1. Class 4: It’s a Small World After All Network Science: Small World February 2012 Dr. Baruch Barzel

2. Milgram’s Six Degrees The first chain letters The destination: Boston, Massachusetts Starting Points: Omaha, Nebraska & Wichita, Kansas Travers and Milgram, Sociometry 32,425 (1969) S IX D EGREES

3. The Exploding Volume of Networks The secret behind the small world effect – Looking at the network volume

4. The Exploding Volume of Networks The secret behind the small world effect – Looking at the network volume Polynomial growth

5. The Exploding Volume of Networks The secret behind the small world effect – Looking at the network volume Polynomial growth

6. The Exploding Volume of Networks The secret behind the small world effect – Looking at the network volume Polynomial growthExponential growth

7. The Exploding Volume of Networks The secret behind the small world effect – Looking at the network volume Polynomial growthExponential growth

8. Random Graphs are not (Exactly) Trees Some of your neighbors neighbors are also your own Exponential growth: Clustering inhibits the small-worldness

9. Random Graphs are not (Exactly) Trees Exponential growth: The exponential growth continues as long as N(d) < N

10. Random Graphs are not (Exactly) Trees Exponential growth: The exponential growth continues as long as N(d) < N

11. Random Graphs are not (Exactly) Trees Exponential growth: The exponential growth continues as long as

12. Clustering vs. Randomness A network can be a small world as long as clustering can be ignored ClusteredRandom Where should we place the social network?

13. What we Really Mean by Clustering RandomLocally Structured Clustering coefficient is zero

14. What we Really Mean by Clustering RandomLocally Structured Clustering implies locality Randomness enables shortcuts

15. Watts Going on with Social Networks Could a network which is so strongly locally structured be at the same time a small world?

16. Watts Going on with Social Networks The solution is to merge structure and randomness Watts and Strogatz, Nature 393,409 (1998) The Watts Strogatz Model : 1.Start with a lattice network. 2.For every edge rewire with a probability 

17. Watts Going on with Social Networks The solution is to merge structure and randomness Watts and Strogatz, Nature 393,409 (1998) For

18. Watts Going on with Social Networks The solution is to merge structure and randomness Watts and Strogatz, Nature 393,409 (1998)

19. Watts Going on with Social Networks The solution is to merge structure and randomness Watts and Strogatz, Nature 393,409 (1998) The Watts Strogatz Model : It takes a lot of randomness to ruin the clustering, but a very small amount to overcome locality

20. Watts Going on with Social Networks Could a network which is so strongly locally structured be at the same time a small world? Yes. You don’t need more than a few random links.

21. Watts Going on with Social Networks Could a network which is so strongly locally structured be at the same time a small world? Yes. You don’t need more than a few random links.

22. Going Beyond Facebook Albert and Barabási, Reviews of Modern Physics 74,47 (2002)

23. Going Beyond Facebook Map of scientific Collaborations

24. Watts Going on with Social Networks Could a network which is so strongly locally structured be at the same time a small world? Yes. You don’t need more than a few random links. The Watts Strogatz Model : o Provides insight on the interplay between clustering and the small world topology o Captures the structural essence of many realistic networks o Accounts for the high clustering observed in realistic networks o Does not lead to the correct degree distribution o Does not enable node targeting

25. Revisiting Milgram’s Experiment How do You Go About Finding the Trail

26. Revisiting Milgram’s Experiment How random are we allowed to really be?

27. Searchability What does it mean for a network to be searchable o A message is given to node S, in order to deliver to the target T o S has only local information, namely its own acquaintances o What is the typical number of steps, t (delivery time) SearchableNon-searchable For Erdős–Rényi For Watts-Strogatz

28. Searchability What does it mean for a network to be searchable o A message is given to node S, in order to deliver to the target T o S has only local information, namely its own acquaintances o What is the typical number of steps, t (delivery time) SearchableNon-searchable For Erdős–Rényi For Watts-Strogatz Kleinberg, Nature 406,845 (2000)

29. Every recipient simply sends it to its neighbor which is closest to the target Searchability We need a bit more structure o We start with a grid o We rewire one of X ’s edges with probability β o We choose to rewire the edge to Y with a probability Kleinberg, Nature 406,845 (2000)

30. The Effect of Structured Shortcuts We need a bit more structure Kleinberg, Nature 406,845 (2000) o When  is small – We are back to Watts and Strogatz o When  is large – We are back to Manhattan At  searchability becomes optimized

31. You are likely to have a contact half way through Why Two of All Numbers Kleinberg, Nature 406,845 (2000) We divide the network into logarithmically growing shells: At  long-range contacts are evenly distributed over distance scales The probability of a rewired edge into the j -th shell

32. How Many People From Over the Ocean Do You Know Saul Steinberg, “View of the World from 9 th Avenue” Just as many as you know from down the street

33. How Many People From Over the Ocean Do You Know Just as many as you know from down the street

34. The Internet Based Experiment 60000 start nodes 18 targets 384 completed chains Average path length between 5 to 7. Dodds, Muhamad and Watts, Science 301,827 (2003)

35. The Internet Based Experiment 60000 start nodes 18 targets 384 completed chains Average path length between 5 to 7. Dodds, Muhamad and Watts, Science 301,827 (2003)