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Lecture 4: Connect (4/4)

COMS 4995-1: Introduction to Social NetworksTuesday, September 18th

How the friendship we form connect us?

Why we are within a few clicks on Facebook?

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This course is now officially “sexy [kinda]”congratulations!

1st assign. due Thursday 4:10pm−Part A+C on papers!−Part B(+raw results of C) on

dropbox−Sign the cover sheet−1 late days: 5% (you have 3 free during

semester)

Some announcements

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Milgram’s “small world” experiment

It’s a “combinatorial small world” It’s a “complex small world” It’s an “algorithmic small world”

Outline

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Main idea: social networks follows a structure with a random perturbation

Formal construction:1. Connect all nodes at distance in a regular

lattice2. Rewire each edge uniformly with probability

p(variant: connect each node to q neighbors, chosen uniformly)

Small-world model

Collective dynamics of ‘small-world’ networks. D. Watts, S. Strogatz, Nature (1998)

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Main idea: social networks follows a structure with a random perturbation

Small-world model

Collective dynamics of ‘small-world’ networks. D. Watts, S. Strogatz, Nature (1998)

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Milgram’s “small world” experiment

It’s a “combinatorial small world” It’s a “complex small world” It’s an “algorithmic small world”

Outline

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Where are we so far?Analogy with a cosmological principle− Are you ready to accept a

cosmological theory that does not predict life?

In other words, let’s perform a simple sanity check

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1. Consider a randomly augmented lattice (N nodes)

A thought experiment

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1. Consider a randomly augmented lattice (N nodes)

2. Perform “small world” Milgram experiment

Can you tell what will happen?(a)The folder arrives in 6 hops(b)The folder arrives in O(ln(N)) hops(c)The folder never arrives(d)I need more information

A thought experiment

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(a)The folder arrives in 6 hopsNOT TRUE

It actually does look like a naive answer More precisely:

−By previous result we know that shortest paths is of the order of ln(N), which contradicts this statement.

A thought experiment

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(b) The folder arrives in O(ln(N))ACCORDING TO OUR PRINCIPLE, OUGHT TO

BE TRUE BECAUSE IT WAS OBSERVED BY MILGRAM

A sufficient condition for this to be true is:−Milgram’s procedure extract shortest path

Answering this critical question boils down to an algorithmic problem

A thought experiment

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(c) The folder never arrivesSEEMS UNLIKELY

unless the procedure is badly designed (cycle)

or we model people droppingor if the grid contains hole

A thought experiment

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(d) I need more information In particular, how to model Milgram’s

procedure “If you do not know a target, forward the

folder to your friend or acquaintance that is most likely to know her.”

A thought experiment

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A mathematical model of what Milgram measured−Participants know where the target is

located−They use grid information + shortcuts

“incidentally”N.B.: Grid “dimensions” can describe geography or other sociological property (occupation, language)

Example:

What is Greedy Routing?

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Does it extract the shortest path?−Not necessarily, this is why we need to

analyze it! Case study: dimension k=1, target t,

starting from u0−We introduce interval:−The greedy routing constructs a path

we denote the end-point of the ith shortcuts as

How does greedy routing perform?

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CLAIM: If none of are in and we start from u0 outside

−Then greedy routing needs at least min(n,l) steps

Analysis of Greedy routing

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Fixing , this event has proba ≤1/2−So with proba ≥1/2, are not in

On this event, assuming s not in −Greedy routing needs more than n steps−Or it has to reach t from boundary of ,

using l steps

How does greedy routing perform?

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In a line Milgram’s uses steps−square root is not

satisfying for small world−Not much better when k>1 ! −even worse, the proof applies to any

distributed alg. Our sanity check test has grandly failed!

−“Small world” results explain that short paths exist … finding them remains a daunting algorithmic task

A thought experiment

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Milgram’s “small world” experiment

It’s a “combinatorial small world” It’s a “complex small world” It’s an “algorithmic small world”

−Beyond uniform random augmentation

Outline

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In a uniformly augmented lattice shortcuts do exist−About shorcuts leads to when

But they are dispersed among nodes Moreover, previous steps do not lead to

progress−So need about N/√N = √N trials

Is there another augmentation?

Autopsy of “Small-world” failure

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The 10 papers that will make you a social expert

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1. S.Milgram, “The small world problem,” Psychology today, 1967.2. M. Granovetter, “The strength of weak ties: A network theory revisited,” Sociological

theory, vol. 1, pp. 201–233, 1983.3. M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a Feather: Homophily in Social

Networks,” Annual review of sociology, vol. 27, pp. 415–444, Jan. 2001.4. M. O. Lorenz, “Methods of measuring the concentration of wealth,” Publications of the

American Statistical Association, vol. 9, no. 70, pp. 209–219, 1905. + H. Simon, “On a Class of Skew Distribution Functions,” Biometrika, vol. 42, no. 3, pp. 425–440, 1955.

5. R. I. M. Dunbar, “Coevolution of Neocortical Size, Group-Size and Language in Humans,” Behav Brain Sci, vol. 16, no. 4, pp. 681–694, 1993.

6. D. Cartwright and F. Harary, “Structural balance: a generalization of Heider's theory.,” Psychological Review, vol. 63, no. 5, pp. 277–293, 1956.

7. M. Granovetter, “Threshold Models of Collective Behavior,” The American Journal of Sociology, vol. 83, no. 6, pp. 1420–1443, May 1978.

8. B. Ryan and N. C. Gross, “The diffusion of hybrid seed corn in two Iowa communities,” Rural sociology, vol. 8, no. 1, pp. 15–24, 1943. + S. Asch, “Opinions and social pressure,” Scientific American, 1955.

9. R. S. Burt, Structural Holes: The Social Structure of Competition. Harvard University Press, 1992.

10. F. Galton, “Vox Populi,” Nature, vol. 75, no. 1949, pp. 450–451, Mar. 1907.

10 sociological must-reads

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People “love those who are like themselves”, “Similarity begets friendship”−Nichomachean Ethics, Aristotle & Phaedrus,

Plato

Do you think homophilyproduces or hindersmall world?

Homophily

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What if the augmentation exhibits a bias−Most of the people you know are near, −Occasionally, you know someone outside

Does this break the lower bound proof?

Does finding a neighborhood of t becomes easier?

Augmenting lattice with a bias

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Formal construction:1. Connect nodes at distance p in a regular

lattice2. Connect each node to q other nodes, chosen

with a biased probability3. p=q=1 to simplify

How to model augmentation bias

The small-world phenomenon: An algorithmic perspective. J. Kleinberg, Proc. of ACM STOC (2000)

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Formal construction:1. Connect nodes at distance p in a regular

lattice2. Connect each node to q other nodes, chosen

with a biased probability

r may be called the clustering coefficient If a node is twice further, probability is

times less

How to model augmentation bias

The small-world phenomenon: An algorithmic perspective. J. Kleinberg, Proc. of ACM STOC (2000)

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Impact of clustering coefficient

Small values of rApproaches uniform

augmentation

Large values of rApproaches original lattice

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(a) Yes, finding a neighborhood of t becomes easier

A PRIORI NOT TRUE−It is easier only if you are already near the

target−In general, it can take a larger number of

steps

Can we break the lower bound?

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(b) Yes, for another reason−All positions are not equal, hence progress is

possible−As shortcut are used recursively, probability

increases−So we need to study the sequence of

progress

Can we break the lower bound?

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Assume r=k (dimension of the grid)−A neighborhood of t of radius d/2−Contains (d/2)k nodes−Each may be chosen with

probability roughly 1/(3d/2)k

−Growth of ball compensatesprobability decreases!

Harmonic distribution.

The critical case

The small-world phenomenon: An algorithmic perspective. J. Kleinberg, Proc. of ACM STOC (2000)

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Augmented lattice

rr=k0

Combinatorial Small world

(Short paths exist)dist. alg. need N(k-r)/(k+1) steps

The small-world phenomenon: An algorithmic perspective. J. Kleinberg, Proc. of ACM STOC (2000)

Not a small world(Short paths do not

exist)alg. need N(r-k)/(r-(k-1))

steps

Navigable small worlddist. alg need O(log2(N))

steps

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Is the analysis of greedy routing tight?−Yes, greedy routing performs in Ω(log2 n)

Can we find path as short as log(n) (shortest path)?−Yes, with extra information on neighboring

nodes−Or another augmentation

Can we build augmentation for an infinite lattice?−See homework exercice (check tomorrow

night)

Theoretical follow ups

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Can we augment other graphs?−G=(V,E) (i.e. a lattice) with distance known−Random augmentation adds one shortcut

per nodeIs routing on G + shortcuts used incidentally efficient?

Indeed all these graphs are polylog augmentable:−Bounded ball growth, Doubling dimensions−Bounded “width” (Trees, bounded treewidth

graphs) What about all graphs? Lower Bound

O(n1/√ln(n))

Theoretical follow ups (cont’d)

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Practical follow upCan we observe harmonic distribution?• Yes, using closeness

rank instead of distance

Can we prove it emerge?• Recent results • Through rewiring,

mobility

Geographic routing in social networks.D. Liben-Nowell et. al. PNAS (2005)

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Milgram’s experiment prove that social networks are navigable −individuals can take advantage of short

paths−with basic information

This is at odds with uniform random graphs The key ingredients to explain navigability

−A space easy to route (e.g. grid, trees, etc.).−A subtle harmonic augmentation (e.g. ball

radius).

Summary