# A Topology Optimization Method for Reducing Communication Overhead in the Kalman Consensus Filter
## 3 Network Topology Optimization Method
### Network Topology Optimization Method  
Combining with Propositions 1-3, we can properly delete and add communication links according to some rules to solve problem (P2). The rules for deleting communication links are:  
For any node \(i\), \(i\)\(\in\)\(\{1,\,2,\ldots,\,l\}\),  
Step 1: To determine the initial deleting communication link at node \(i\):  
(1-1) The links set \(\mathcal{E}_{i,1}\) with local maximum sum of node degree \(d\), is found, and  
\[\mathcal{E}_{i,1}=\left\{(i,j)|\underset{j\in N_{i}}{\max}(d_{i}+d_{j})\right\} \tag{3}\]  
(1-2) The links set \(\mathcal{E}_{i,2}\) with the average local maximum variation of average path length \(l\) is determined, and  
\[\mathcal{E}_{i,2}=\left\{(i,j)|\underset{(i,j)\in\mathcal{E}_{i,1}}{\max} \frac{\Delta l_{i}+\underset{k\in\mathcal{N}_{i}}{\sum}\Delta l_{k}}{d_{i}+1}\right\} \tag{4}\]  
where \(\Delta l_{i}\) denotes the variation of \(l\) at node \(i\) before and after deleting link \((i,j)\). (1-3) The links set \(\mathcal{E}_{i,3}\) with the average local minimum variation of clustering coefficient \(c\) is determined, and  
\[\mathcal{E}_{i,3}=\left\{(i,j)|\underset{(i,j)\in\mathcal{E}_{i,2}}{\min} \frac{\Delta c_{i}+\underset{k\in\mathcal{N}_{i}}{\sum}\Delta c_{k}}{d_{i}+1}\right\} \tag{5}\]  
where \(\Delta c_{i}\) denotes the variation of \(c\) at node \(i\) before and after deleting link \((i,j)\). (1-4) The first link belonging to set \(\mathcal{E}_{i,3}\) is taken as the initial deleting communication link at node \(i\), and  
\[\text{del}^{\prime}_{i}=\left\{(i,j),d_{i}+d_{j},\frac{\Delta l_{i}+ \underset{j\in\mathcal{N}_{i}}{\sum}\Delta l_{j}}{d_{i}+1},\frac{\Delta c_{i} +\underset{j\in\mathcal{N}_{i}}{\sum}\Delta c_{j}}{d_{i}+1}\right\} \tag{6}\]  
Step 2: To determine the final deleting communication link at node \(i\):  
(2-1) The initial deleting communication link at node \(i\) is exchanged with its neighboring nodes, and \(\text{del}^{\prime}_{\mathcal{N}_{i}}\triangleq\left\{\text{del}^{\prime}_{ j}\right\}_{j\in\mathcal{N}_{i}}\). (2-2) The same operations in step 1 are used to determine the final deleting communication link at node \(i\), and  
\[\text{del}_{i}=\left\{(a,b),d_{a}+d_{b},\frac{\Delta l_{a}+\underset{c\in \mathcal{N}_{a}}{\sum}\Delta l_{c}}{d_{a}+1},\frac{\Delta c_{a}+\underset{c\in \mathcal{N}_{a}}{\sum}\Delta c_{c}}{d_{a}+1}\right\} \tag{7}\]  
The rules for adding communication links are:  
For any node \(p\), \(p\)\(\in\)\(\{1,\,2,\,\ldots,\,l\}\),  
Step \(1^{\prime}\): To determine the initial adding communication link at node \(p\):(1\({}^{\prime}\)-1) The links set \(\mathcal{L}_{p,1}\) with local minimum sum of node degree \(d\) within the communication range of node \(p\), \(D_{p}\), is found, and  
\[\mathcal{L}_{p,1}=\left\{(p,s)|\min\limits_{s:(s\
otin\mathcal{N}_{p})\cap I\in (p,s)\leq D_{p}}\bigl{(}d_{p}+d_{s}\bigr{)}\right\} \tag{8}\]  
(1\({}^{\prime}\)-2) The links set \(\mathcal{L}_{p,2}\) with the average local maximum variation of average path length \(l\) is determined, and  
\[\mathcal{L}_{p,2}=\left\{(p,s)|\max\limits_{(p,s)\in\mathcal{L}_{p,1}}\frac{ \Delta l_{p}+\sum\limits_{q\in\mathcal{N}_{p}}\Delta l_{q}+\Delta l_{s}}{d_{p} +2}\right\} \tag{9}\]  
where \(\Delta l_{p}\) denotes the variation of \(l\) at node \(p\) before and after adding link \((p,s)\). (1\({}^{\prime}\)-3) The links set \(\mathcal{L}_{p,3}\) with the average local minimum variation of clustering coefficient \(c\), is determined, and  
\[\mathcal{L}_{p,3}=\left\{(p,s)|\min\limits_{(p,s)\in\mathcal{L}_{p,2}}\frac{ \Delta c_{p}+\sum\limits_{q\in\mathcal{N}_{p}}\Delta c_{q}+\Delta c_{s}}{d_{p} +2}\right\} \tag{10}\]  
where \(\Delta c_{p}\) denotes the variation of \(c\) at node \(p\) before and after adding link \((p,s)\).