher-order substructures. Counting and enumerating the butterflies offer significant benefits across various applications, including fraud detection, graph embedding, and community search. While the corresponding motif, the triangle, in the unipartite graphs has been widely studied in both static and temporal settings, the extension of butterfly to temporal bipartite graphs remains unexplored. In this paper, we investigate the temporal butterfly counting and enumeration problem: count and enumerate the butterflies whose edges establish following a certain order within a given duration. Towards efficient computation, we devise a non-trivial baseline rooted in the state-of-the-art butterfly counting algorithm on static graphs, further, explore the intrinsic property of the temporal butterfly, and develop a new optimization framework with a compact data structure and effective priority strategy. The time complexity is proved to be significantly reduced without compromising on space efficiency. In addition, we generalize our algorithms to practical streaming settings and multi-core computing architectures. Our extensive experiments on 11 large-scale real-world datasets demonstrate the efficiency and scalability of our solutions.

Figures (23)

• Figure 1: 6 types of non-isomorphic temporal butterfly. The vertices from the same layer have the same color (grey in $U$ and white in $L$), and the number denotes the temporal order.
• Figure 2: A user-website network.
• Figure 3: All possible static butterflies while $u$ is the start-vertex and the vertex priority follows $p_V^1 > p_V^2 > p_V^3 > p_V^4$.
• Figure 4: All possible temporal orderings of two temporal wedges $\angle_i$ and $\angle_j$, represented by red and black arcs above/under a time axis. The arcs point from $t_s$ to $t_a$.
• Figure 5: The wedge sets construct from Figure \ref{['fig:example']} while $u_2$ is the start-vertex and $u_1$ is the end-vertex.

Theorems & Definitions (18)

• Definition 1: Temporal Wedge
• Definition 2: Butterfly wang2014rectangle
• Definition 3: Temporal Butterfly
• Definition 4: Vertex Priority
• Theorem 1
• Definition 5: Wedge Set
• Definition 6: Wedge Priority
• Example 1
• Lemma 1
• Lemma 2