# Improving Traffic Load Distribution Fairness in Mobile Social Networks
## 4 TraLDA Design
### Inherent Popularity Calculation
The Kalman filter is widely used in control system design to estimate unmeasured process conditions. It can calculate the best estimates of the current states of a dynamic system defined in a state vector. The state is updated based on periodic observations of the system. We use a typical state space model [46] to express the problem in our model. Furthermore, we only investigate the case when a seasonal component dominates the time series (see [47] for the discussion of Kalman prediction for a complete model). The state space model is constituted by two scalar equations, namely the observation equation and the state equation. For our model with (only) a seasonal component, the observation equation is given as follows  
\[y_{t}=S_{t}+W_{t}\,\ \ \ \ \ \ t=1,2,\ \ldots \tag{3}\]  
where \(S_{t}\) is a state variable and \(W_{t}\) is an additive white noise with zero mean and variance \(\sigma_{w}^{2}\) (\(W_{t}=WN(0,\sigma_{w}^{2})\)). Furthermore, when we consider \(S_{t}\) representing a seasonal component  
Figure 4: **(left)** The changes of degree of an illustrative hub node in Reality (measured by node degree in a 6-h time window), and (**right**) the detected periodicities of the node’s degree. This describes that the node popularity in the mobile social network fluctuates over time and has a weekly period. with a period \(d\) such that \(S_{t+d}=S_{t}\) and \(\sum_{t=1}^{d}S_{t}=0\), it is therefore possible to determine \(S_{t+1}\) as  
\[S_{t+1}=-S_{t}-S_{t-1}\ldots-S_{t-d+2} \tag{4}\]  
For a more general expression of \(S_{t}\) allowing random deviations to exist in the periodicity, a white noise term \(V_{t}\) (\(V_{t}=WN\left(0,\sigma_{w}^{2}\right)\)) is added in the right hand side of (4). Afterwards, regarding only the seasonal effect on the series, in order to obtain the state equation for our model, we introduce \((d-1)\) dimensional state vector \(X_{t}\) defined as  
\[X_{t}=\left[ \begin{array}{cccc}S_{t}&S_{t-1}&\ldots&S_{t-d+2} \end{array} \right]^{T} \tag{5}\]  
and the series \(S_{t}\) is determined as  
\[S_{t}=\left[1 \begin{array}{cccc}0&0&0&\ldots&0\end{array}  \right]X_{t} \tag{6}\]  
For the purpose of the derivation of [PERSON] prediction, the observation equation in (3) is now rewritten in a general form as follows  
\[Y_{t}= \begin{array}{c}G_{t}X_{t}+W_{t}\end{array}  \tag{7}\]  
with \(G_{t}=\left[1 \begin{array}{cccc}0&0&0&\ldots&0\end{array} \right]\), and \(X_{t}\) satisfies the state equation  
\[X_{t+1}= \begin{array}{c}F_{t}X_{t}+V_{t}\end{array}  \tag{8}\]  
with \(V_{t}=\left[\sigma_{v}^{2}&0&0&\ldots&0\end{array}\right]^{T}\), and \(F_{t}= \begin{bmatrix}-1-1&\ldots&-1-1\\ 1&0&\ldots&0&0\\ 0&1&\ldots&0&0\\ \vdots&\vdots&\ldots&\vdots&\vdots\\ 0&0&\ldots&1&0\end{bmatrix} \). Given the observation equation in (7) and the state equation in (8), the recursive equations of Kalman-filter for the estimation of the values of the series are defined as follows. Considering the initial settings as  
\[\hat{X}_{1}=P(X_{1}|Y_{0})\]  
Figure 5: **(upper)** Long-term seasonal data, and **(lower)** deseasonalized data of the hub node’ degree in Figure 4. These figures show that node popularity in mobile social networks typically comprises a periodic (seasonal) component along with a random noise component. \[\Omega_{1}=E(X_{1}-\hat{X}_{1})(X_{1}-\hat{X}_{1})^{T} \tag{10}\]  
the Kalman recursive equations are then given as  
\[\hat{X}_{t+1}=F_{t}\hat{X}_{t}+\Theta_{t}\Delta_{t}^{-1}(Y_{t}-G_{t}\hat{X}_{t}) \tag{11}\]  
\[\Omega_{t+1}=F_{t}\Omega_{t}F_{t}^{T}+Q_{t}-\Theta_{t}\Delta_{t}^{-1}\Theta_{t} ^{T} \tag{12}\]  
\[\text{where }\Delta_{t}=G_{t}\Omega_{t}G_{t}^{T}+R_{t},\Theta_{t}=F_{t}\Omega_{t}G_ {t}^{T},Q_{t}= \begin{bmatrix}\sigma_{\
u}^{2}&0&\dots&0\\ 0&\sigma_{\
u}^{2}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&\sigma_{\
u}^{2}\end{bmatrix}