lecture6 notes - 6.207/14.15 Networks Lecture 6 Growing...

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6.207/14.15: Networks Lecture 6: Growing Random Networks and Power Laws Daron Acemoglu and Asu Ozdaglar MIT September 28, 2009 1
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Networks: Lecture 6 Outline Growing random networks Power-law degree distributions: Rich-Get-Richer effects Models: Uniform attachment model Preferential attachment model Reading: EK, Chapter 18. Jackson, Chapter 5, Sections 5.1-5.2. 2
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Networks: Lecture 6 Growing Random Networks So far, we have focused on static random graph models in which edges among “fixed” n nodes are formed via random rules in a static manner. Erd¨os-Renyi model has small distances, but low clustering and a rapidly falling degree distribution. Configuration model generates arbitrary degree distributions. Small-world model provides a tractable model that has small distances and high clustering. Most networks form dynamically whereby new nodes are born over time and form attachments to existing nodes when they are born. Example: Consider the creation of web pages. When a new web page is designed, it includes links to existing web pages. Over time, an existing page will be linked to by new web pages. The same phenomenon true in many other networks: Networks of friendships, citations, professional relationships. Evolution over time introduces a natural heterogeneity to nodes based on their age in a growing network. 3
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Networks: Lecture 6 Emergence of Degree Distributions These considerations motivate dynamic or generative models of networks. These models also provide foundations for the emergence of natural linkage structures or degree distributions. What degree distributions are observed in real-world networks? In social networks, degree distributions can be viewed as a measure of “popularity” of the nodes. Popularity is a phenomenon characterized by extreme imbalances : while almost everyone goes through life known only to people in their immediate social circles, a few people achieve wide visibility. Let us focus on the concrete example of World Wide Web (WWW), i.e., network of web pages. In studies over many different Web snapshots taken at different points in time, it has been observed that the degree distribution obeys a power law distribution, i.e., the fraction of web pages with k in-links (or out-links) is approximately proportional to k 2.1 (or k 2.7 ). 4
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Networks: Lecture 6 Power Law Distribution—1 Many social and biological phenomena also governed by power laws. Population sizes of cities observed to follow a power law distribution. Number of copies of a gene in a genome follows a power law distribution. Some physicists think these correspond to some “universal laws”, as illustrated by the following quote from Barabasi that appeared in the April 2002 issue of the Scientist : “What do proteins in our bodies, the Internet, a cool collection of atoms, and sexual networks have in common? One man thinks he has the answer and it is going to transform the way we view the world.” A nonnegative random variable X is said to have a power law distribution if P ( X x ) cx α , for constants c > 0 and α > 0. (Here f ( x )
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