Understanding Social Networks: Key Theories and Concepts in Network Theory

1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

2. Last week… Overview of class Networks – why they have low diameter Game theory – dominant strategy/Nash equil. Markets – bidding in 1st price auctions And in the blogsphere…

3. Blog Posts Week 1

4. Bush vs. Kerry • Poster: Alexander Sheu • About: Pure Nash and mixed Nash equilibria • Link: http://www.slate.com/id/2108640/

5. Difference Between Social Networking and Social Networks • Difference between this course and Facebook. • One is about connecting with your friends • One is a group of theories about how things connect.

6. Difference Between Social Networking and Social Networks • Difference between this course and Facebook. • One is about connecting with your friends • One is a group of theories about how things connect. • One you will get points for posting about.

7. Difference Between Social Networking and Social Networks • Difference between this course and Facebook. • One is about connecting with your friends • One is a group of theories about how things connect. • One you will get points for posting about. • The other is a good way of connecting with your friends.

8. “Very Useful Website” • Power of “memes”

9. Conficker Worm

10. Next three weeks Social networks diameter, decentralized search, preferential attachment, PageRank, information cascades

11. Lecture Four: The diameter of a random graph.

12. Six degrees of separation Last time: The diameter of a social network is typically small.

13. Argument Each person has two new friends 1 2 22 + 2d 2d+1 - 1  diameter = log n

14. Argument Each person has two new friends  diameter = log n … but friends are likely to overlap.

15. Understanding social networks These networks are complex, … but they have a simple story for creation The interplay of fate and chance.

16. A random explanation Random links make short paths e.g., if you take a graph and “perturb” it, long paths are likely to reconnect

17. A random graph Each person knows 3 random others KEY: = a person = her rolodex

18. A random graph Each person knows 3 random others People meet at random, write names into rolodexes. Relationships are reciprocal. Each rolodex has 3 distinct names.

19. A random graph Collapse big nodes to get graph.

20. A random graph Collapse big nodes to get graph.

21. Diameter of a random graph Consider growing tree while size of current tree is small enough Interior of current tree Leaves of current tree

22. Breadth-first search tree How many new leaves? Interior of current tree

23. Breadth-first search tree How many new leaves? Interior of current tree

24. Breadth-first search tree How many new leaves? Interior of current tree

25. Doubling argument When size of current tree is small enough # of leaves approximately doubles (doubling fails if new friend of a leaf node falls inside current tree or collides with new friend of another leaf node)

26. Doubling argument What is small enough? Suppose current tree T has size x. Pr[1st new friend isin T] < x/n Pr[neither new friend isin T] > (1 – x/n)2 Pr[all new friends outside of T] > [(1 – x/n)2]x/2

27. Time for Math Corner

28. Doubling argument What is small enough? Suppose current tree T has size x. Pr[all new friends outside of T] > (1 – x/n)x This is constant for x = √n.

29. Bounding number of steps Doubling number of leaves each time, it takes ? steps to reach √n nodes.

30. Bounding number of steps Doubling number of leaves each time, it takes log √n stepsto reach √n nodes. But we still haven’t reached most nodes!

31. Good ideas are worth repeating To compute distance from some node 1 to another node 2, Idea: grow 2 trees! Each tree gets √n nodes in time log √n; argue that the trees intersect.

32. Growing two trees Random graph Tree 1 node 1 node 2 Tree 2

33. The birthday paradox Experiment: Your index card contains a random number between 1 and 100. Find someone in the same row as you that has your number and you will both earn a point. Find someone in an adjacent row that has your number and you will get ½ a point.

34. The birthday paradox Suppose you have d people, and each has a random number between 1 and n. Prob[no two people have same #] = = 1 x (1 – 1/n) x (1 – 2/n) x … x (1 – (d-1)/n) > (1 – d/n)d Constant for d = √n!

35. Good ideas are worth repeating Tree of size x has about x/2 leaves. Each leaf chooses two random neighbors. What is prob. two trees don’t intersect? Birthday paradox!

36. Intersecting trees Two trees of size √n (so √n/2 leaves each, or leaves in total √n).

37. Intersecting trees By birthday paradox, with constant probability 2 leaves pick same neighbor.

38. Intersecting trees With constant probability, these 2 leaves are from different trees, and so the trees intersect.

39. Bounding distance bt. two nodes Two trees of size √n intersectwith constant probability, … and so we can combine the trees.

40. Diameter of a random graph Hence the expected distance between any two nodes … is about 2 log √n = log (√n)2= log n. Diameter of this class should be about 4!

41. Next time decentralized search