## 4 Methodology
### Global Transition Flow Graph  
While traditional hypergraphs can effectively capture high-order and undirected associations among entities, they are inherently limited in representing directional semantics,such as location-to-location transitions in user visit trajectories. In contrast to the collective preference hypergraph, which models co-visitation, we define a Global Transition Flow Graph \(\mathcal{G}_{T}=(\mathcal{V}_{T},\mathcal{E}_{T},\mathbf{H}_{T},\mathbf{W}_{T})\) to encode directed transition dynamics between locations. Here, the vertex set \(\mathcal{V}_{T}=\{l_{1},l_{2},\ldots,l_{|L|}\}\) denotes the set of locations, while the edge set \(\mathcal{E}_{T}=\{e_{u}^{T}\mid u\in U\}\) captures the sequential transitions observed in all user visit records, where each \(e_{u}^{T}\) corresponds to an ordered set of location transitions \((l_{i}\to l_{j})\) made by user \(u\). Hyperedges represent transitions between locations based on sequential visit data. Each hyperedge in this graph connects a pair of locations, representing the direct transitions from one location to another in a user's trajectory. These transitions are directed to preserve the movement dynamics of users. To represent such directional transitions, we define the incidence matrix \(\mathbf{H}_{T}\in\mathbb{R}^{|\mathcal{E}|\times|\mathcal{E}_{T}|}\), where \(\mathbf{H}_{T}(l_{i},e_{u}^{T})=1\) if location \(l_{i}\) serves as a source or target in the transition \(e_{u}^{T}\), and its role is further differentiated via a pair of incidence matrices \((\mathbf{H}_{T}^{s},\mathbf{H}_{T}^{t})\) that explicitly capture source and target locations, respectively. Specifically, they are defined as follows:  
\[\mathbf{H}_{T}^{s}(l_{i},e_{u}^{T})=\begin{cases}1,&\text{if $l_{i}$ is a source in $e_{u}^{T}$}\\ 0,&\text{otherwise}\end{cases}, \tag{4}\]  
\[\mathbf{H}_{T}^{t}(l_{j},e_{u}^{T})=\begin{cases}1,&\text{if $l_{j}$ is a target in $e_{u}^{T}$}\\ 0,&\text{otherwise}\end{cases}. \tag{5}\]  
Using these directed incidence matrices, we define a transition-aware matrix as \(\mathbf{A}_{T}=\mathbf{H}_{T}^{s}\mathbf{W}_{T}(\mathbf{H}_{T}^{t})^{\top}\), where \(\mathbf{W}_{T}=\text{diag}(w_{1},\ldots,w_{|\mathcal{E}_{T}|})\) is a diagonal matrix assigning weights to each transitional edge, commonly normalized by the number of transitions observed in each user's trajectory. It captures the directed transition strength from source locations to target locations.  
To enable spectral reasoning over this directed global transition flow graph \(\mathcal{G}_{T}\), we define a transition-aware Laplacian matrix as follows:  
\[\mathbf{L}_{T}=\mathbf{D}_{T}^{s}-\mathbf{H}_{T}^{s}\mathbf{W}_{T}\mathbf{D}_ {T}^{-1}\mathbf{W}_{T}(\mathbf{H}_{T}^{t})^{\top}, \tag{6}\]  
where \(\mathbf{D}_{T}^{s}\) and \(\mathbf{D}_{T}\) are the diagonal degree matrices of source nodes and transition edges, respectively. This spectral operator allows us to propagate preferences along directed paths in the trajectory space, thus enabling a fine-grained analysis of transition dependencies. The transition-aware Laplacian matrix captures the directed transition dynamics between locations by incorporating both source and target location information. Spectral reasoning over this matrix allows for the propagation of user preferences along directed paths in the global transition flow graph, enabling the model to capture the directed nature of user mobility. This spectral analysis provides a means to analyze global transition patterns and predict future locations based on the directionality of previous transitions.  
The proposed global transition flow graph \(\mathcal{G}_{T}\), therefore, captures the global structure of location transitions across all users and facilitates the discovery of dynamic behavioral patterns and next-location prediction signals.