03-small_world_annot

# 9282010 jure leskovec stanford cs224w social and

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9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 21

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Measure of clustering (local structure i e (local structure, i.e., triangles in a graph) Clustering coefficient C of node i is: i k degree of node i i … degree of node Clustering coefficient: C = 1/n C C i =0 C i =1/3 C i =1 C = 1/n i 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 22
1/n C i ient, C = 1 ng coeffic Clusterin Prob. of rewiring, p 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 23

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Collaborations between actors (IMDB): 225,226 nodes, avg. degree k=61 Electrical power grid: 4 941 nodes k=2 67 4,941 nodes, k=2.67 Network of neurons 282 nodes, k=14 L ... Average shortest path length C ... Average clustering coefficient 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 24
When we add one random connection out of each node we get short paths. Why? Suppose we build random edges by giving every node half edge and randomly pair them C id h h t t Consider a graph where we contract 2x2 subgraphs into supernodes Now we have 4 edges sticking out of each supernode From Thm. we have short paths between super nodes, we can turn this into a path in a real graph by dd h adding at most 2 steps per hop: O(2log n) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 25

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Ok so paths are short Ok, so paths are short And people are able to find them! (without the global knowledge of the network) 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 26
s only knows locations of its friends s only knows locations of its friends and location of the target t s does not know links of anyone but itself Geographic navigation: s forwards the message to the node closest to t 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 27

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Model: Grid where each node has one random edge This is a small world. Fact: A decentralized algorithm in Watts Strogatz model needs n 2/3 steps to reach t in expectation (even though paths of length log(n) exist). Proof: Let’s do this in 1 dim. n nodes on i l d di t d d a ring plus one random directed edge per node. Lower bound on search time is now n 1/2 d/d Lower bound for d dim.: n d/d+1 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 28
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• Fall '09
• Jure Leskovec, Clustering coefficient, Small-world network, Information Network Analysis

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