Class 4: It’s a Small World After All

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

2. Milgram’s Six Degrees The first chain letters The destination: Boston, Massachusetts Starting Points: Omaha, Nebraska & Wichita, Kansas SIX DEGREES Travers and Milgram, Sociometry32,425 (1969)

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 Exponential growth Polynomial growth

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

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

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 Where should we place the social network? Clustered Random

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

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

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 • The Watts Strogatz Model: • Start with a lattice network. • For every edge rewire with a probability b. Watts and Strogatz, Nature 393,409 (1998)

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

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 • The Watts Strogatz Model: • It takes a lot of randomness to ruin the clustering, but a very small amount to overcome locality Watts and Strogatz, Nature 393,409 (1998)

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: • Provides insight on the interplay between clustering and the small world topology • Captures the structural essence of many realistic networks • Accounts for the high clustering observed in realistic networks • Does not lead to the correct degree distribution • 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 • A message is given to node S, in order to deliver to the target T • S has only local information, namely its own acquaintances • What is the typical number of steps, t (delivery time) • Searchable • Non-searchable • For Erdős–Rényi • For Watts-Strogatz

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

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

30. The Effect of Structured Shortcuts • We need a bit more structure • When ais small – We are back to Watts and Strogatz • When ais large – We are back to Manhattan At a = 2 searchability becomes optimized Kleinberg, Nature 406,845 (2000)

31. Why Two of All Numbers We divide the network into logarithmically growing shells: The probability of a rewired edge into the j-th shell At a = 2 long-range contacts are evenly distributed over distance scales You are likely to have a contact half way through Kleinberg, Nature 406,845 (2000)

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

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)