Theory of Rumor Spreading in Complex networks

Today in this blog we will deal with the world most beautiful and dangerous invention named as rumor. Rumors spreading play an important part in shaping of the world.  The spread of rumors can shape the public opinion in a country, greatly impact financial markets and also can cause panic in a society during wars. Thus one can possibly imagine the viral marketing of the rumor one’s it goes over internet or any social networking site. Most of the corporate world now uses WWW to spread rumor over Internet and “world-of-email”.

Rumor can be viewed as an “infection of mind”. Here we will primarily focus our discussion on Graph node rumor spreading by standard model of Daley and Kendal or the DK model and Maki-Thomson model. DK model divides the population into three major categories namely: Ignorant, Stiflers and Spreaders.  Rumors are spread via pair-wise contact of spreaders and others in the population.

S: People who are ignorant of the rumor;

I: People who actively spread the rumor;

R: People who have heard the rumor, but no longer are interested in spreading it.

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.

In social networking let us consider the Graph G (V, E). Following Maki-Thomson model graph consider a population consisting of N individuals and rumor can only spread by direct contact along the links..

Whenever a spreader contacts an ignorant, the ignorant becomes a spreader at a rate λ .

When a spreader contacts another spreader or a stifler the initiating spreader becomes a stifler at a rate α

In the above, the first rule models the tendency of individuals to accept a rumor only with a certain probability which, loosely speaking, depends on the urgency or credibility of a rumor. The second rule, on the other hand, models the tendency of individuals to lose interest in spreading a rumor when they learn, through contacts with others, that the rumor has become stale news, or is false. In both the DK and the MK rumor models, and their variants, stifling is the only mechanism that results in cessation of rumor spreading.

We will describe above model using IMC framework (mean field equations). IMC was initially introduced to handle means for modeling social processes involving several agents. It consists of N nodes and internal transition is not only depended on the current node but also on the node adjacent to the  current node.

Consider now a node j which is in the ignorant state at time t. We denote with p_ii^j  the probability that this node stays in the ignorant state in the time interval[t +∆t ] and with p_is^j = 1- p_ii^j   the probability that it makes a transition to the spreader state. It then follows that

p_ii^j  =(1-∆t λ)^g ,

where g=g(t) denotes the number of neighbors of node j which are in the spreader state at time t.

The corresponding probability for a transition from the spreader to the stifler state, p_sr(k,t) is given by

p_sr(k,t)=1- p_ss(k,t).

The final size of the rumor, R is shown as a function of the spreading rate λ for the ER network of size 10⁶. The results are shown for several values of the stifling parameter α.
References:
[1] Theory of rumor spreading in complex networks. M.Nekovee, Y.Moreno, G.Binaconi, M.Marsili
[2] Rumor spreading in social network-Wikipidea, http://en.wikipedia.org/wiki/Rumor_spread_in_social_network