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Social networks

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

- Introduction: what are they, why important …- Small world networks- Four basic network arguments- Kinds of network data (collection)- Business networks

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Two approaches to network theory

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?

Answer: often “small-world”,

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

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:

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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)

Remember him?

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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”

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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 address

Follow up studies typically small scale

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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.

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Some Milgram follow-ups…

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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: 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]

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A search for high Kevin Bacon numbers…

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The center of the movie universe (sept 2013)

Nr 370 Nr 136

Nr 39

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The best centers… (2013 + 2011)

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(Kevin Bacon at place 444 in 2011)(Rutger Hauer at place 39, J.Krabbé 935)

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Where visual analysis fails: IMDB network

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“Elvis has left the building …”

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Small world networks

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

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

Caenorhabditis Elegans959 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

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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.

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

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APL is small in random networks

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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:

A short Average Path Length and

A high clustering coefficient

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

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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

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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

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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)

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Examples of interesting

properties of

small world networks

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Synchronizing fireflies …

<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. 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”)

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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?

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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

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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And if we consider all three…

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… then we find this:

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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The scale-free

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Scale-free networks are:

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

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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

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Some related issues

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“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 you get cascades suddenly as networks get more heterogenous, a sudden

jump in the likelihood of cascades as networks get even more heterogenous, the

likelihood of cascades decreases* 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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Malcolm Gladwell

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

Duncan Watts

(scientist, Yahoo,Microsoft Research)

http://www.youtube.com/watch?v=AtnR5H6AVVU

http://www.fastcompany.com/641124/tipping-point-toast

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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 patternsfrom micro behavior

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The general approach … understand

how STRUCTURE can arise from

underlying MICRO-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, …

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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:

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