P1: qVa Trim: 6.125in×9.25in Top: 0.5in Gutter: 0.75in
CUUS2079-03 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 16:45
74 Network Measures=
∑
k(Ai,kAj,k−A ̄iA ̄j−A ̄iA ̄j+A ̄iA ̄j)=
∑
k(Ai,kAj,k−Ai,kA ̄j−A ̄iAj,k+A ̄iA ̄j)=
∑
k(Ai,k−A ̄i)(Aj,k−A ̄j), (3.63)whereA ̄i=^1 n∑
kAi,k. The term∑
k(Ai,k−A ̄i)(Aj,k−A ̄j) is basically the
covariance betweenAiandAj. The covariance can be normalized by the
multiplication of variances,σpearson(vi,vj)=σsignificance(vi,vj)
√∑
k(Ai,k−A ̄i)^2√∑
k(Aj,k−A ̄j)^2=∑
√ k(Ai,k−A ̄i)(Aj,k−A ̄j),
∑
k(Ai,k−A ̄i)^2√∑
k(Aj,k−A ̄j)^2, (3.64)
PEARSON which is called thePearson correlation coefficient. Its value, unlike the
CORRELATION other two measures, is in the range [− 1 ,1]. A positive correlation value
denotes that whenvibefriends an individualvk,vjis also likely to befriend
vk. A negative value denotes the opposite (i.e., whenvibefriendsvk,itis
unlikely forvjto befriendvk). A zero value denotes that there is no linear
relationship between the befriending behavior ofviandvj.3.4.2 Regular Equivalence
In regular equivalence, unlike structural equivalence, we do not look at
the neighborhoods shared between individuals, but at how neighborhoods
themselves are similar. For instance, athletes are similar not because they
know each other in person, but because they know similar individuals, such
as coaches, trainers, and other players. The same argument holds for any
other profession or industry in which individuals might not know each
other in person, but are in contact with very similar individuals. Regular
equivalence assesses similarity by comparing the similarity of neighbors
and not by their overlap.
One way of formalizing this is to consider nodesviandvjsimilar when
they have many similar neighborsvk andvl. This concept is shown in
Figure3.15(a). Formally,σregular(vi,vj)=α∑
k,lAi,kAj,lσRegular(vk,vl). (3.65)