Rumor spread in social network: Difference between revisions

Revision as of 19:42, 6 April 2024

Revision as of 19:44, 6 April 2024

Line 13:

==== Epidemic models====

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

*I: people who are ignorant of the rumor;

Line 51:

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

{{ordered list

| 1 = We initial rumor to a single node <math>i</math>;

| 2 = We pick one of its neighbors as given by the [[adjacency matrix]], so the probability we will pick node <math>j</math> is <br />

|2= We pick one of its neighbors as given by the [[adjacency matrix]], so the probability we will pick node <math>j</math> is <br />

<math>p_j={A_{ji} \over k_i}</math> <br />

where <math>A_{ji}</math> is from the adjacency matrix and <math>A_{ji}=1</math> if there is a tie from <math>i</math> to <math>j</math>, and <math>k_i= \textstyle \sum_{j=1}^N A_{ij}</math> is the [[Degree (graph theory)|degree]] for node <math>i</math>;

| 3 = Then have the choice: {{ordered list|list_style_type=lower-alpha

|3= Then have the choice: {{ordered list|list_style_type=lower-alpha

|1= If node <math>j</math> is an ignorant, it becomes a spreader at a rate <math>\alpha</math>;

|2= If node <math>j</math> is a spreader or stifler, then node <math>i</math> becomes a stifler at a rate <math>\beta</math>.

}}

| 4 = We pick another node who is a spreader at random, and repeat the process.

}}