# Packet-Forwarding Algorithm in DTN Based on the Pheromone of Destination Node
## 3 The Model of Pheromone and Community
### Pheromone-Spreading Model  
If the pheromone \(P_{a,b}\) of node \(b\) increases during time \(T\), then it will be recomputed as \(P\cdot\omega\) and spread to the neighbors of node \(a\), \(\omega\) is the coefficient of spread, and its range is [0,1]. When the pheromone of the node \(b\) decreases to zero, a community based on the pheromone of the node \(b\) is constructed. In this special community, node \(b\) is the core node, and the pheromone will become strong from the edge node to the core. Hence, if we find the pheromone of node \(b\), then we can enter the  
Figure 2: Communication coverage radius of a node.  
community of node \(b\). We can easily obtain the best forwarding path along the direction of the stronger pheromone. The derivation of \(\omega\) is follows.  
On the basis of the six degrees of separation theory [6], we consider that the distance between the edge and core node should not be more than four relay nodes. We suppose that the network topology does not change before the maximum community with four relayers between the edge node and the core node. The four relayers between the edge node and the core node are shown in Figure 3.  
In Figure 3, \(P_{0,1}\), \(P_{0,2}\), \(P_{0,3}\), \(P_{0,4}\) and \(P_{0,5}\) represent the pheromones of \(v_{1},v_{2},v_{3},v_{4}\) and \(v_{5}\) which are spread from the core node \(v_{0}\), respectively. They also represent the capabilities of the current node to forward a packet to the core node.  
To ensure that the distance from the edge node to the core node is less than six hops in the community, the pheromone \(P_{0,5}\) that was received by the outer node \(n_{5}\) must be decreased to zero during time \(T\), and this premise can be expressed as:  
\[P_{0,5}\leq T, \tag{5}\]  
If \(P_{0,5}\) exists, then \(P_{0,1}\), \(P_{0,2}\), \(P_{0,3}\), and \(P_{0,4}\) must also exist. The topology between nodes \(v_{4}\) and \(v_{5}\) must remain stable during \(T\), the topology between nodes \(v_{3}\) and \(v_{4}\) must remain stable during \(3T\),.., and the topology between nodes \(v_{0}\) and \(v_{1}\) must remain stable during \(5T\). Accordingly, the pheromone of the core node is spread from \(v_{1}\) to \(v_{5}\) needs to be spread by nodes \(v_{2},v_{3}\), and \(v_{4}\). The pheromone is received by \(v_{1},v_{2},v_{3},v_{4}\), and \(v_{5}\) and can be calculated as follows:  
\[P_{0,1}=Y_{1}, \tag{6}\]  
\[P_{0,2}=P_{0,1}\cdot\omega, \tag{7}\]  
\[P_{0,3}=P_{0,2}\cdot\omega, \tag{8}\]  
\[P_{0,4}=P_{0,3}\cdot\omega, \tag{9}\]  
\[P_{0,5}=P_{0,4}\cdot\omega, \tag{10}\]  
If the topology of the community remains stable in the process of spreading \(P_{0,1}\) to \(v_{5}\), then the maximum community is constructed; thus, \(B_{1}=B_{2}=B_{3}=B_{4}=B_{5}=0\). Then, \(P_{0,1}\) can be calculated using:  
\[P_{0,1}=T_{1}+T_{2}+T_{3}+T_{4}+T_{5}\;, \tag{11}\]  
Given that the topology of the community remains stable during the spreading process, each statistical time interval is equal to \(T\), Therefore, \(T_{1}=T_{2}=T_{3}=T_{4}=T_{5}=T\). Then Equation (11) can be simplified as:  
\[P_{0,1}=5T\]  
According to Equations (1)-(7), \(P_{0,5}\) can be calculated as Equation (12)  
\[P_{0,5}=5T\cdot\omega^{4}, \tag{12}\]  
Finally, we can calculate the spread coefficient \(\omega\) as:  
\[\omega\leq\sqrt[4]{\frac{1}{5T}}, \tag{13}\]  
Figure 3: Six relays between the edge node and the core node.