Transcript Document

Innovation in networks
and alliance management
Small world networks
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Course aim
knowledge about concepts in
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 …
Network properties (and a bit on trust)
Four basic network arguments
Kinds of network data (collection)
Business networks
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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 and
in the trust topic that will follow later
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?
(what a weird question, actually)
Answer: often “small-world”,
and often also scale-free
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3 important network properties
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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)
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(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?)
people in Nebraska and Kansas.
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)
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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
address
 Follow up studies
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)
Email to <address>, otherwise same rules. Addresses were
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
mentioned about 10% more often)
 Chain size 5, 6 or 7.
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Some Milgram follow-ups…
6.6!
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The Kevin Bacon experiment –
Tjaden (+/- 1996)
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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(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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Bacon / Hauer / Connery
(numbers now changed a bit)
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The best centers… (2011)
(Kevin Bacon at place 444)
(Rutger Hauer at place 43, J.Krabbé 867)
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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?
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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
lead to these small-world properties
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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:
hyperlinks
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
Airline, road, river
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
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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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?
(potential answer: through random
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
Start rewiring links randomly
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,
research on the spread of
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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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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Potential answer: Perhaps through “preferential
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)
Question: when do you get cascades of adoption?
Answer: two phase transitions or tipping points:
 in sparse networks no cascades
 as networks get more dense, a sudden jump in
the likelihood of cascades
 as networks get more dense, the likelihood of
cascades decreases and suddenly goes to zero
* 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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Malcolm Gladwell
(journalist/writer: wrote
“Blink” and “The tipping point”
Duncan Watts
(scientist, Yahoo,
Microsoft Research)
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Hmm ...
<try Netlogo Small Worlds>
We will see that you do not always end up with
small worlds!
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The bigger picture
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The general approach … understand
how STRUCTURE can arise from
underlying 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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More material on the website ...
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To Do:
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Read and comprehend the papers on small world
networks, scale-free networks (see website).
Think about applications of these results
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