## 4 Average end-to-end delay  
This parameter represents the network delay of routing Prelearn messages, the delay of phase transition and the delay of forwarding Prelearn messages. In this simulation, we first evaluate the impact of the aging variable on the probability of the destination profile receiving Prelearn messages (\(POV\)). As shown in Figure 5, when the value of the aging variable changes, the corresponding value of \(POV\) always exceeds 0.7. Moreover, the \(POV\) reaches a maximum value of 0.85 when the aging variable is 0.3. Figure 6 shows the impact of the phase transition value on the probability of the destination profile receiving Prelearn messages (\(POV\)). As it can be seen, when the phase transition value increases from \(N/4\) to \(N\), the fluctuation range of the \(POV\) is from 0.3 to 0.8. Particularly, when the phase transition is at \(N/3.5\), the Prelearn message has not only obtained enough statistics about the state of the network but also found the destination profile by the message delivery function. Therefore, the message carrier can make the suitable delivery decisions for Prelearn messages, the efficient data transmission performs, and the \(POV\) gets the maximum value of 0.8. Figure 7 exhibits the impact of the phase transition value on delivery ratio. As it can be seen, the broken line first rises sharply and then falls rapidly. This means the phase transition value plays a significant role in the delivery ratio. A short Predict phase may miss the optimal candidate in the data transmission process, while a long Predict phase may imply that a large number of irrelevant nodes participate in the forwarding process of Prelearn messages. When the phase transition occurs at the optimal moment \(n/3.5\), message carriers select the suitable relay nodes based on reliable information about the state of the network, and the destination profile is more likely to have appeared in the Predict phase, so the delivery ratio reaches the maximum value of 0.95. Figure 8 shows the influence of the phase transition value on network overhead. As it can be seen, the broken line first drops rapidly and then rises slowly. The later the phase transition occurs, the more \begin{table}
\begin{tabular}{c c} \hline
**Simulation Environment** & **Description** \\ \hline Simulator & Opportunistic Network Environment (ONE) \\ Mobility model & HCMM \\ Communication area (m\({}^{2}\)) & \(3000\times 3000\) \\ A period of time (min) & 40 \\ Simulation time (h) & 1–12 \\ Number of nodes & 1000 \\ Number of Prelearn messages & 100 \\ Time to create a Prelearn message (s) & 1–10 \\ Cache space of a node (Mb) & 10 (Initial value), 15, 20, 25, 30, 35, 40 \\ Speech of a node (m/s) & 1–4 \\ Number of data packets carried by a node & 10 \\ \hline \end{tabular}
\end{table} 
Table 1: The simulation settings for Predict and Forward algorithm. the nodes take part in the forwarding process of Prelearn messages. Meanwhile, the number of nodes unrelated to the process of data transmission also increases, and more time and resources are wasted in transmitting Prelearn messages to the unrelated nodes, so the network overhead presents a rising trend. However, When the phase transition occurs at the optimal moment \(n/e\), the Prelearn message acquires sufficient information about the state of the network, the node obtains the exact probability of future meeting with other nodes, and the routing and delivering of the Prelearn message are at their optimal state, so the network overhead gets the minimum value of 85. Figure 9 exhibits the influence of the phase transition value on average end-to-end delay. As it can be seen, the broken line first declines slowly and then rises rapidly. In other words, when the number of the nodes routed by Prelearn messages increases, the number of the nodes participating in the data transmission process also goes up, so the average end-to-end delay in Predict and Forward algorithm decreases constantly.