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In fact with the non probabilistic version of the

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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 non-probabilistic 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. ISBN-13: 9780691134406. ISBN-10: 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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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.

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