03-small_world_annot

If we want to disconnect l nodes we need to cut α l

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if we want to disconnect l nodes, we need to cut α l edges l Low expansion: α = = High expansion: α = Social networks: C iti Communities 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 14

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d regular graph (every node has deg. d ): b d ( h S d ) Expansions it at best . (when S is 1 node) Is there a graph on n nodes ( n  ), max deg. d (const) so that expansion α remains const? d (const), so that expansion Examples: Grid: d=4: α =2n/(n 2 /4) 0 d 4: 2n/(n /4) (n/2 by n/2 square in the center) Complete binary tree: α 0 for |S|=(n/2)-1 Fact: for a random 3 regular graph on n nodes there is nodes, there is some const α ( α >0 , indep. of n ) such that w.h.p. the expansion of the graph is α 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 15
Fact: In a graph on n nodes with expansion α for all pairs of nodes s and t there is a path connecting them of O((log n) / α ) edges. Proof: Let S j be a set of all nodes found within j steps of BFS from t . Then: |S j+1 | |S j | + α |S j |/d = |S j | (1 + α /d ) In how many steps of BFS we reach >n/2 nodes? Need j so that: (1 + α /d ) j > n/2 , set j=d log( n)/ α So, in O(log n) steps |S j | grows to Θ (n) . A d th di t f G i O(l ( )/ ) And the diameter of G is O(log(n)/ α 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 16

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Consequence of expansion: Short paths: O(log n) between each pair Working definition of a “short path”: O(log n) This is the “best” we can do if the graph has constant degree Pure exponential growth and n nodes But social networks have local structure: Triadic closure: Friend of a friend is my friend Maybe grid is a better model? 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, Triadic closure reduces growth rate 17
Just before the edge (u v) is placed how many (u,v) hops is between u and v ? 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 18

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Just before the edge (u v) is placed how many (u,v) hops is between u and v ? Fraction of triad Network % Δ F 66% G np (D) closing edges D 28% A 23% (F) w L 50% (L) (A) u v 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 19
[Watts Strogatz Nature ‘98] How to have local edges (lots of triangles) and small diameter? Small world model [Watts Strogatz ‘98] : Strogatz 98] Start with a low dimensional regular lattice Rewire: Add/remove edges to create shortcuts to join remote parts of the lattice For each edge with prob. p move the other end to a random vertex 9/28/2010 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 20

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[Watts Strogatz Nature ‘98] High clustering High clustering Low clustering Rewiring allows to interpolate between High diameter Low diameter Low diameter regular lattice and a random graph
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