Unformatted text preview: eferral by j , and P ((i , j ) ∈ R ) denote the probability of this event. 29 Networks: Lecture 1 A Little Bit of Analysis The Strong Triadic Closure (continued)
Then
P((i , j ) ∈ R 
=
<
= (i , j ) ∈ E � )
�
�
P (i , k ) ∈ E and (j , k ) ∈ E �  (i , j ) ∈ E �
/
�
�
P (i , k ) ∈ E and (j , k ) ∈ E �  (i , j ) ∈ E \ E �
/
�
�
P (i , j ) ∈ R  (i , j ) ∈ E \ E � , potentially explaining Granovetter’s ﬁndings. In fact, with the nonprobabilistic version of the property, P ((i , j ) ∈ R  (i , j ) ∈ E � ) = 0! 30 Networks: Lecture 1 A Little Bit of Analysis Power in a Network The Medicis emerged as the most inﬂuential family in 15th century
Florence. Cosimo de Medici ultimately formed the most politically
powerful and economically prosperous family in Florence, dominating
Mediterranean trade.
The Medicis, to start with, were less powerful than many other
important families, both in terms of political dominance of Florentine
institutions and economic wealth.
How did they achieve their prominence?
It could just be luck (in social science, we have to be very careful to
distinguish luck from a systematic pattern, and correlation from
causation).
An interesting explanation, eschewing luck, is oﬀered by Padgett and
Ansell (1993) “Robust Action and the Rise of the Medici”— they
were the most powerful family because of their situation in the
social network of Florence.
31 Networks: Lecture 1 A Little Bit of Analysis Power in a Network (continued) Image by MIT OpenCourseWare. Adapted from Figure 1.1 on p. 4 in Jackson, Matthew O.
Social and Economic Networks. Princeton, NJ: Princeton University Press, 2008.
ISBN13: 9780691134406. ISBN10: 0691134405. Figure: Political and friendship blockmodel structure (Padgett and Ansell 1993)
32 Networks: Lecture 1 A Little Bit of Analysis Power in a Network (continued)
One measure of power that takes into account the “location” of the
family with the network is the “betweenness” measure deﬁned as
follows.
Let P (i , j ) be the number of shortest paths connecting family i to
family j .
Let Pk (i , j ) be the number of shortest paths connecting these two
families that include family k .
The measure of betweenness (for a network with n nodes) is then
deﬁned as Bk ≡ Pk (i , j ) /P (i , j ) ,
(n − 1)(n − 2)/2
(i ,j )∈E :i �=j ,k ∈{i ,j }
/ ∑ with the convention that Pk (i , j ) /P (i , j ) = 0 if P (i , j ) = 0.
Intuitively, this measure gives, for each pair of families, the fraction of
the shortest paths that go through family k .
33 Networks: Lecture 1 A Little Bit of Analysis Power in a Network (continued)
It turns out that this measure over betweenness Bk is very high for
the Medicis, 0.522.
No other family has Bk greater than 0.255.
So the Medicis may have played a central role in holding the network
of inﬂuential families in Florence together and thus gained “power”
via this channel.
Is this a good measure of “social power”? Of political power? Is this
a pla...
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 Fall '09
 Acemoglu
 strong triadic closure

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