1

Innovation in networks and alliance management

Small world networks

TU/e - 0ZM05/0EM15/0A150 2

Course aim

knowledge about concepts in network theory, and being able to apply that knowledge

TU/e - 0ZM05/0EM15/0A150 3

The setup in some more detail

Network theory and background

- Introduction: what are they, why important …- Network properties (and a bit on trust)- Four basic network arguments- Kinds of network data (collection)- Business networks

TU/e - 0ZM05/0EM15/0A150

Two approaches to network theory

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)

4

TU/e - 0ZM05/0EM15/0A150

What kind of structures do networks have, empirically?

(what a weird question, actually)

Answer: often “small-world”,

and often also scale-free

5

TU/e - 0ZM05/0EM15/0A150

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

Clustering coefficient (“cliquishness”)Number of closed triplets / Total number of triplets (or: probability that two of my ties are connected)

(Shape of the) degree distributionA distribution is “scale free” when P(k), the proportion of nodes with degree k follows this formula, for some value of gamma:

6

TU/e - 0ZM05/0EM15/0A150 7

Example 1 - Small world networks

NOTE- Edge of network theory- Not fully understood yet …- … but interesting findings

TU/e - 0ZM05/0EM15/0A150 8

Enter: Stanley Milgram (1933-1984)

Remember him?

TU/e - 0ZM05/0EM15/0A150 9

The small world phenomenon – Milgram´s (1967) original study

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”

TU/e - 0ZM05/0EM15/0A150 10

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 address

Follow up studies typically small scale

TU/e - 0ZM05/0EM15/0A150 11

The small world phenomenon (cont.)

“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.

TU/e - 0ZM05/0EM15/0A150

Some Milgram follow-ups…

12

6.6!

TU/e - 0ZM05/0EM15/0A150 13

TU/e - 0ZM05/0EM15/0A150 14

The Kevin Bacon experiment – Tjaden (+/- 1996)

Actors = actors Ties = “has played in a movie with”

TU/e - 0ZM05/0EM15/0A150 15

The Kevin Bacon game

Can be played at:http://oracleofbacon.org

Kevin Bacon number (data might have changed by now)

Jack Nicholson: 1 (A few good men)

Robert de Niro: 1 (Sleepers)

Rutger Hauer (NL): 2 [Nick Stahl]

Famke Janssen (NL): 2 [Nick Stahl]

Bruce Willis: 2 [Patrick Michael Strange]

Kl.M. Brandauer (AU): 2 [Robert Redford]

Arn. Schwarzenegger: 2 [Kevin Pollak]

TU/e - 0ZM05/0EM15/0A150 16

A search for high Kevin Bacon numbers…

3 2

TU/e - 0ZM05/0EM15/0A150 17

Bacon / Hauer / Connery (numbers now changed a bit)

TU/e - 0ZM05/0EM15/0A150

The best centers… (2011)

18(Kevin Bacon at place 444)(Rutger Hauer at place 43, J.Krabbé 867)

TU/e - 0ZM05/0EM15/0A150 19

“Elvis has left the building …”

TU/e - 0ZM05/0EM15/0A150 20

Small world networks=

short average distance between pairs … … but relatively high “cliquishness”

TU/e - 0ZM05/0EM15/0A150 21

We find small average path lengths in all kinds of places… Caenorhabditis Elegans

959 cellsGenome sequenced 1998Nervous system mapped low average path length

+ cliquishness = small world network

Power grid network of Western States5,000 power plants with high-voltage lines low average path length +

cliquishness = small world network

TU/e - 0ZM05/0EM15/0A150 22

How weird is that?

TU/e - 0ZM05/0EM15/0A150

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.

23

NB Erdos was a “Kevin Bacon” long before KevinBacon himself!|

TU/e - 0ZM05/0EM15/0A150

APL is small in random networks

24[Slide copied from Jari_Chennai2010.pdf]

TU/e - 0ZM05/0EM15/0A150 25[Slide copied from Jari_Chennai2010.pdf]

TU/e - 0ZM05/0EM15/0A150

But let’s move on to the second network characteristic …

26

TU/e - 0ZM05/0EM15/0A150 27

TU/e - 0ZM05/0EM15/0A150

This is how small-world networks are defined:

A short Average Path Length and

A high clustering coefficient

… and a randomly “grown” network does NOT lead to these small-world properties

28

TU/e - 0ZM05/0EM15/0A150

Networks of the Real-world (1) Information networks:

World Wide Web: hyperlinks

Citation networks Blog networks

Social networks: people + interactions

Organizational networks Communication networks Collaboration networks Sexual networks Collaboration networks

Technological networks: Power grid Airline, road, river

networks Telephone networks Internet Autonomous systems

Florence families Karate club network

Collaboration networkFriendship network

Source: Leskovec & Faloutsos

TU/e - 0ZM05/0EM15/0A150

Networks of the Real-world (2)

Biological networks metabolic networks food web neural networks gene regulatory

networks Language networks

Semantic networks Software networks …

Yeast proteininteractions

Semantic network

Language network Software network

Source: Leskovec & Faloutsos

TU/e - 0ZM05/0EM15/0A150

And if we consider all three…

31

TU/e - 0ZM05/0EM15/0A150 32

… then we find this:

Wang & Chen (2003) Complex networks: Small-world, Scale-free and beyond

TU/e - 0ZM05/0EM15/0A150 33

The scale-free

TU/e - 0ZM05/0EM15/0A150 34

Small world networks … so what? 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 hencewe might want to create them)

TU/e - 0ZM05/0EM15/0A150

Examples of interestingproperties of

small world networks

35

TU/e - 0ZM05/0EM15/0A150

Synchronizing fireflies …

<go to NetLogo>

Synchronization speed depends on small-world properties of the network

Network characteristics important for “integrating local nodes”

36

TU/e - 0ZM05/0EM15/0A150 37

Combining game theory and networks – Axelrod (1980), Watts & Strogatz (1998?)

1. Consider a given network.

2. All connected actors play the repeated Prisoner’s Dilemma for some rounds

3. 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

4. Repeat 2 and 3 until a stable state is reached.

5. Conclusion: to sustain cooperation, you need a short average distance, and cliquishness (“small worlds”)

TU/e - 0ZM05/0EM15/0A150 38

And another peculiarity ... 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?

TU/e - 0ZM05/0EM15/0A150

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)

39

TU/e - 0ZM05/0EM15/0A150 40

Strogatz and Watts 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?

TU/e - 0ZM05/0EM15/0A150 41

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

TU/e - 0ZM05/0EM15/0A150

And now the third characteristic

42

TU/e - 0ZM05/0EM15/0A150 43

Same thing … we see “scale-freeness” all over

TU/e - 0ZM05/0EM15/0A150

… and it can’t be based on an ER-network

44

TU/e - 0ZM05/0EM15/0A150 45

Scale-free networks are:

Robust to random problems/mistakes ... ... but vulnerable to selectively targeted attacks

TU/e - 0ZM05/0EM15/0A150 46

Another BIG question:How do scale free networks arise?

Potential answer: Perhaps through “preferential attachment”

< show NetLogo simulation here>

(Another) critique to this approach: it ignores ties created by those in the network

TU/e - 0ZM05/0EM15/0A150

Some related issues

47

TU/e - 0ZM05/0EM15/0A150 48

“The tipping point” (Watts*)

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

TU/e - 0ZM05/0EM15/0A150 49

Malcolm Gladwell

(journalist/writer: wrote “Blink” and “The tipping point”

Duncan Watts

(scientist, Yahoo,Microsoft Research)

TU/e - 0ZM05/0EM15/0A150 50

Hmm ...

<try Netlogo Small Worlds>

We will see that you do not always end up with small worlds!

TU/e - 0ZM05/0EM15/0A150 51

The bigger picture

TU/e - 0ZM05/0EM15/0A150

The general approach … understandhow STRUCTURE can arise from underlying DYNAMICS

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, …

… 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, …

TU/e - 0ZM05/0EM15/0A150 53

More material on the website ...

TU/e - 0ZM05/0EM15/0A150 54

To Do:

Read and comprehend the papers on small world networks, scale-free networks (see website).

Think about applications of these results