# Green Small Cell Operation of Ultra-Dense Networks Using Device Assistance +
## 3 Belief Propagation
### Belief Propagation Optimization Algorithm
The message is defined by: \[b_{j\to i}(t+1,x_{j})=\frac{1}{z^{\prime}}\prod_{i^{\prime}\in\mathcal{N}(j) \setminus\{i\}}b_{i^{\prime}\to j}(t,x_{j})\] (13) where \(z^{\prime}\) is the normalizing constant making \(\sum_{x_{j}\in\{0,1\}}b_{j\to i}(t+1,x_{j})=1\). The belief message (13) is the product of the belief messages from UE \(i^{\prime}\in\mathcal{N}(j)\setminus\{i\}\), all neighboring factor nodes, except \(i\). Then, the steps 'UE update' and 'AP update' are repeated until the number of iteration is satisfied. 4. **Decision-making. ** After finishing the iteration procedure, the estimated marginal probability distribution with respect to \(X_{j}\) is finalized as: \[b_{j}(x_{j})=\frac{1}{z^{\prime\prime}}\prod_{i\in\mathcal{N}(j)}b_{i\to j}(t, x_{j})\] (14) where \(z^{\prime\prime}\) is the normalizing constant to make \(b_{j}(x_{j})\) as a probabilistic distribution. Then, as a final step, to describe our association rule \(S\), we define an integer-valued function \(\Phi_{j}(\cdot)\) that determines the position of the \(j\)-th element when the elements of the input vector are sorted in increasing order. Then, we deliberately use the following rank-based association rule of UE \(i\) given by: \[j^{*}(i)=\text{argmax}_{j}\left[\Phi_{j}\left(\left\{SINR_{i,j}(\mathbf{x}) \right\}\right)+\eta\,\Phi_{j}\left(\left\{p_{j}(x_{j})\right\}\right)\right]\] (15) where \(x_{j}=1\), \(\forall j\in J\), and \(\eta\) is a weighting parameter considering the tradeoff between the channel condition and the estimated marginal probability from BP. The intuition of (15) is as follows. If each UE is associated with AP \(j\) solely considering energy efficiency, then most UEs are associated with a few APs having a higher value of \(p_{j}(x_{j})\). High \(p_{j}(x_{j})\) may indicate that there is a global consensus that energy efficiency can be increased when the BS \(j\) is turned on. However, connecting too many UEs to one AP can degrade the data rate, which may decrease energy efficiency. In this sense, user association cannot be determined solely relying on \(p_{j}(x_{j})\). Thus, we exploit \(SINR_{i,j}\) to determine the user association, so each UE can decide its association along with the global perspective of \(p_{j}(x_{j})\) and also personal perspective \(SINR_{i,j}\) to preserve its service quality. Our user association rule can be seen as an extension of the convention user association rule that is based on SINR. For example, if \(\eta=0\), each UE is associated with the AP that provides the highest SINR among APs in \(\mathcal{N}(i)\), which is the conventional user association rule. On the other hand, if \(\eta\to\infty\), the UE is associated with the AP \(j\) whose \(p_{j}(x_{j})\) is the greatest. It could be possible that SCC determines \(\eta\) to maximize energy efficiency by using adaptive learning methods. After user association is done for all UEs, the APs with no UEs are turned off. While the BP framework is readily well defined on many graphical models, the drawback of BP framework is its complexity, which is given by \(O(2^{\mathcal{N}(i)})\) from the view point of the UE. Hence, the complexity of the BP could be a barrier to large-scale ultra-dense networks of APs and UEs. Thus, we next propose online algorithms for dense cellular networks in the next section.