Social Media Mining: An Introduction

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

7.3 Diffusion of Innovations 197

of the cumulative number of adopters at different timesA(t),

i(t)=α+α 0 A 0 +···+αtA(t)=α+

∑t

i=t 0

αiA(i), (7.17)

whereαi’s are the weights for each time step. Often a simplified version

of this linear combination is used. In particular, the following three models

for computingi(t) are considered in the literature:

i(t)=α, External-Influence Model (7.18)

i(t)=βA(t), Internal-Influence Model (7.19)

i(t)=α+βA(t), Mixed-Influence Model (7.20)

whereαis theexternal-influence factorandβ is theimitation factor. EXTERNAL

INFLUENCE

FACTOR

Equation7.18describesi(t) in terms ofαonly and is independent of

the current number of adoptersA(t); therefore, in this model, the adoption

only depends on the external influence. In the second model,i(t) depends

on the number of adopters at any time and is therefore dependent on the

internal factors of the diffusion process.βdefines how much the current

adopter population is going to affect the adoption and is therefore denoted

as theimitationfactor. The mixed-influence model is a model between the IMITATION

two that uses a linear combination of both previous models. FACTOR

External-Influence Model

In the external-influence model, the adoption coefficient only depends on

an external factor. One such example of external influence in social media

is when important news goes viral. Often, people who post or read the news

do not know each other; therefore, the importance of the news determines

whether it goes viral. The external-influence model can be formulated as

dA(t)

dt

=α[P−A(t)]. (7.21)

By solving Equation7.21,

A(t)=P(1−e−αt), (7.22)

whenA(t=t 0 =0)=0. TheA(t) function is shown in Figure7.7. The

number of adopters increases exponentially and then saturates nearP.