In this subsection, we discuss the time evolution of information dissemination process based on our model. First, we investigate the time evolutions of the number of ignorant, dissemination, and recovered nodes. We only change д = 0.0001 and T = 15,000. As shown in Fig. 2.4, the number of spreading nodes S(t) sharply increases at first, and then reaches a peak value. Finally, the number of spreading nodes keeps decreasing until reaching 0. The number of ignorant nodes I (t) sharply falls within an extremely short time. The number of recovered nodes R (t) grows from 0 at a fast rate in a short period. Obviously, these numerical results are consistent with our analysis in Sect. 2.4.

We study the pre-immunity using the presented model. In Fig. 2.4,the valueof S(t) becomes around 0 when dissemination time t = 10, 000, which indicates that there is no spreading node in the network. In other words, the information dissemination process is stopped at this moment. At the same time, there are still a few ignorant nodes in the network, which would lose the interest and become recovered nodes gradually. Obviously, the process of transition is considerably slow since the selfimmune parameter д is very small. For example, in Fig. 2.4, the value of I (t) is reduced about 0.2 % in each unit of time when the value of S(t) is almost 0. In order to further exhibit pre-immunity, we set д = 0 to study time evolutions of the number of three types of nodes without pre-immunity in Fig. 2.5. We can observe that three curves become stable when there is no spreading node in the network. The reason is that ignorant nodes maintain interests all the time if д = 0, and thus ignorant nodes cannot directly become recovered nodes.

Fig. 2.5 Time evolutions of the number of ignorant nodes, spreading nodes, and recovered nodes, where ^ = 0