Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

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

Blog Posts Week 1

Bush vs. Kerry Poster: Alexander Sheu About: Pure Nash and mixed Nash equilibria Link:

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.

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.

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.

“Very Useful Website” Power of “memes”

Conficker Worm

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

Lecture Four: The diameter of a random graph.

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

Argument Each person has two new friends d2d +  diameter = log n 2 d+1 - 1

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

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

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

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

A random graph Each person knows 3 random others 1.People meet at random, write names into rolodexes. 2.Relationships are reciprocal. 3.Each rolodex has 3 distinct names.

A random graph Collapse big nodes to get graph.

A random graph Collapse big nodes to get graph.

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

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

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

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

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)

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

Time for

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.

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

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

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.

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

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.

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!

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!

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

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

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

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

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!

Next time decentralized search