analysis of social information networks
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Analysis of Social Information Networks. Thursday January 20 th , Lecture 1: Small World. What is the small world effect?. “You’r e in NY, b y the way, do you happen to know x? - In fact, yes, what a small world!” W hat seems remotely distant is indeed socially close. - PowerPoint PPT PresentationTRANSCRIPT
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Analysis of Social Information
NetworksThursday January 20th, Lecture 1: Small World
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What is the small world effect?“You’re in NY, by the way, do you happen to know x?- - In fact, yes, what a
small world!”
What seems remotely distant is indeed socially close. The small world problem,
S. Milgram, Psychology today (1967)
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Small world hypothesis is necessary for−Fast and wide information propagation−Tipping point: small change has large effect−Network robustness and consistency
Studying the conditions of the “small world” effects tells us a lot about how we are connected?
Man, by nature, an animal of a society−“ζῷον πολιτικὸν” Aristotle, Politics 1
Why study small world?
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Remember how the King lost his fortune to the chess player?
What would you take?−An increasing series {1¢,2¢,4¢, etc.} over a
month?−Against $1,000−Against $100,000−Against $1,000,000−Against $100,000,000
Small world: a simplistic argument
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How many people would you recognize by name?−‘67 M. Gurevitch (MIT): about 500
Roughly, how many are socially related to you?
Small world: a simplistic argument
how close to you? Compares to %. US pop.500 direct acquaintance C.S. dept 0.00017%
250,000
share an acquaintance with
you
Harlem district 0.083%
125m share an acquaintance with a
friend of yours
Northeast + Midwest
42%
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The previous model is way too optimistic!−Reason #1: it assumes acquaintance set are
disjoint whereas they are related as part of a social graph searching through the graph creates inbreeding.
−Reason #2: social acquaintances are biasedGeography, occupation & social status, race Favors clusters, inbreeding. Increases social
distance−Other reasons: unbalanced degrees
Small world: The skeptics
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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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A direct experimental approach−Main issue: we do not know the graph (This is
1967!)−Can we still find a method to exhibit small chains
of acquaintance between two arbitrary people? At least we can try it out:
Pick a single “target” and an arbitrary sample of peopleSend a “folder” with target description, asking participants to Add her name, and forward to her friend or acquaintance
who she thought would be likely to know the target, Send a “tracer” card directly to a collection point
Milgram’s experiment
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Would this work?First folders arrived using 4 days and 2 intermediaries
2 successful experiments− 1st: 44 folders out of
160− 2nd : 64 folders on 296− Average distance = 6.2.
The small world problem, S. Milgram, Psychology today (1967)
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More observations− Chains conditioned
locally: gender, family/ friendsglobally: hometown/job
− Some received multiple folders (up to 16)
− Drop out-rate ~25%May artificially reduce distance and can be corrected
The small world problem, S. Milgram, Psychology today (1967)
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Are these results reproducible?−Using new forms of communication
P. Dodds, R. Muhamad, and D. Watts. An experimental study of search in global social networks. Science (2003).
Another method: assuming graph is known −Exhaustive search:− J. Leskovec and E. Horvitz. Planetary-scale views on a large
instant-messaging network. WWW (2008)− Erdös number, Kevin Bacon number− More evidence of short paths, including in different
domains
Experiment sequels
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Confirmation using email− Larger scale
18 targets, 24,000 starting points
− Drop-out rates : 65%success rate is lower: 1.5%median distance: 4 raw data and 7 corrected data
− Role of social statussuccess rate changes, but not mean distance
An experimental study of search in global social networks. P. Dodds, R. Muhamad, and D. Watts. Science (2003).
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Erdös numberConnections = collaboration− 2 persons connected if
they co-authored a paper.
How far are you from Erdös?− 94% within distance 6− Median: 5− Small number prestigious
Field medalists: 2-5; Abel: 2-4
The Erdös Number Project, www.oakland.edu/enp/
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Kevin Bacon numberConnections = collaboration− 2 actors connected if
they appear in one movie
− ~ 88% actors connected
− 98.3% actors: 4 and less Maximum number is 8.
The oracle of Bacon, oracleofbacon.org
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Instant MessagingDifferent scales• 180m nodes, 13b
edges Connections = collaboration− 180m nodes, 1.3b
edges
Planetary-scale views on a large instant-messaging network. J. Leskovec and E. Horvitz. WWW (2008)
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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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Yes, according to our (simplistic) combinatorial argument shown before.−Can we extend it to more realistic model?
Approach #1: 1. construct a graph from model of social
connections2. Characterizes connectivity, distance,
diameter −Main difficulty: connections between people
are difficult to predict
Do short paths exist?
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Uniform random graphs (Erdös-Rényi), 3 flavors1. Each edge exists independently (with fixed prob.
p)2. Choose a subset of m edges3. Choose for each node d random neighbors
Main merit: properties are established rigourously for large scale asymptotic (size n going to ∞)−Properties of connected components−Presence/absence of isolated nodes−Small diameter
Random Graphs
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Flavor 1 and 2 : series of phase transitions
All monotone properties have sharp threshold
Properties of Unif. Rand. Graph
p11/n ln(n)(1+ω(1))/n0
Conn. Comp.<ln(n)
1 large connected component; others <ln(n)Isolated nodes
graph is connected
ω(ln(n))/n
Diameter O(ln(n))
E. Friedgut, G. Kalai, Proc. AMS (1996) following Erdös-Rényi’s results
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Flavor 3 (each node with d random neighbors) How to construct?
1. Assign d “semi-edge” to each node2. Pair “semi-edge” into edge randomly3. Remove self-loops and multi-edges (rejection)(NB: This construction can be used for any degree dist.)
Degree constant: should we expect connectivity?
Properties of Unif. Rand. Graph
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Transitions is even faster:−d=1: collection of disjoint pairs (not
connected)−d=2: collection of disjoint cycles (a.s.
disconnected)−d=3: connected, small diameter, in fact
much more ... Thm: for d≥3 there exists γ>0 such that
for any sizeG is a γ_expander with high
probability:
Properties of Unif. Rand. Graph
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Proposition 1: If G is a γ_expander with degree at most d, its diameter is at most −Proof: combinatorial geometric expansion
Expanders have other properties (more later)−Robustness w.r.t. edges removal−Behavior of random walks
Properties of expanders
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Are these graphs expanders?
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Assuming random uniform connections−Phase transitions (i.e. tipping points) are the rule.−Connectivity, small diameter, even expander
property can be guaranteed without strict coordination… this may outclass deterministic structure!
−Elegant and precise mathematical formulation
Is small world only a reflection of the combinatorial power of randomness?
Summary
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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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Do not underestimate the effect of the inbreeding!−How many of your friends are friends (strong
ties)?−Which are friends with different people
(weak ties)? Several evidences that these play different
roles−Strong ties: resistance to innovation and
danger(e.g., reduce risk of teen suicide)
−Weak ties: propagate information, connect groups(e.g., who to ask when looking for a job)
Social networks are not flat
Suicide and Friendships Among American Adolescents. P. Bearman, J. Moody, Am. J. Public Health (2004)
The Strength of Weak Ties, M. Granovetter Am. J Soc. (1983)
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A metric of strong ties−Clustering coefficient−Also writes
Structured networks: rings, grids, torus−Clustering remains constant in large graph−As indeed, in most empirical data set
Random graphs−Clustering coefficient vanishes in large graph
… hence most results are biased towards weak ties
Can we quantify the difference?
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Main idea: social networks follows a structure with a random perturbation
Formal construction:1. Connect nearby nodes in a regular lattice2. Rewire each edge uniformly with probability
p(variant: add a new uniform edge with probability p)
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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Between order and randomnessAs a function of p:− C(p): clustering
coefficient− L(p): median length of
the shortest path− Both normalized by
p=0.For small p (~ 0.01)− large clustering − small diameterA few rewire suffice!
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Thm: ∀p>0, augmented lattice has small diameter,
there is A such that P[D≤A ln(N)] → 1−Elementary proof if assuming regular augmentation−Otherwise proof similar to Uniform Random Graph
Statistical analysis randomly perturbed structures−Degree distribution, assortative mixing,
hypergeometry
A new statistical mechanics
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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