Wrapup

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Wrapup
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Topics

• Photoshoppe hints/clarifications
• Social networks, graphs, and matrices

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Photoshoppe

• Image Types
• Crop
• Zoom
• Posterize

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Image Types

• PNG images come in several forms
• n x k x 3 image(p, q, ) is a color-triple
• n x k, with a 256 x 3 color table
• image(p,q) 12 means look at row 12 in color
  table to get the real color
• n x k with no colortable for greyscale (?)
• ONLY need to handle the first form
• If you want to handle others, feel free

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Crop

• When crop is selected, user must click-and-drag
  to complete the operation.
• crop-button callback needs to set up the
  click-n-drag
• waitforbuttonpress
• rbbox
• Both of these return ONLY after the users done
  something.
• To get mouse-location, use get(,
  CurrentPoint)
• Be sure to trim the values returned to within the
  image range!

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Zoom

• Two options
• Implement with a zoom in/out toggle
• use the builtin zoom function
• Zoom in/out toggle
• Double pixel size
• Double edge length or
• Double pixel area (mutliply edge by sqrt(2))
• Both are OK choices

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Posterize

• Youll use kmeans to do posterization.
• Sometimes kmeans comes up with fewer clusters of
  data than you requested
• Use kmeans(data, k, emptyaction, drop)
• Tells kmeans to ignore the fact that one or more
  clusters are empty.

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Social networks and graphs

• We all have friendsand other people were not
  friends with
• Can draw diagrams like this

Anna
Cho
Mike
Robin
Sanjaya
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Encoding social networks

• Replace names with numbersBuild matrix
• 1 1 0 0 0
• 1 1 1 0 0
• 0 1 1 1 0
• 0 0 1 1 0
• 0 0 0 0 1

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Graphs and matrices

• The matrix is called an adjacency matrix
• Shows all places you can reach in 0 or 1 step
  from a node
• Information for node i stored in row i (or column
  i)
• Matrix is symmetric
• 1 1 0 0 0
• 1 1 1 0 0
• 0 1 1 1 0
• 0 0 1 1 0
• 0 0 0 0 1

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• Matrix has 1s on diagonal to say you can get
  from person k to person k by taking no steps at
  all

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M 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0
0 0 1

• Whats M2?
• 1 2 1 0 0
• 2 2 2 1 0
• 1 2 3 2 0
• 0 1 2 2 0
• 0 0 0 0 1
• Each entry says number of ways to get from i to j
  in 2 or fewer steps.
• 3,2 entry 3-2-2 and 3-3-2.

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Whats Mk?

• The (i,j) entry counts the number of ways to get
  from i to j in k or fewer steps
• let v 0 1 0 0 0. What does Mv represent?
• All friends of person 2

Mv 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 x 0 0
0 1 1 0 0 0 0 0 0 1 0
1 1 1 0 0
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Whats M(Mv)?

• M(Mv) represents paths from person 2 to any
  friend of a friend
• And M(M(M(v))) represents friends-of-friends-of-fr
  iends
• If we computed u M(M(M(M(M(Mv))))), wed find
  all people reachable by 6 degrees of separation
• If conjecture is right, this would be everyone.
• u should be a vector with no zeroes.

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Six degrees

• To make that test, wed need a very big matrix M
• For US population, 250,000,000 x 250,000,000.
• Too big to store practically
• Even Rhode Island is 1,000,000 x 1,000,000.

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Matrix size

• 1012 entries too many to store
• Almost all entries are zero
• Use sparse matrix representation!
• Assuming 500 friends per person, RI matrix
  becomes workable!

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Weighted matrices

• We could write down numbers between 0 and 1 for
  each edge, representing strength of connection
• You, roomate connection of strength 1
• You, CS4 TA strength 0.1
• You, parking enforcement officer for Brown
  strength 0.001

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Weighted matrices

• Can be used to study rumor propagation
• If you hear a rumor from many untrusted sources,
  its like hearing it from a trusted source
• Mkv tells how much a rumor started by person 2
  has influenced each person after k steps of rumor
  propagation
• Probably best to include some discount factor
  along the way

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Discounted rumors

• When you hear a rumor and spread it to 500
  friends, maybe each believes it only with a 1
  chance.
• Compute .01 Mv.
• Eventual result
• v 0.01 Mv .012M2v

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Tipping points

• When does a rumor become believed?
• When does a riot start?
• When does a shoe-brand become popular?
• When does everyone seem to come down with the
  same cold?

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• Can study these questions by looking at matrix
  powers
• When Mkv is mostly nonzero (or mostly above some
  threshold), we have a tipping point

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Conjectures

• The tipping point happens when someone with lots
  of friends hears the rumor
• In any sufficiently connected community, even
  without superfriendlies, it happens after a
  certain time
• In communities with enough weak links, even if
  there are no strong links, it happens
• Strong links prevent tipping

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Testing conjectures

• Repeatedly
• Generate random graph M of some density
• Check whether M6v is all-positive
• Compute successes/total trials.
• When this is gt 50, youve found critical density

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Test2

• Do the same test, but after removing
  superfriendlies
• Any difference in required density?

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Test3

• Generate friendships with a better model
• Each person gets a few friends to start with
• Thereafter, new friendships happen in proportion
  to the number of friends you have
• Friendly people get more friends!
• Test connection model with/without
  superfriendlies.

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Generating numbers with a particular distribution

• Id like to generate a number from 1 to 5
• But Id like to have the likelihoods of
  generating each one to be different in
  proportion to 3 4 2 6 2
• Solution write down1 1 1 2 2 2 2 3 3 4 4 4 4 4
  4 5 5
• Pick a random from 117 look at that element of
  the list!

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Code

• function y randDist(dist)
• pick a number from 1..size(dist)
• probability of picking k is proportional to
  dist(k)
• u sum(dist)
• k randint(1,1, 1 u)
• for p 1size(dist)
• if k lt dist(p)
• y p
• return
• else
• k k dist(p)
• end
• end
• y size(dist)

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