Social Media Mining: An Introduction

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CUUS2079-08 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:22

224 Influence and Homophily

Equation8.17, these expectations are as follows:

E(XL)=E(XR)=

∑

i(XL)i

2 m

=

∑

idixi

2 m

(8.20)

E(XLXR)=

1

2 m

∑

i

(XL)i(XR)i=

∑

ijAijxixj

2 m

. (8.21)

By plugging Equations8.20and8.21into Equation8.19, the covariance

betweenXLandXRis

σ(XL,XR)=E[XLXR]−E[XL]E[XR]

=

∑

ijAijxixj

2 m

−

∑

ijdidjxixj

(2m)^2

=

1

2 m

∑

ij

(Aij−

didj

2 m

)xixj. (8.22)

Similar to modularity (Section8.1.1), we can normalize covariance.

PEARSON Pearson correlationρ(XL,XR) is the normalized version of covariance:

CORRELATION

ρ(XL,XR)=

σ(XL,XR)

σ(XL)σ(XR)

. (8.23)

From Equation8.18,σ(XL)=σ(XR); thus,

ρ(XL,XR)=

σ(XL,XR)

σ(XL)^2

,

=

1

2 m

∑

ij(Aij−

didj

2 m )xixj

E[(XL)^2 ]−(E[XL])^2

=

1

2 m

∑

ij(Aij−

didj

2 m )xixj

1

2 m

∑

ijAijx

2

i−

1

2 m

∑

ij

didj

2 mxixj

. (8.24)

Note the similarity between Equations8.9and8.24. Although modularity

is used for nominal attributes and correlation for ordinal attributes, the major

difference between the two equations is that theδfunction in modularity is

replaced byxixjin the correlation equation.

Example 8.2.Consider Figure8.4with values demonstrating the attributes

associated with each node. Since this graph is undirected, we have the

following edges:

E={(a,c),(c,a),(c,b),(b,c)}. (8.25)