## 3 Message Passing for JP-CoMP Clustering
### Problem Formulation  
In the main idea described above, we approximated the original nonlinear form into linear-form objective function. With this transformation, multi-layer message passing proposes a mixed-form objective function to provide optimal cell clustering in JP-CoMP. This approach compromises the performance improvement by releasing the cell size limit and mismatch of the approximated function. The mixed-form objective function consists of two parameters: the optimal sum capacity and the closest distance. The systematical work of multi-layer message passing is defined, as follows. In the cluster, the base station will exchange messages with neighboring cells and choose the exemplar based on the sum capacity. To characterize the throughput improvement of cooperative manner, the CoMP gain, \(s_{ij}\), is defined in our model. The CoMP gain maximization will increase the efficiency of JP-CoMP throughput improvement by determining the tendency of one base station to coordinate with another base station based on sum capacity. This term can be defined, as follows,  
\[s_{ij}=R^{comp}_{(i,j)}\,, \tag{4}\]  
where \(R^{comp}_{(i,j)}\) represents the sum capacity of a specific user when the serving base stations, in this case, the \(i\)th base station and the \(j\)th base station, operate in a cooperative manner. In the super-cluster, the exemplar will exchange messages to other exemplars and choose the super-exemplar based on the closest distance. This distance presents a simple quantitative parameter for JP-CoMP. Although the distance does not have a direct relationship with any increases in the throughput, minimizing the distances between the members of a cluster increases the efficiency of JP-COMP throughput improvements. The inverse distance is used in our model, and this term can be defined, as follows,  
\[l_{iC_{ii}}=\mathcal{T}(D_{k},D_{i})^{-1},\forall\;k\;\in\;i^{\prime}, \tag{5}\]  
where \(D_{k}\) is the position of the \(k\)th base station as an exemplar, \(D_{i}\) is the position of the \(i\)th base station as an exemplar, and \(\mathcal{T}(\cdot,\cdot)\) denotes the distance between two positions. Finally, all of the base stations in the same super-cluster will cooperate and ignore the cluster limitation. The objective function, in this case the sum capacity, can be represented by using \(s_{ij}\), \(l_{iC_{ii}}\), and the assignment matrix, as follows,  
\[S_{ij}(C_{ij})= \begin{cases}s_{ij},&\text{if }i\
eq j\text{ and }\;C_{ij}\
eq 0\\ s_{ii}+l_{iC_{ii}},&\text{if }i=j\text{ and }\;C_{ij}=0\;,\\ 0,&\text{otherwise},\end{cases}  \tag{6}\]  
where the \([C_{ij}]_{N\times N}\) assignment variables consist of two assignment matrices of the non-diagonal variable \(C_{ij}\in\{0,1\}\) and the diagonal variable \(C_{ii}\in\{0,\,1,\,\ldots,\,N\}\). For the non-diagonal variable, \(C_{ij}=1\) implies that the \(j\)th base station is an exemplar of the \(i\)th base station. For the diagonal variable, \(C_{ii}=k\), where \(k\in\{1,\,\ldots,\,N\}\) implies that the \(k\)th base station is a super-exemplar of the \(i\)th base station. Otherwise, the \(i\)th base station is not an exemplar. The sum capacity maximization problem in (3) is approximated into a linear optimization problem, as shown below,  
\[\max\ \sum_{i}\sum_{j}S_{ij}\left(C_{ij}\right),\] (7) subject to \[\sum_{j\in\phi\left(i\right)}\left[C_{ij}\
eq 0\right]=1, \tag{8}\] \[\sum_{i\in\phi\left(j\right)}C_{ij}\leq\left[C_{jj}\ =j\right],\] (9) \[\text{if }\exists\ i:C_{ii}=k\text{ then }C_{kk}=k,\ \ i\in\ \theta\left(k\right). \tag{10}\]  
Each constraint has own special meaning in forming multi-layer clustering. Constraint (8) explains that the \(i\)th BS should choose only one BS as its exemplar. The notation \(\phi(i)\) is defined as the set of neighboring BSs adjacent to the \(i\)th BS. Constraint (9) explains that each BS only cooperate with at most one BS in the first layer clustering, where \(\phi(i)\) is defined as the set of neighboring BSs adjacent to the \(j\)th BS. This constraint only enforces two possible cases.