## 3 Influence Metrics
### Global Influence
#### 3.2.1 Dynamic Closeness Metric  
One centrality measure that relies on distance is Closeness. It is determined by averaging the shortest distances (involving the fewest nodes, thus minimizing overhead) from a specific node to all other nodes in the network. This is equivalent to summing the shortest distances (\(d_{short}\)) and dividing by the number of nodes (referred to as the network order, denoted as \(|V|\)), minus one, as node \(j\) itself is excluded from this calculation:  
\[ \begin{array}{c}\textit{Average path lengths}(v)=\frac{\sum_{k=1}^{|V|-1} min(D(v,k))}{|V|-1}=\\ =\frac{\sum_{k=1}^{|V|-1}min\left(\sum_{i=1}^{|V|-1}\sum_{j=2}^{|V|}\left(\{x _{ij}\}_{t_{ij}}\right)+\lambda\right)}{|V|-1}\end{array}  \tag{10}\]  
where \(x_{ij}=1\) if the link \(\{i,j\}\) is used in a route between the source node \(v\) and the destination node \(k\) and \(x_{ij}=0\) otherwise. The lower the above value, the closer a node is to the center of the network. For this reason, closeness is defined as the reciprocal of Equation (10), so that the more centered a node \(v\) is in the network, the higher its closeness metric is:  
\[C_{CLO}(v)=\frac{|V|-1}{\sum_{k=1,k\
eq\
u}^{|V|-1}min\left(\sum_{i=1}^{|V|-1} \sum_{j=2}^{|V|}\left(\{x_{ij}\}_{t_{ij}}\right)+\lambda\right)} \tag{11}\]  
The Closeness measure in the network is strongly related to the rate of information propagation between devices as well as the timeframes at which messages are transmitted over the network. This metric offers a means to evaluate how accessible a device is within the network. Let us examine an example calculation for the connected devices \(\{A,B,C,D,F\}\) during \(G_{7}\), as shown in Figure 4.  
Figure 4: Network connections during instant \(G_{7}\), based on the example of Figure 1.  
\[C_{CLO}(A) =\frac{6-1}{\lambda(2+3+1+3)+\lambda}=\frac{5}{10\lambda}\] \[C_{CLO}(B) =\frac{6-1}{\lambda(2+1+1+1)+\lambda}=\frac{5}{6\lambda}\] \[C_{CLO}(C) =\frac{6-1}{\lambda(3+1+2+1)+\lambda}=\frac{5}{8\lambda}\] \[C_{CLO}(D) =\frac{6-1}{\lambda(1+1+2+2)+\lambda}=\frac{5}{7\lambda}\] \[C_{CLO}(F) =\frac{6-1}{\lambda(3+1+1+2)+\lambda}=\frac{5}{8\lambda}\]  
Therefore, the ranking of nodes according to their closeness index is as follows:  
\[Ranking=\{B,D,C,F,A\} \tag{12}\]  
Regrettably, calculating the Closeness metric (as indicated in Equation (11)) for each node in a network requires knowledge of the distances between all pairs of vertices. In disconnected networks, where nodes belong to distinct components or subnetworks that lack any larger linked subnetwork, the distance between two nodes is traditionally considered infinite, as depicted in the example shown in Figure 1, rendering Closeness inapplicable. As a result, the reciprocal becomes 0, and the sum in the equation (Equation (11)) diverges. Devices often belong to various components, rendering Closeness values irrelevant for all devices in the network except those within the largest component. Consequently, the computation of the Closeness metric must exclude devices that are part of smaller components. Through the utilization of dynamic shortest paths, we can overcome the limitation of the conventional Closeness metric in disconnected networks. By employing a method that accumulates the reciprocal of path costs instead of the reciprocal of the total path cost, we can redefine the Closeness metric. This approach takes into consideration communication costs or overhead. As a result, the Dynamic Closeness metric is defined as the sum of the reciprocals of distances, rather than the reciprocal of the sum of distances:  
Figure 4.0  
The adoption of the Dynamic Closeness measure prevents situations where an infinite distance dominates over other distances. Additionally, this measure can be standardized by considering that in a network with a star topology, the maximum value is achieved by the central node, which is equal to \(\mid V\mid-1\) (the longest distance possible in a network with \(\mid V\mid\) nodes is \(\mid V\mid-1\), i.e., in a chain-connected network).