Proposed Model with a Novel Selfishness Division
It is assumed that there is only one node having information initially, and then this node moves randomly in a finite region. When this node encounters its friends who do not have the information, the information can be delivered.
Ik (t) is denoted as the number of k -degree nodes which do not have information at time t. Sk (t) is defined as the number of k -degree nodes which obtain information at time t. Rk (t) means the number of k -degree nodes which refuse accepting the information at time t. Zk (t) is defined as the number of k-degree nodes which have received the information but don’t want to forward it to the next node at time t. Let Nk (t) denote the number of k -degree nodes at a given time t. Then, the fractions of these four types of nodes in Nk (t) becomes as follows at time t.
Within the interval [t, t + At], we can have the variation of ik(t) by
Here P(G, k) is defined as the probability that a k-degree node ni without information receives it or refuses accepting it from any one within At.
We consider two aspects to derive P(G, k). On one hand, node ni receiving the information from other friends is classified by sk (t). On the other hand, node ni which may refuse accepting information is classified by rk (t).
As mentioned above, only when an opportunistic link exists between two nodes, they are able to have a chance to exchange information. In addition, since mobile nodes forward information only to their friends, the social tie between two nodes should also exist. As the inter-meet between two nodes follows an exponential distribution, the probability that node ni encounters other nodes within the interval of [t, t + At] becomes 1 - e-x(N-1)At. In fact, if the interval is short enough, node ni can only encounter one node. Thus, probability that the node ni meets the other node (e.g., nj.) within [t, t + At] is 1 - e-x(N-1)At. Since the degree of node ni is k, the probability that node nj is a friend of ni can be obtained by k/(N - 1). Besides, the probability that node nj has information is XN=I nNTP(k')sk (t), where the degree of a node is the k'. Then, node ni may get information successfully from one of its friends when these two nodes encounter with the probability 1 - pnf.
Therefore, according to the above description, the probability P(A, k) that node ni can receive the information from its friend within At can be obtained by
Here, the above shows the necessary conditions that node ni can receive information. Firstly, node ni should encounter node nj, and this node nj is a friend of node ni. Secondly, node nj should have information and be willing to forward information to node ni.
Then, about the second aspect, the probability P(not receive) that node ni does not accept information absolutely within At becomes
By combining with (3.3) and (3.5), we obtain From (3.3), we have
Similarly, for a short time interval At, it can be obtained by,
where P(not forward) denotes the probability that a k -degree node having the information refuses to forward it to any of other nodes within [t, t + At], which can be computed as follows:
In addition, given a short time interval At, we can derive Therefore, we have
Given a short time interval At, we have Then, we have
Therefore, the ordinary differential equations (ODEs) to model the dynamic information dissemination among the mobile individuals are shown as follows: