## 3 Influence Metrics
### Global Influence
#### 3.2.2 Social Closeness Metric  
Human social relationships typically display greater stability than transmission links between mobile devices due to the complex network conditions present, for instance, in Opportunistic Mobile Social Networks (OppMSNs), characterized by intermittent connectivity that results in unstable end-to-end paths between devices. As a result, OppMSN routing decisions may be made more efficient using social indicators. It is noticed that people keep both regular and sporadic interactions within their social surroundings. Information propagation greatly depends on the degree of contact between nodes. If the sender often communicates with the destination device, the sender may be aware of the times when they are most likely to run across the destination or nodes that are very likely to cross paths with the destination in the future [29; 30]. Conversely, the likelihood of two devices knowing one another improves if they have a greater number of friends in common. We examine the devices' past contacts in order to develop the Opportunistic Relationship Index (ORI), a social metric that is derived from important structural characteristics of a complex network, specifically the contact durations between devices, their shared neighbors, and distances [31]. In order to reflect the possibility of establishing a connection between devices \(v\) and \(k\), the score is calculated as shown in Equation (17) for each pair of unconnected devices \(v\) and \(k\). In this equation, \(\sum_{i=1}^{T}\left\{\varphi_{vk}x_{vk}\right\}_{t_{i}}\) represents the frequency of contact occurrences between devices \(v\) and \(k\) within the time span \(T\), and \(\sum_{i=1}^{|V|-1}\sum_{j=2}^{|V|}x_{ij}\) denotes the distance matrix of existing paths between the two devices:  
\[ORI_{T}(v,k)= \begin{cases}\left(\sum_{i=1}^{T}\left\{\varphi_{vk}x_{vk}\right\} _{t_{i}}\right)^{\frac{|T(v)\cdot T(k)|+1}{2}}_{t_{i}}\text{ if }\Gamma(v)\cap\Gamma(K) \
eq 0\\ \left(\sum_{i=1}^{T}\left\{\varphi_{vk}x_{vk}\right\}_{t_{i}}\right)^{\frac{ 1}{min\left[\sum_{i=1}^{|V|-1}\sum_{j=2}^{|V|}\right]}}_{t_{i}}\text{ otherwise},\end{cases}  \tag{18}\]  
where \(\Gamma(v)=\sum_{i=1}^{T}\sum_{i=1,l\
eq v}^{|V|}\left\{x_{vl}\right\}_{t_{i}}\) and \(\Gamma(k)=\sum_{i=1}^{|V|}\sum_{l=1,l\
eq k}^{|V|}\left\{x_{vl}\right\}_{t_{i}}\) represent the respective sets of neighbors for devices \(v\) and \(k\), respectively, over the time span \(T\). Figure 5 in this context shows the subnetwork created from Appendix A Figure A1, only showing the four current routes connecting device \(A\) and device \(B\), with the weights denoting the calculated \(ORI\). Then, we present the shortest path based on opportunistic relationships, which include both opportunistic relationships and communication costs. We invert the weights to find the path with the lowest weight since the weight denotes the \(ORI\). We use a tuning parameter to ensure that the ORI and the number of intermediary devices affect the choice of the best path, incorporating both communication cost and \(ORI\). The following is the social measure used in Equation (18) to determine the opportunistic cost (\(Cost_{opp}\)) of a path between a source device (s) and a destination device (\(d\)):  
\[Cost_{opp}(s,s+1,\ldots,d)=\sum_{i=s}^{d-1}\left(\frac{1}{ORI_{T}(i,i+1)} \right)^{\alpha}, \tag{19}\]  
subject to \(t_{s,s+1}\leq t_{s+1,s+2}\leq\cdots\leq t_{d-1,d}\). The \(ORI\) between devices \(i\) and \(j\) before their interaction with the next relay device is indicated by the symbol \(ORI_{T}\) in Equation (18). The moment at which device \(i\) and device \(j\) first made contact is represented by \(t_{ij}\), and the sequence in which connections between intermediary devices are made is indicated by \(t_{s,s+1}\leq t_{s+1,s+2}\leq\cdots\leq t_{d-1,d}\). We further normalize the series by dividing the geometric mean of \(ORI_{T}\) by the highest \(ORI_{T}\), as the routing choice may not function effectively with a low \(ORI_{T}\) value across devices.