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Lecture 5 - Generalized Random Graphs and Small-Wo

Lecture 5 - Generalized Random Graphs and Small-Wo -...

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6.207/14.15: Networks Lecture 5: Generalized Random Graphs and Small-World Model Daron Acemoglu and Asu Ozdaglar MIT September 23, 2009 1
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Networks: Lecture 5 Outline Generalized random graph models Graphs with prescribed degrees – Configuration model Emergence of a giant component in the configuration model Small-world model Clustering Average path lengths Reading: Jackson, Sections 4.1.2, 4.1.4-4.1.6, 4.2.1, 4.2.6, 4.2.7. EK, Chapter 20. 2
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Networks: Lecture 5 Configuration Model—1 We have seen that the Erd¨ os-Renyi model has a Poisson degree distribution, which falls off very fast. Our next goal is to generate random networks with a “given degree distribution”. One of the most widely method used for this purpose is the configuration model developed by Bender and Canfield in 1978. The configuration model is specified in terms of a degree sequence , i.e., for a network of n nodes, we have a desired degree sequence ( d 1 , . . . , d n ) , which specifies the degree d i of node i , for i = 1, . . . , n . Given a degree distribution P ( d ) , we can generate the degree sequence for n nodes by sampling the degrees independently from the distribution P ( d ) , i.e., d i P ( d ) . A law of large numbers argument establishes that the frequency of degrees P ( n ) ( d ) converges to the degree distribution P ( d ) as n goes to infinity. 3
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Networks: Lecture 5 Configuration Model—2 Given ( d 1 , . . . , d n ) , we construct a sequence where node 1 is listed d 1 times, node 2 is listed d 2 times, and so on: 1, 1, 1, 1, . . . , 1 | {z } d 1 entries 2, 2, . . . , 2 | {z } d 2 entries · · · n , n , n . . . , n | {z } d n entries . We can think of this as giving each node i in the graph d i “stubs” sticking out of it, which are ends of edges-to-be. We randomly pick two elements of the sequence and form a link between the two nodes corresponding to those entries. We delete those entries from the sequence and repeat. Remarks: The sum of degrees needs to be even (or else an entry will be left out at the end). It is possible to have more than one link between two nodes (thus generating a “multigraph”). Self-loops are possible. 4
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Networks: Lecture 5 Distribution of the Degree of a Neighboring Node—1 We will use a branching process approximation to study the giant component in the configuration model.
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