## 3 System Model and Problem Formulation
### Model of Social Dynamic Mobility  
As we can see in Figure 1, the mobility of social sensors such as mobile phones and tablets is determined by the decisions and behaviors of their owners. Accordingly, their mobility is consistent with the human mobility. In the literature [27; 28], several models are proposed to characterize human mobility trajectories. However, these models lack the capability of capturing the features of human dynamic mobility. In [25], the analytical results show that the human movements are not random. Thus, a novel model is proposed to characterize the mobility of humans. The proposed model presents high predictability of human movement. In the following, we present the detailed model of social dynamic mobility. From the work in [25; 26], we know the human trajectories can be characterized by two probability distributions, related to the jump length (\(\Delta t\)) and waiting time (\(\Delta t\)), respectively, given by [25; 26]:  
\[\mathcal{P}(\Delta r) =(\Delta r+\Delta r_{0})^{-1-\alpha}e^{-\Delta r/k_{1}} \tag{1}\] \[\mathcal{P}(\Delta t) =(\Delta t)^{-1-\beta}e^{-\Delta t/k_{2}} \tag{2}\]  
where the jump length \(\Delta r\) is the distance from current location to the next location. The waiting time \(\Delta t\) denotes the spending time of individual at the same location. \(\alpha\) and \(\beta\) are the parameters controlling the jump distance and wait time, respectively, while \(k_{1}\) and \(k_{2}\) are the cutoff values of jump distance and wait time, respectively. The value of \(\Delta r_{0}\) specifies the minimum jump distance. According to the above distributions and the analysis in [25], the individual will change the current location after a waiting time \(\Delta t\). The individual mobility process is described by two generic mechanisms, i.e., exploration and preferential return. For exploration, each social sensor node intends to move to a new location with probability \(\mathcal{P}_{new}\)  
\[\mathcal{P}_{new}=\rho S^{-7}. \tag{3}\]  
Figure 1: Sensing scenario of WSN in CPSS. In Equation (3), the \(S\) denotes the number of visited locations, while \(\rho\),\(\gamma\) refer to the parameters of exploring a new location and returning to a visited location, respectively. The social sensor node visits the new location based on the probability \(\mathcal{P}_{new}\). After a waiting time \(\Delta t\), the social sensors return to the previously visited location based on the following probability \(\mathcal{P}_{ret}\)  
\[\mathcal{P}_{ret}=1-\rho S^{-\gamma}. \tag{4}\]  
The analytical results for the model in [25] show that, over time, the number of visited locations \(S\) should follow the law:  
\[S(t)\sim t^{\mu}. \tag{5}\]  
where \(\mu=\beta/(1+\gamma)\), and \(\gamma\) refers to the parameter of returning to a visited location, \(\gamma>0\). \(\beta\) denotes the parameter of controlling wait time, \(0<\beta<1\). We know that \(\mu<1\), which means that the \(S(t)\) has a decreasing tendency of visiting unvisited locations for the individual. According to the above definition of location in Section 3.1, the number of locations in a city or urban area is limited. When the simulation time \(t\) tends to be infinity, the number of visited location will be constant. Then, based on the initial time moment, we can determine the number of visited locations \(S\). According to the analysis in [25], we can count the frequency of visiting location \(i\) and calculate the corresponding transition probability between any two visit locations \(j\) and \(k\), i.e., \(\mathcal{P}_{i}(s_{j}|s_{k})=\mathcal{F}_{j,k}^{i},i\in\{1,\cdots,O\},j,k\in \{1,2,\cdots,S\},j\
eq k\). \[\mathcal{P}_{i}(s_{j}|s_{k})= \begin{bmatrix}0&\mathcal{F}_{(1,2)}^{i}&\cdots &\mathcal{F}_{(1,S)}^{i}\\ \mathcal{F}_{(1,2)}^{i}&0&\cdots&\mathcal{F}_{(1,S)}^{i}\\ \vdots&\vdots&\ddots&\vdots\\ \mathcal{F}_{(S,1)}^{i}&\mathcal{F}_{(S,2)}^{i}&\cdots&0\end{bmatrix}  \tag{6}\]  
From the work in [25], we know that the model can effectively capture the basic features of human social mobility.