# 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.

*I _{k} (t)* is denoted as the number of

*k*-degree nodes which do not have information at time t.

*S*is defined as the number of

_{k}(t)*k*-degree nodes which obtain information at time t.

*R*means the number of

_{k}(t)*k*-degree nodes which refuse accepting the information at time t.

*Z*is defined as the number of k-degree nodes which have received the information but donâ€™t want to forward it to the next node at time t. Let

_{k}(t)*N*denote the number of

_{k}(t)*k*-degree nodes at a given time t. Then, the fractions of these four types of nodes in

*N*becomes as follows at time t.

_{k}(t)

Within the interval *[t, t + At],* we can have the variation of *i _{k}(t)* by

Here *P(G, k)* is defined as the probability that a k-degree node *n _{i}* 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 *n _{i}* receiving the information from other friends is classified by

*s*On the other hand, node

_{k}(t).*n*which may refuse accepting information is classified by

_{i}*r*

_{k}(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

*n*can only encounter one node. Thus, probability that the node

_{i }*ni*meets the other node (e.g.,

*nj*.) within

*[t, t + At]*is 1 -

*e*-1)At. Since the degree of node

^{-x(N}*n*is k, the probability that node

_{i}*nj*is a friend of

*n*can be obtained by

_{i}*k/(N -*1). Besides, the probability that node

*nj*has information is X

^{N}=I n

^{N}T

*P(k')s*where the degree of a node is the

_{k}(t),*k'.*Then, node

*n*may get information successfully from one of its friends when these two nodes encounter with the probability 1 -

_{i}*p*.

_{n}fTherefore, 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 *n _{i}* should encounter node

*nj*, and this node

*nj*is a friend of node

*n*. Secondly, node

_{i}*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 *n _{i}* 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: