# A Special Structural Based Weighted Network Approach for the Analysis of Protein Complexes
## 3 Methods
### Identifying Overlapping Structures  
To identify the overlapping structure, let \(v\in V\), \(N(v)=\{u|u\in V,(v,u)\in E\}\) be set of neighbour protein \(v\) and \(deg(v)=|N(v)|\) be the number of neighbours of protein \(v\). Given protein \(v\in V\) we can define the neighborhood network \(\mathcal{GN_{V}}=(V_{v},E_{v})\) as sub-network of protein \(v\) and its direct neighbours interacting in network \(\mathcal{G}\). Hence \(V_{v}=\{v\}\cup\{u|u\in V,(v,u)\in E\}\) and \(E_{v}=\{(u_{i},u_{j})\big{|}(u_{i},u_{j})\in E,u_{i},u_{j}\in V_{v}\}\). Thus, the weighted degree average of a local neighborhood sub-network \(\mathcal{GN_{V}}\) is defined by the equation  
\[Avg(deg(\mathcal{GN_{V}}))=\frac{\sum\limits_{u\in V_{v}}deg(u)}{|V_{v}|}, \tag{5}\]  
To calculate the global importance of a protein \(v\), we calculate the shortest paths between all protein pairs that pass through the target proteins by defining the betweenness of node \(v\) by  
\[B(v)=\sum\limits_{\begin{subarray}{c}s\
eq v,t\
eq v\\ s,t,v\in V\end{subarray}}\frac{\delta_{s,t}(v)}{\delta_{s,t}}\,, \tag{6}\]  
where \(\delta_{s,t}\) is the number of shortest paths between protein \(s\) to \(t\) and \(\delta_{s,t}(v)\) is the number of shortest paths between protein \(s\) to \(t\) that pass through the intermediate (bridge) protein \(v\). Thus, using Equation (6) the average betweenness of its local neighborhood sub-network \(\mathcal{GN_{v}}\) is calculated by  
\[Avg(B(\mathcal{GN_{v}})=\frac{\sum_{u\in V}B(u)}{|V_{v}|}, \tag{7}\]  
where \(AvgB(\mathcal{GN_{V}})\) is the average of \(B(u)\) for all \(u\in V_{v}\) in local neighborhood sub-network \(\mathcal{GN_{v}}\) and \(|V_{v}|\) denotes the total number of nodes in the _PPI network_. Then, using Equations (5) and (7) we defined overlapping protein structure in the PPI network by  
\[\mathcal{OVP}(\mathcal{GN_{V}})=\begin{cases}1&if\ deg(v)\geq Avg(deg( \mathcal{GN_{V}}))\wedge B(v)>Avg(B(\mathcal{GN_{V}}))),\\ 0&otherwise\end{cases} \tag{8}\]where \(\mathcal{GN}_{v}\) is candidate overlapping protein complex with \(\mathcal{OVP}(\mathcal{GN}_{v})=1\). To measure the degree of overlap between sets of candidate overlapping protein complexes we calculate the overlapping score between the two sets by  
\[OS\big{(}\mathcal{OVP}_{i},\mathcal{OVP}_{j}\big{)}=\frac{\big{|}\mathcal{OVP}_{ i}\cap\mathcal{OVP}_{j}\big{|}}{\big{|}\mathcal{OVP}_{i}\big{|}+\big{|}\mathcal{OVP}_{j} \big{|}-\big{|}\mathcal{OVP}_{i}\cap\mathcal{OVP}_{j}\big{|}}, \tag{9}\]  
where \(OS\big{(}\mathcal{OVP}_{i},\mathcal{OVP}_{j}\big{)}\) is overlapping score between \(\mathcal{OVP}_{i}\) and \(\mathcal{OVP}_{j}\) ranging from \([0,1)\) in which 0 indicates no overlap between the sets and 1 indicates that the sets are identical; \(|\mathcal{OVP}_{i}|\) and \(\big{|}\mathcal{OVP}_{j}\big{|}\) denote the sizes of sets \(\mathcal{OVP}_{i}\) and \(\mathcal{OVP}_{j}\), respectively. \(\big{|}\mathcal{OVP}_{i}\cap\mathcal{OVP}_{j}\big{|}\) denotes the intersection of sets \(\mathcal{OVP}_{i}\) and \(\mathcal{OVP}_{j}\). In this paper, we identify the candidate overlapping protein complex when \(OS\big{(}\mathcal{OVP}_{i},\mathcal{OVP}_{j}\big{)}\geq\pi\), where \(\pi\) is predefined overlap threshold ranging in \((0,1]\).