32 Small world networks

```Social networks
Small world networks
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Course aim
network theory, and being able to
apply that knowledge
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The setup in some more detail
Network theory and background
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Introduction: what are they, why important …
Small world networks
Four basic network arguments
Kinds of network data (collection)
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Two approaches to network theory
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Bottom up (let’s try to understand network
characteristics and arguments)
as in … “Four network arguments” by Matzat
(lecture 3)
Top down (let’s have a look at many networks,
and try to deduce what is happening from what
we see)
as in “small world networks” (now)
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What kind of structures do
networks have, empirically?
and often also scale-free
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3 important network properties

Average Path Length (APL) (<l>)
Shortest path between two nodes i and j of a network,
averaged across all (pairs of) nodes
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Clustering coefficient (“cliquishness”)
Number of closed triplets / Total number of triplets
(or: probability that two of my ties are connected)

(Shape of the) degree distribution
A distribution is “scale free” when P(k), the proportion of
nodes with degree k follows this formula, for some value of
gamma:
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Example 1 - Small world networks
NOTE
- Edge of network theory
- Not fully understood yet …
- … but interesting findings
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Enter: Stanley Milgram (1933-1984)
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Remember him?
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The small world phenomenon –
Milgram´s (1967) original study
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Milgram sent packages to several (60? 160?)
Aim was “get this package to <address of person
in Boston>”
Rule: only send this package to someone whom
you know on a first name basis. Aim: try to make
the chain as short as possible.
Result: average length of a chain is only six
“six degrees of separation”
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Milgram’s original study (2)

An urban myth?
Milgram used only part of
the data, actually mainly
the ones supporting his
claim
 Many packages did not
end up at the Boston
typically small scale

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The small world phenomenon (cont.)
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“Small world project” has been testing this assertion (not
anymore, see http://smallworld.columbia.edu)
American college professor, Indian technology consultant,
Estonian archival inspector, …
Conclusion:
 Low completion rate (384 out of 24,163 = 1.5%)
 Succesful chains more often through professional ties
 Succesful chains more often through weak ties (weak ties
 Chain size 5, 6 or 7.
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Some Milgram follow-ups…
6.6!
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The Kevin Bacon experiment –
Actors = actors
Ties = “has played in a movie with”
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The Kevin Bacon game
Can be played at:
http://oracleofbacon.org
Kevin Bacon
number
(data might have changed by now)
Jack Nicholson:
Robert de Niro:
Rutger Hauer (NL):
Famke Janssen (NL):
Bruce Willis:
Kl.M. Brandauer (AU):
Arn. Schwarzenegger:
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1
2
2
2
2
2
(A few good men)
(Sleepers)
[Nick Stahl]
[Nick Stahl]
[Patrick Michael Strange]
[Robert Redford]
[Kevin Pollak]
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A search for high Kevin Bacon numbers…
3
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2
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The center of the movie universe
Nr 370
(sept 2013)
Nr 136
Nr 39
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The best centers… (2013 + 2011)
(Kevin Bacon at place 444 in 2011)
(Rutger Hauer at place 39, J.Krabbé 935)
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“Elvis has left the building …”
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We find small average path lengths in all kinds
of places…
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Caenorhabditis Elegans
959 cells
Genome sequenced 1998
Nervous system mapped
 low average path length
+ cliquishness = small world network
Power grid network of Western States
5,000 power plants with high-voltage lines
 low average path length +
cliquishness = small world network
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How weird is that?
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Could there be a simple explanation?

Consider a random network: each pair of
nodes is connected with a given probability
p.
This is called an Erdos-Renyi network.
NB Erdos was a “Kevin
Bacon” long before Kevin
Bacon himself!|
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APL is small in random networks
[Slide copied from Jari_Chennai2010.pdf]
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[Slide copied from Jari_Chennai2010.pdf]
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But let’s move on to the second network
characteristic …
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This is how small-world networks
are defined:
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A short Average Path Length and
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A high clustering coefficient
… and a randomly “grown” network does NOT
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Source: Leskovec & Faloutsos
Networks of the Real-world (1)
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Information networks:
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Social networks: people +
interactions
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World Wide Web:
Citation networks
Blog networks
Florence families
Organizational networks
Communication networks
Collaboration networks
Sexual networks
Collaboration networks
Karate club network
Technological networks:
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Power grid
networks
Telephone networks
Internet
Autonomous systems
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Friendship network
Collaboration network
Source: Leskovec & Faloutsos
Networks of the Real-world (2)
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Biological networks
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Language networks
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metabolic networks
food web
neural networks
gene regulatory
networks
Yeast protein
interactions
Semantic network
Semantic networks
Software networks
…
Language network
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Software network
Small world networks … so what?
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
You see it a lot around us: for instance in road
maps, food chains, electric power grids,
metabolite processing networks, neural networks,
telephone call graphs and social influence
networks  may be useful to study them
They seem to be useful for a lot
of things, and there are reasons
to believe they might be useful
for innovation purposes (and hence
we might want to create them)
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Examples of interesting
properties of
small world networks
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Synchronizing fireflies …
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<go to NetLogo>
Synchronization speed depends on small-world
properties of the network
 Network characteristics important for “integrating
local nodes”
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Combining game theory and networks –
Axelrod (1980), Watts & Strogatz (1998?)
1.
2.
3.
4.
5.
Consider a given network.
All connected actors play the repeated Prisoner’s Dilemma
for some rounds
After a given number of rounds, the strategies “reproduce”
in the sense that the proportion of the more succesful
strategies increases in the network, whereas the less
succesful strategies decrease or die
Repeat 2 and 3 until a stable state is reached.
Conclusion: to sustain cooperation, you need a short
average distance, and cliquishness (“small worlds”)
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And another peculiarity ...
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Seems to be useful in “decentralized computing”
 Imagine a ring of 1,000 lightbulbs
 Each is on or off
 Each bulb looks at three neighbors left and right...
 ... and decides somehow whether or not to switch to on
or off.
Question: how can we design a rule so that the network can
tackle a given GLOBAL (binary) question, for instance the
question whether most of the lightbulbs were initially on or
off.
- As yet unsolved. Best rule gives 82 % correct.
- But: on small-world networks, a simple majority rule gets
88% correct.
How can local knowledge be used to solve global problems?
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If small-world networks are so
interesting and we see them
everywhere, how do they arise?
rewiring of a given structure)
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Strogatz and Watts
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6 billion nodes on a circle
Each connected to nearest 1,000 neighbors
Calculate average path length and clustering as
the network starts to change
Network changes from structured to random
APL: starts at 3 million, decreases to 4 (!)
Clustering: starts at 0.75, decreases to zero
(actually to 1 in 6 million)
Strogatz and Watts asked: what happens along
the way with APL and Clustering?
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Strogatz and Watts (2)
“We move in tight circles yet
we are all bound together by
remarkably short chains”
(Strogatz, 2003)
 Implications for, for instance,
diseases...
The general hint:
-If networks start from relatively
structured …
-… and tend to progress sort of
randomly …
-- … then you might get small
world networks a large part of the
time
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And now the third characteristic
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And if we consider all three…
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… then we find this:
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Wang & Chen (2003) Complex networks: Small-world, Scale-free and beyond
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Same thing … we see “scale-freeness” all over
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… and it can’t be based on an ER-network
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Scale-free networks are:
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Robust to random problems/mistakes ...
... but vulnerable to selectively targeted attacks
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Another BIG question:
How do scale free networks arise?
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attachment”
< show NetLogo simulation here>
(Another) critique to this approach:
it ignores ties created by those in the network
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Some related issues
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“The tipping point” (Watts*)
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Consider a network in which each node determines
whether or not to adopt, based on what his direct
connections do.
Nodes have different thresholds to adopt
(randomly distributed)
Answer: two phase transitions or tipping points:
 in sparse networks no cascades, as networks get
more dense you get cascades suddenly
 as networks get more heterogenous, a sudden
jump in the likelihood of cascades
 as networks get even more heterogenous, the
* Watts, D.J. (2002) A simple model of global cascades on random networks. Proceedings of the National Academy of Sciences USA 99, 5766-5771
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“Find the influentials”
(or not?)
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(journalist/writer: wrote
http://www.fastcompany.com/641124
/tipping-point-toast
H6AVVU
Duncan Watts
(scientist, Yahoo,
Microsoft Research)
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"All they'll ever say," Watts insists, is that a) there are people who are more influential than others, and
b) they are disproportionately important in getting a trend going.
That may be oversimplifying it a bit, but last year, Watts decided to put the whole idea to the test by
building another Sims-like computer simulation. He programmed a group of 10,000 people, all governed
by a few simple interpersonal rules. Each was able to communicate with anyone nearby. With every
contact, each had a small probability of "infecting" another. And each person also paid attention to what
was happening around him: If lots of other people were adopting a trend, he would be more likely to
join, and vice versa. The "people" in the virtual society had varying amounts of sociability--some were
more connected than others. Watts designated the top 10% most-connected as Influentials; they could
affect four times as many people as the average Joe. In essence, it was a virtual society run--in a very
crude fashion--according to the rules laid out by thinkers like Gladwell and Keller.
Watts set the test in motion by randomly picking one person as a trendsetter, then sat back to see if the
trend would spread. He did so thousands of times in a row.
The results were deeply counterintuitive. The experiment did produce several hundred societywide
infections. But in the large majority of cases, the cascade began with an average Joe (although in cases
where an Influential touched off the trend, it spread much further). To stack the deck in favor of
Influentials, Watts changed the simulation, making them 10 times more connected. Now they could
infect 40 times more people than the average citizen (and again, when they kicked off a cascade, it was
substantially larger). But the rank-and-file citizen was still far more likely to start a contagion.
Why didn't the Influentials wield more power? With 40 times the reach of a normal person, why couldn't
they kick-start a trend every time? Watts believes this is because a trend's success depends not on the
person who starts it, but on how susceptible the society is overall to the trend--not how persuasive the
early adopter is, but whether everyone else is easily persuaded. And in fact, when Watts tweaked his
model to increase everyone's odds of being infected, the number of trends skyrocketed.
"If society is ready to embrace a trend, almost anyone can start one--and if it isn't, then almost no one
can," Watts concludes. To succeed with a new product, it's less a matter of finding the perfect hipster to
infect and more a matter of gauging the public's mood. Sure, there'll always be a first mover in a trend.
But since she generally stumbles into that role by chance, she is, in Watts's terminology, an "accidental
Influential."
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The bigger picture:
Understanding macro patterns
from micro behavior
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The general approach … understand
how STRUCTURE can arise from
underlying MICRO-DYNAMICS
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Scientists are trying to connect the structural
properties …
Scale-free, small-world, locally clustered, bow-tie,
hubs and authorities, communities, bipartite cores,
network motifs, highly optimized tolerance, …
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… to processes
(Erdos-Renyi) Random graphs, Exponential random
graphs, Small-world model, Preferential
attachment, Edge copying model, Community guided
attachment, Forest fire models, Kronecker graphs, …
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RECAP
• Six degrees of separation: average path length is often small
• Many real-world networks have properties that are similar:
small-world and scale-free.
• We do not really understand yet how these properties emerge.
• We saw two clues: the Watts-Strogatz model for small-worlds
and the preferential attachment model for scale-freeness.
• Small-world and scale-free networks have some nice properties
(which might explain why they exist)
• Considerable controversy over what these kinds of results imply,
for instance for marketing purposes
• Hot scientific topic: connecting micro-behavior to
macro-properties
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To Do:
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Read and comprehend the papers on small world
networks, scale-free networks (see website, there
is extra material too).
Think about implications and applications of these
results
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