CS 322: (Social and Information) Network Analysis Jure Leskovec Stanford University

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1 CS 322: (Social and Information) Network Analysis Jure Leskovec Stanford University

2 Course website: Slides will be available online Reading material will be posted online: Chapters from the new (released today!) book from Jon Kleinberg and David Easley from Cornell Whole book is available at: book /h /kl / t k b k

3 G(V, ( E) ) has expansion α: if S V: #edges leaving S α min( S, S ) S All edges with exactly one node in S S Or equivalently: α is the minimum ratio: # edges leaving min( S over all sets S, S ) S

4 Expansion gives us a measure of robustness if we want to disconnect l nodes, we need to cut α l edges Low expansion: α = High expansion: α = Star graph: α =

5 d regular graph (every node has deg. d): Expansions it at best d. (when S is 1 node) d) Is there a graph on n nodes (n ), max deg. d (const), so that expansion α remains const? Examples: Grid: d=4: α =2n/(n 2 /4) 0 (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 some const α (α>0, indep. of n) such that w.h.p. the expansion of the graph is α

6 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). And the diameter of G is O(log(n)/ α)

7 Just tbf before the edge (u,v) is placed dhow many hops is between u and v?

[Watts Strogatz Nature 98] How to have local edges

edges to create shortcuts to join remote parts of the

8 [Watts Strogatz Nature 98] How to have local edges (lots of triangles) and small diameter? Small world model [Watts Strogatz 1998]: 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 [Watts Strogatz Nature 98] High clustering High clustering Low clustering High diameter Low diameter Low diameter Rewiring allows to interpolate between regular lattice and a random graph

10 Measure of clustering (local structure, i.e., ie triangles in a graph) Clustering coefficient C i of node i is: k i degree of node i C i =0 C i =1/3 C i =1 Clustering coefficient: C =1/n C i

11 1/n C i ng coefficient, C = 1 Clusterin Prob. of rewiring, p

12 Collaborations between film actors (IMDB) 225,226 nodes, avg. degree k=61 Electrical power grid: 4,941 nodes, k=2.67 Network of neurons 282 nodes, k=14 L... Average shortest path length C... Average clustering coefficient

13 People are able to find them! (without the global knowledge of the network)

14 People use different strategies: geography vs. occupation Criticism: Funneling: 31 of 64 chains go passed through 1 of 3 people p ass their final step Not all links/nodes are equal Is this reproducible in you vary conditions: High status person High participation rates: study from Columbia: 18 targets, 65kstartest, 75% dropout per step, only384 chains got through (about half to a stockbroker)

16 f relative frequency i... incomplete c... completed

17 s only knows locations of its friends and location of the target t. s does not know random links of anyone but itself. Geographic navigation: s forwards the message to the node closest to to t.

18 Model: Grid where each node has one random edge Why is this 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: Do this in 1 d. n nodes on a ring plus one random directed edge per node. Lower bound on search time is now n 1/2.

19 Principle of deferred decision: assume random long range links aeo are only yceatedo created once you get to them. E i = event that long range link out of the new node points to some node in I. Then: P(E i )= 2x/n (haven t seen the random node i yet) Pick k, let E=event that any of first k nodes you see has long range link to I: P(E) = P(U k i E i ) = 2kx/n

20 P(E)=P(UP(U k i E i ) P(E i ) = 2kx/n Need k, x s.t. 2kx/n < 1 Choose: k=x= ½n ½ So, P(E) 2kx/n =2(½n ½ )/n= ½ Suppose s is outside I and E does not happen, then the algorithm must take min(k,x) steps to get to I.

21 Claim: Getting from s to t takes k ½n ½ steps. If we don t take a long random link, we must traverse ½n ½ steps to get in I. Expected dtime to get to I ½ + ½n 1/2 P(E doesn t occur) = ¼n 1/2

22 [Kleinberg 2001] Nodes still on a grid Each node has one long range link Prob. of long link: P(u v) ~ d(u,v) -α d(u,v) grid distance between u and v α parameter 0

23 Claim: For 2 d grid: α=22 we can get from s to t in O(log(n) 2 ) steps. StI Set I = d/2 Now: P(long range link points to I) = 1/log(n)

24 Small α: too many long links Big α: too many short links

25 Why distribution P(u v) ~ d(u,v) -dim works? Approx uniform over scales of resolution # points at distance d grows as d dim, prob. d -dim of each edge const. prob. of a link, independent of d

26 h(u,v) = tree distance (height of the least common ancestor) -α h(u,v) P(u v) ~ b How many nodes are at dist. h? (b-1)b h-1 P(u v) is approx uniform at all scales of resolution Path must exist a node creates d=k log 2 n edges Fact: For any direct subtree T one of v s d links points to T. By induction on T we can reach any node t in O(log(n)) steps.

27 Extension [Watts Dodds Newman 2002]: Multiple hierarchies geography, professor, religion, Q: how to analyze the model? Simulations: works for a range of alphas Biggest range of searchable hbl alphas for 2 or 3 hierarchies

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University http://cs224w.stanford.edu 10/4/2011 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu