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Unformatted text preview: quite fragile, there is great sensitivity to unpredictable
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
9 Networks: Lecture 6 History of Power Laws—3
In 1965, Price applied these ideas to networks, with a particular focus on
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
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¨s-Renyi model, in which nodes are born over time and
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 continuous-time mean-ﬁeld 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
The change at time t > i of the expected degree of node i is given by...
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- Fall '09