Social Media Mining: An Introduction

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

202 Information Diffusion in Social Media

assume that (1) no contact network information is available and (2) the

process by which hosts get infected is unknown. These models can be

applied to situations in social media where the decision process has a

certain uncertainty to it or is ambiguous to the analyst.

7.4.1 Definitions

Since there is no network, we assume that we have a population where the

disease is being spread. LetNdefine the size of this crowd. Any member

of the crowd can be in either one of three states:

1. Susceptible: When an individual is in the susceptible state, he or she
   can potentially get infected by the disease. In reality, infections can
   come from outside the population where the disease is being spread
   (e.g., by genetic mutation, contact with an animal, etc.); however, for
   CLOSED- simplicity, we make aclosed-world assumption, where susceptible
   WORLD
   ASSUMPTION

individuals can only get infected by infected people in the population.

We denote the number of susceptibles at timetasS(t) and the fraction

of the population that is susceptible ass(t)=S(t)/N.

2. Infected: An infected individual has the chance of infecting suscep-
   tible parties. LetI(t) denote the number of infected individuals at
   timet, and leti(t) denote the fraction of individuals who are infected,
   i(t)=I(t)/N.


3. Recovered (or Removed): These are individuals who have either
   recovered from the disease and hence have complete or partial immu-
   nity against the infection or were killed by the infection. LetR(t)
   denote the size of this set at timetandr(t) the fraction recovered,
   r(t)=R(t)/N.
   Clearly,N=S(t)+I(t)+R(t) for allt. Since we are assuming that
   there is some level of randomness associated with the values ofS(t),I(t),
   andR(t), we try to deal with expected values and assumeS,I, andR
   represent these at timet.

7.4.2 SI Model

We start with the most basic model. In this model, the susceptible individuals

get infected, and once infected, they will never get cured. Denoteβas the

contact probability. In other words, the probability of a pair of people

meeting in any time step isβ. So, ifβ=1, everyone comes into contact

with everyone else, and ifβ=0, no one meets another individual. Assume