Unformatted text preview: in common (i.e., the λ websites that are linked to my website are not
linked among themselves). Then in two steps, I can reach λ2 other
nodes.
Repeating the same reasoning (and maintaining the same unrealistic
assumption), in d steps I can reach λd other nodes.
Now imagine that this network has n = λd nodes.
This implies that the “degrees of separation” (average distance) is
d= ln n
.
ln λ
13 Networks: Lecture 1 Introduction Interpreting Small Worlds (continued) But our unrealistic assumption rules out the reasonable triadic
relations and clustering phenomena, which are common both in social
networks, web links, and other networks.
4 4 2 2
5a 1 5 1
5b 3 3
6 6 Interestingly, however, in Poisson (ErdosRenyi) random graphs, we
will see that average distance can be approximated for large n by
d = ln n / ln λ (where λ is the expected degree of a node).
This is because triadic relations shown in the ﬁgure are relatively rare
in such graphs.
14 Networks: Lecture 1 Introduction Interpreting Small Worlds (continued)
This last result in fact can be interpreted as stating that Poisson
(ErdosRenyi) random graphs, though mathematically convenient, will
not be good approximations to social networks.
This can be seen from the above numbers as well.
The Karinthy conjecture, under the Poisson assumption, would
require that each person should have had approximately 68
“independent” friends. (exp[ln(1, 500, 000, 000)/5] � 68.5).
The Milgram conjecture, of six degrees of separation in the 1960s,
would require that each person should have had approximately 41
friends.
Instead, most people would be connected to others in remote parts
through “special links” (or “connectors”), such as their political
representatives, village head, or cousin in a diﬀerent city etc.
Models of small world networks try to capture this pattern (albeit not
always perfectly).
15 Networks: Lecture 1 Introduction Social and Economic Networks Most “networked” interactions involve a human element, hence much
of network analysis must have some focus on social and economic
networks (even when the main interest may be on understanding
communication networks).
E.g., social network structures, such as Facebook, superimposed over
the Internet. In this course, social and economic networks will be our main focus.
An important feature of social and economic networks is that they are
not only characterized by a pattern of linkages, but also by the
interactions that take place over the network structure.
Will you lend money to your friend? Will you follow their advice? Will
you imitate their behavior? Will you trade with other ﬁrms that you are
potentially “connected to”? Most of these decisions are strategic, hence the use of game theory.
16 Networks: Lecture 1 Introduction A Central Question
What are the commonalities in diﬀerent (social, economic and other)
networks?
Diﬀusion of new technologies and spread of epidemics have certain
common features when one looks at their dynamics.
Does this mean th...
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 Fall '09
 Acemoglu
 strong triadic closure

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