Social Media Mining: An Introduction

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CUUS2079-07 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:17

184 Information Diffusion in Social Media

of being either majority blue or majority red is 50%. During the experiment,

each student comes to the urn, picks one marble, and checks its color in

private. The student predicts majority blue or red, writes the prediction

on the blackboard (which was blank initially), and puts the marble back

in the urn. Other students cannot see the color of the marble taken out,

but can see the predictions made by the students regarding the majority

color and written on the blackboard. Let the BOARD variable denote the

sequence of predictions written on the blackboard. So, before the first

student, it is

BOARD:{}

We start with the first student. If the marble selected is red, the prediction

will be majority red; if blue, it will be majority blue. Assuming it was blue,

on the board we have

BOARD:{B}

The second student can pick a blue or a red marble. If blue, he also

predicts majority blue because he knows that the previous student must

have picked blue. If red, he knows that because he has picked red and the

first student has picked blue, he can randomly assume majority red or blue.

So, after the second student we either have

BOARD: {B,B}or BOARD: {B,R}

Assume we end up with BOARD:{B, B}. In this case, if the third student

takes out a red ball, the conditional probability is higher for majority blue,

although she observed a red marble. Hence, a herd behavior takes place, and

on the board, we will have BOARD:{B,B,B}. From this student and onward,

independent of what is being observed, everyone will predict majority blue.

Let us demonstrate why this happens based on conditional probabilities and

our problem setting. In our problem, we know that the first student predicts

majority blue ifP(majority blue|student’s obervation)> 1 /2 and majority

red otherwise. We also know from the experiments setup that

P(majority blue)=P(majority red)= 1 / 2 , (7.1)

P(blue|majority blue)=P(red|majority red)= 2 / 3. (7.2)