# Discovering Moving Flock Patterns in Movement Data: A Reeb Graph-Based Approach
## 1 Introduction
[28] proposed a clustering method to exploiting the flocking movements for activity of daily living learning and recognition in human movement data. [PERSON] et al. [29] described an improved strategy for tracking loose group companions, which can avoid the strict requirement for members in a flock. They then developed a technique to identify such flock patterns in which members can be varying. [PERSON] et al. [30] developed a novel graph-based movement pattern mining algorithm, \"_EvolvingClusters_\", to discover flock patterns online, and evaluated the efficiency and effectiveness of the proposed algorithm on four representative real-world datasets. [PERSON] et al. [15] proposed a hierarchical collective motion pattern inspired by sheep for multiple unmanned aerial vehicle (UAV) cooperative control, and designed a method to mine the flocking collective behavior of UAVs. Compared to the extensive research on flock patterns, there has been limited specifically focus on moving flock patterns. The typical work on detecting moving flock patterns is outlined below. [PERSON] [31] and [PERSON] et al. [32] proposed the first formal definition of \"moving flock\" and developed a corresponding algorithm based on collective coherence. They then tested the algorithm on two datasets that tracked pedestrian movements in recreational areas. [PERSON] et al. [33] presented an approach for mining frequent patterns to discover moving flock patterns in large spatiotemporal datasets. This approach was integrated with a visual interface of the space-time cube, enabling users to interactively explore the results. [PERSON] et al. [34] defined \"freedom moving flock\" and developed a corresponding algorithm for extracting these patterns. The algorithm was tested on two real pedestrian trajectory datasets, and the results demonstrated the usefulness of extracting such patterns. [PERSON] et al. [35] extended classical trajectory-oriented flock mining algorithms to discover moving flock patterns of transit passengers using massive map data, and developed a map-centric visual analysis approach to interactively analyze the discovered moving flock patterns. Summarizing the research discussed above, we find that although researchers have proposed various types of flock patterns from different perspectives, there is still no unified classification system for flock patterns. Therefore, the development of a taxonomy for flock patterns is necessary. Additionally, while [PERSON] [31] and [PERSON] et al. [32] introduced the first formal definition of moving flock based on the flock definition previously presented by [PERSON] et al. (2013), there are still specific drawbacks, particularly the potential confusion between stationary flock and moving flock in certain scenarios. Thus, the definition of moving flock should be improved. Furthermore, to facilitate the data mining process, a new tool that better integrates both spatial and temporal information of moving objects is needed. In this paper, we address these issues by first developing an improved definition of moving flock to clarify moving flock patterns; second, we present a taxonomy of flock patterns that can be used to derive various types of moving flock patterns; then, we focus on four distinct types of moving flock patterns that we find particularly interesting; and finally, the Reeb graph is used as a novel tool to transfer movement data, and an approach based on the Reeb graph is proposed to discover the four distinct types of moving flock patterns. The remainder of this paper is organized as follows. Section 2 introduces the improved definition of moving flock. Section 3 presents the taxonomy of flock patterns. Section 4 details the methodology for discovering the four distinct types of moving flock patterns, including a presentation of the Reeb graph and methods for constructing Reeb graphs. Section 5 describes a case study using movement data from a real football match to validate the effectiveness of the proposed approach.