## 3 Component-Wise Details of LPaMI
### Graph Construction  
Since we have a time-stamped collection of OOIs, the domain of time is a linearly ordered set \(T\). A time-dependent network is a directed graph \(G=(V,E)\), where every edge \(e\) has an associated link-traversal function \(f_{e}:T\to T\). We consider two same OOIs as one instance in LPaMI if the time gap between them \(\triangle t=(t_{i+1}-t_{i})\) is below a given threshold _TimeThreshold_. We consider such photographs to be taken in same place and represent them as a single node. Furthermore, if \(v_{i}=v_{i+1}\) and \(\triangle t<\textit{TimeThreshold}\), they can be converged into one record such as \(v=v_{i}=v_{i+1};t=(t_{i}+t_{i+1})/2\). If \(v_{i}\
eq v_{i+1}\) or \(\triangle t>\textit{TimeThreshold}\), we create an edge \(e=(v_{i},v_{i+1})\). We illustrate the sample travel data of a person and display it in a graph format in the Figure 7.  
To support applying versatile graph mining operations in LPaMI, we propose generating graphs of two types: (1) a set of sequence graphs; and (2) a cumulated graph. We define both of them below. **Sequence Graph Model:** Let \(f\) be the function to create the set of sequence graphs \(S_{t}(V_{t},E_{t})\in S\) from given graph \(G\) and time set \(T\). If \(f\) satisfies \(f(t)\leq f(t+1)\) then the total numbers of vertices and edges are monotonically increasing over time. Suppose \(S_{t}(V_{t},E_{t})\) is a sequence graph, where \(V_{t}\subseteq V\) is the set of vertices \(v_{1},v_{2},...,v_{k}\in V_{t}\), and \(E_{t}\subseteq E\) is the set edges for the time-instance \(t\). Each edge connects the consecutive two vertices \(e(v_{i},v_{i+1})\in E_{t},i=1,\ldots,k-1\). The vertex \(v_{1}\) is called the source, \(t_{1}\) the starting time, \(v_{k}\) the destination, and \(t_{k}\) the arrival time in time instance \(t\). The set of sequence graphs is homogeneous in nature, where all the nodes are of same type. Figure 7: Modelling scene data from photographs in a graph format. Each node is an OOI, like a visited place, and an edge denotes two places visited one after the other. **Definition 1**. : _(Set of Sequence Graphs). We define \(S\), \(S=S_{t_{1}}\to S_{t_{2}}\rightarrow\cdots\to S_{t_{k}}\), as a sequence graph where \(t_{i}<t_{i+1}\)._  
For instance, a user visits every place in a sequence in a real-world scenario. Using the given data, we create a set of sequence graphs and align them to generate cumulated graphs based on the defined granularity level of a user (granularity level can be either year, quarter, month, weekly, or day) using the MUSCLE algorithm. A set of sequence graphs is illustrated in Figure 1 in the graph modelling phase, where the size of each sequence is of one month. **Cumulated Graph Model:** Our model takes as input a sequence of edge sets \(\{E_{1},E_{2},\ldots E_{n}\}\) arriving at times \(\{T_{1},T_{2},\ldots T_{n}\}\) and a function \(fW(G)\), which outputs the value of a specified statistic on graph \(G\). At each time \(T_{i}\), \(g\) is applied to the current aggregated graph \(G_{i}\) to obtain value \(w(e)_{i}\) for edge weight. Each vertex \(v_{i}\) in \(G_{i}\) is merged by node type i.e, if they are same OOIs. Given a large graph and the granularity level, the proposed model creates a set of heterogeneous graphs \(G_{t}(V_{t},E_{t},T_{V},g)\) that consists of various types of nodes. Member graphs at each time interval are combined to create an aggregated graph. Here, \(T_{V}\) is a finite set of node types and the function \(g\) is applied to the current aggregated graph \(G_{t}\) to obtain edge weight value \(w(e)_{t}\) for each edge. In each graph \(G_{t}\), the number of vertices is much smaller because the vertex \(v_{i}\) in \(G_{t}\) is merged by node type. **Definition 2**. : _(Cumulated Graph): The cumulated graph \(G(V_{t},E_{t},T_{V},g)\) is a single large graph, where \(V_{t}\in V\) for the set of edges \(E_{t}\in E\).