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

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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…

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Blog Posts Week 1

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

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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.

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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.

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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.

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“Very Useful Website” Power of “memes”

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Conficker Worm

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

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Lecture Four: The diameter of a random graph.

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

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

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

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

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

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

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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.

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A random graph Collapse big nodes to get graph.

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A random graph Collapse big nodes to get graph.

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

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

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

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

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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)

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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

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Time for

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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.

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

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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!

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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.

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

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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.

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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!

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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!

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

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

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

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

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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!

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Next time decentralized search

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