Social Media Mining: An Introduction

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8.2 Influence 233

Volume

u

Iu

tu tv

t–tu

t–tv

t–tw

tw t

Σ

Iv

Iw

vw

Time

Figure 8.6. The Size of the Influenced Population as a Summation of Individuals Influ-

enced by Activated Individuals (from [Yang and Leskovec, 2010]).

coefficientscuandαufor anyuby methods such asmaximum likelihood

estimation (see [Myung, 2003] for more details).

This is called theparametricestimation, and the method assumes that

all users influence others in the same parametric form. A more flexible

approach is to assume a nonparametric function and estimate the influence

function’s form. This approach was first introduced as the linear influence

model (LIM) [Yang and Leskovec, 2010]. LINEAR

INFLUENCE

MODEL (LIM)

In LIM, we extend our formulation by assuming that nodes get deacti-

vated over time and then no longer influence others. LetA(u,t)=1 denote

that nodeuis active at timet, andA(u,t)=0 denote that nodeuis either

deactived or still not influenced. Following a network notation and assum-

ing that|V|is the total size of the population andTis the last time step, we

can reformulate Equation8.33for|P(t)|as

|P(t)|=

∑|V|

u= 1

∑T

t= 1

A(u,t)I(u,t), (8.34)

or equivalently in matrix form,

P=AI. (8.35)

It is common to assume that individuals can only activate other indi-

viduals and cannot stop others from becoming activated. Hence, negative

values for influence do not make sense; therefore, we would like measured