Rumor spread in social network

Spread of rumors

A standard model of rumor spreading was introduced by Daley and Kendall, which is called DK model. Assume there are N people in total. And those people in the network are categorized into three groups: ignorants, spreaders and stiflers, which are denoted as S, I, and R respectively hereinafter:

The rumor is propagated through the population by pair-wise contacts between spreaders and others in the population. Any spreader involved in a pair-wise meeting attempts to “infect” the other individual with the rumor. In the case this other individual is an ignorant, he or she becomes a spreader. In the other two cases, either one or both of those involved in the meeting learn that the rumor is known and decided not to tell the rumor anymore, thereby turning into stiflers.

One famous variant is Maki-Thompson(MK) model. In this model, rumor is spread by directed contacts of the spreaders with others in the population. Furthermore, when a spreader contacts another spreader only the initiating spreader becomes a stifler. Therefore, three types of interactions can happen with certain rates.

Of course we always have conservation of individuals:

The change in each class in a small time interval is:

Since we know S , I and R sum up to N , we can reduce one equation from the above, which leads to a set of differential equations using relative variable x = I / N and y = S / N as follows

which we can write

Compared with the ordinary SIR model, we see that the only difference to the ordinary SIR model is that we have a factor α + β in the first equation instead of just α . We immediately see that the ignorants can only decrease since x , y ≥ 0 and d y d t ≤ 0 . Also, if

which means

the rumour model exhibits an “epidemic” even for arbitrarily small rate parameters.

Rumor spread in social network

We model the process introduced above on a network in discrete time, that is, we can model it as a DTMC. Say we have a network with N nodes, then we can define X i ( t ) to be the state of node i at time t. Then X ( t ) is a stochastic process on S = { S , I , R } N . At a single moment, some node i and node j interact with each other, and then one of them will change its state. Thus we define the function f so that for x in S , f ( x , i , j ) is when the state of network is x , node i and node j interact with each other, and one of them will change its state. The transition matrix depends on the number of ties of node i and node j, as well as the state of node i and node j. For any y = f ( x , i , j ) , we try to find P ( x , y ) . If node i is in state I and node j is in state S, then P ( x , y ) = α A j i / k i ; if node i is in state I and node j is in state I, then P ( x , y ) = β A j i / k i ; if node i is in state I and node j is in state R, then P ( x , y ) = β A j i / k i . For all other y , P ( x , y ) = 0 .
The procedure on a network is as follows:

We would expect that this process spreads the rumor throughout a considerable fraction of the network. Note however that if we have a strong local clustering around a node, what can happen is that many nodes become spreaders and have neighbors who are spreaders. Then, every time we pick one of those, they will recover and can extinguish the rumor spread. On the other hand, if we have a network that is small world, that is, a network in which the shortest path between two randomly chosen nodes is much smaller than that one would expect, we can expect the rumor spread far away.

Also we can compute the final number of people who once spread the news, this is given by
r ∞ = 1 − e − ( α + β β ) r ∞
In networks the process that does not have a threshold in a well mixed population, exhibits a clear cut phase-transition in small worlds. The following graph illustrates the asymptotic value of r ∞ as a function of the rewiring probability p .