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Unformatted text preview: quite fragile, there is great sensitivity to unpredictable
initial ﬂuctuations.
Empirically studied by Salganik, Dodds and Watts (2006): They
created a music download site with 48 obscure songs. A visitor to the
site can listen to the songs and also is shown the “current” download
count for each song.
Each visitor at random is assigned to 8 “parallel copies” of the site,
which started out identically.
Market share of diﬀerent songs varied considerably across diﬀerent
copies.
9 Networks: Lecture 6 History of Power Laws—3
In 1965, Price applied these ideas to networks, with a particular focus on
citation networks.
Price studied the network of citations between scientiﬁc papers and found
that the in degrees (number of times a paper has been cited) have power
law distributions.
His idea was that an article would gain citations over time in a manner
proportional to the number of citations the paper already had.
This is consistent with the idea that researchers ﬁnd some article (e.g. via
searching for keywords on the Internet) and then search for additional papers
by tracing through the references of the ﬁrst article.
The more citations an article has, the higher the likelihood that it will be
found and cited again.
Price called this dynamic link formation process cumulative advantage.
Today it is known under the name preferential attachment after the
inﬂuential work of Barabasi and Albert in 1999.
10 Networks: Lecture 6 Uniform Attachment Model Before studying the preferential attachment model, we discuss a dynamic
variation on the Erd¨sRenyi model, in which nodes are born over time and
o
form edges to existing nodes at the time of their birth.
Index the nodes by the order of their birth, i.e., node i is born at date i ,
i = 0, 1, . . ..
A node forms undirected edges to existing nodes when it is born. Let di (t )
be the degree of node i at time t .
Start the network with m + 1 nodes (born at times 0, . . . , m) all connected
to one another.
Thus, the ﬁrst newborn node is the one born at time m + 1.
Assume that each newborn node uniformly randomly selects m nodes from
the existing set of nodes and links to them (ignore repetitions). 11 Networks: Lecture 6 Evolution of Expected Degrees
We will use a continuoustime meanﬁeld analysis to track the evolution of
the “expected degrees of nodes”.
We have the initial condition di (i ) = m for all i , every node has m links at
their birth.
The change at time t > i of the expected degree of node i is given by...
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 Fall '09
 Acemoglu

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