Rumor spread in social network: Difference between revisions

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[[Rumor]] is an important form of social communications[[communication]]s, and spread of rumors plays a significant role in a variety of human affairs. There are two rumor models that are widely used, i.e. DK model and MK model. Particularly, we can view rumor spread as a [[stochastic process]] in social networks.

== Model ==

=== Spread of Rumorsrumors ===

AnA standard model of rumor spreading was introduced by Daley and Kendall,<ref>Daley, D.J., and Kendal, D.G. 1965 Stochastic rumors, J. Inst. Maths Applics 1, p42.</ref> 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:

*S: people who are ignorant of the rumor;

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The change in each class in a small time interval is:<br />

:<math> \Delta S \approx - \Delta t \alpha IS/N</math> <br />

:<math> \Delta I \approx \Delta t [{\alpha IS \over N} - {\beta I^2 \over N} - {\beta IR \over N}]</math> <br />

:<math> \Delta R \approx \Delta t [{\beta I^2 \over N}+{\beta IR \over N}] </math>

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

:<math> {dx \over dt} = x \alpha y - \beta x^2 - \beta x(1-x-y) </math> <br />

:<math> {dy \over dt} = - \alpha xy</math><br />

which we can write

:<math> {dx \over dt} = (\alpha + \beta)xy - \beta x </math> <br />

:<math> {dy \over dt} = - \alpha xy</math>

Compared with the ordinary [[Compartmental models in epidemiology|SIR model]], we see that the only difference to the ordinary [[Compartmental models in epidemiology|SIR model]] is that we have a factor <math>\alpha + \beta</math> in the first equation instead of just <math>\alpha</math>. We immediately see that the ignorants can only decrease since <math>x,y\ge 0</math> and <math>{dy \over dt}\le 0</math>. Also, if<br />

:<math>R_0={\alpha +\beta \over \beta} >1 </math><br />

which means<br />

:<math>{\alpha \over \beta}>0</math><br />

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

=== Rumor Spreadspread in Socialsocial Networknetwork ===

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 <math> X_i(t)</math> to be the state of node i at time t. Then <math>X(t)</math> is a stochastic process on <math>S=\{S,I,R\}^N</math>. 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 <math>f</math> so that for <math>x</math> in <math>S</math>,<math>f(x,i,j)</math> is when the state of network is <math>x</math>, 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 <math>y=f(x,i,j)</math>, we try to find <math>P(x,y)</math>. If node i is in state I and node j is in state S, then <math>P(x,y)=\alpha A_{ji}/k_i</math>; if node i is in state I and node j is in state I, then <math>P(x,y)=\beta A_{ji}/k_i</math>; if node i is in state I and node j is in state R, then <math>P(x,y)=\beta A_{ji}/k_i</math>. For all other <math>y</math>, <math>P(x,y)=0</math>.<br />

The procedure<ref>Brockmann, D. 2011 Complex Networks and Systems, Lecture Notes, Northwestern University</ref> on a network is as follows: