## 3 Problem Formulation
### Time-Weighted Decay Function  
Intuitively, users' interests change over time, and the more recent the attribute, the more representative it is, as with the location attributes. In addition, the higher the attribute \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(G=(V,E)\) & a graph with vertex set \(V\) and edge set \(E\) \\ \(H\subseteq G\) & \(H\) is a subgraph of \(G\), and \(V_{H}\) and \(E_{H}\) are the vertex set and \\ edge set of \(H\), respectively \\ \(deg_{H}(v)\) & the degree of vertex \(v\) in \(H\) \\ AvgDegree(\(H\)) & the average degree of nodes in graph \(H\) \\ \(T_{j}\) & a check-in timestamp \\ \(A(v)=\{A_{1}(v),A_{2}(v),\ldots,A_{l}(v)\}\) & text attribute set of node \(v\) \\ \(W(v)=\{W_{1}(v),W_{2}(v),\ldots,W_{l}(v)\}\) & attribute weight set of node \(v\), \(\sum_{i=1}^{l}W_{i}(v)=1\) \\ \(l=|A(v)|\) & size of attribute set \(A(v)\) \\ \(A(u,v)=A(u)\cup A(v)\) & union of attribute sets \(A(u)\) and \(A(v)\) without duplicate \\ \(Vec(v)=(Vec_{1}(v),Vec_{2}(v),\ldots,Vec_{l\_dimen}(v))\) & attribute weight vector of \(A(u,v)\) \\ \(l\_dimen=|A(u,v)|\) & size of attribute set \(A(u,v)\) \\ \(P(v)=\big{(}P_{x}(v),P_{y}(v)\big{)}\) & position of node \(v\), and \(P_{x}(v)\), \(P_{y}(v)\) are \(v\)’s latitude and \\  & longitude coordinates \\ \(P_{j}(v)=\Big{(}P_{xj}(v),\ P_{yj}(v)\Big{)}\) & check-in coordinates of node \(v\) at \(T_{j}\), and \(P_{xj}(v)\), \(P_{yj}(v)\) are \(v\)’s latitude and longitude at \(T_{j}\) \\ \(CosTime(u,v)\) & the textual similarity score of node \(u\) and \(v\) \\ \(GSim(u,v)\) & the spatial similarity score of node \(u\) and \(v\) \\ \(TScore(q,v)\) & the similarity score of node \(v\) and \(q\) \\ \(TScore(q,H)\) & the graph score of \(H\) containing \(q\) \\ \(r\) & the time decay factor \\ \(\alpha\) & the balance factor of different type of attribute scores \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and meanings. frequency, the stronger the ability to express interest. The user's interest in attributes is the largest at check-in and then decays at a certain speed. As shown in Figure 2, assuming that attribute \(A_{i}(v)\) appears in \(v\)'s check-in records at \(T_{0}\), \(T_{1}\) and \(T_{2}\), the user \(v\)'s interest in \(A_{i}(v)\) is enhanced every time. This is consistent with the law of human memory, known as the Ebbinghaus memory curve. Let \(\gamma\) be the decay factor, representing the decay rate of interest. The larger the \(\gamma\), the faster the attenuation. To simplify subsequent operations, let \(\gamma\) remain constant at each check-in. We designed decay functions for single check-in and repeated check-in occasions of attributes, as shown in Formulas (1)-(3). \[f\big{(}A_{i}(v),\ T_{j}\big{)}=e^{-r\Delta t_{j}} \tag{1}\]  
\[f(A_{i}(v))=\sum_{j=0}^{N_{v}^{i}}f\big{(}A_{i}(v),\ T_{j}\big{)} \tag{2}\]  
\[W_{i}(v)=\frac{f(A_{i}(v))}{\sum_{m=1}^{l}f(A_{m}(v))} \tag{3}\]  
Formula (1) expresses the attenuation of user \(v\)'s interest in \(A_{i}(v)\) from time \(T_{j}\), where \(\Delta t_{j}=\frac{T_{end}-T_{j}}{T_{end}-T_{start}}\), \(T_{j}\) is the check-in time of \(A_{i}(v)\), and \(T_{start}\) and \(T_{end}\) are the start and end timestamps of the whole query window. As can be seen from Formula (1), the fresher the \(T_{j}\) is, the smaller the \(\Delta t_{j}\) and the larger the \(f\big{(}A_{i}(v),\ T_{j}\big{)}\), indicating that the user is more interested in the attribute \(A_{i}(v)\). Since \(A_{i}(v)\) may appear in several check-in records at different times \(T_{j}\), Formula (2) describes \(v\)'s overall interest in \(A_{i}(v)\), where \(N_{v}^{i}\) is the sum of the occurrence times of \(A_{i}(v)\) in the historical check-in records of user \(v\). For the convenience of comparison, we normalize \(v\)'s interest in all attributes and finally obtain the weight of attribute \(A_{i}(v)\) named as \(W_{i}(v)\), which is \(v\)'s interest in attribute \(A_{i}(v)\). The larger the \(W_{i}(v)\), the greater the user \(v\)'s interest in \(A_{i}(v)\).