Social networks Small world networks

1. Social networksSmall world networks

2. Course aim knowledge about concepts in network theory, and being able to apply that knowledge

3. 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

4. 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)

5. What kind of structures do networks have, empirically?Answer: often “small-world”,and often also scale-free

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

7. Example 1 - Small worldnetworks NOTE • Edge of network theory • Not fully understood yet … • … but interesting findings

8. Enter: Stanley Milgram (1933-1984) • Remember him?

9. The small worldphenomenon – Milgram´s (1967) originalstudy • Milgram sent packages toseveral (60? 160?) peoplein Nebraska and Kansas. • Aim was “get this package to <address of person in Boston>” • Rule: onlysendthis package tosomeonewhomyouknow on a first name basis. Aim: tryto make the chain as short as possible. • Result: averagelength of a chain is onlysix “sixdegrees of separation”

10. Milgram’s original study (2) • An urbanmyth? • Milgramusedonly part of the data, actuallymainly the onessupporting his claim • Many packages didnot end up at the Boston address • Follow up studies typicallysmall scale

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.

12. Some Milgramfollow-ups… 6.6!

14. The Kevin Bacon experiment – Tjaden(+/- 1996) Actors = actors Ties = “has played in a movie with”

15. The Kevin Bacon game Canbeplayed at: http://oracleofbacon.org Kevin Bacon number (data might have changedbynow) 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]

16. A search for high Kevin Bacon numbers… 3 2

17. The center of the movie universe (sept 2013) Nr 370 Nr 136 Nr 39

18. The best centers… (2013 + 2011) (Kevin Bacon at place 444 in 2011) (RutgerHauer at place 39, J.Krabbé 935)

19. Where visual analysis fails: IMDB network

20. “Elvis has left the building …”

21. Small world networks = short average distance between pairs … … but relatively high “cliquishness”

22. We find small average path lengths in all kinds of places… • CaenorhabditisElegans 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

23. How weird is that?

24. 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. NBErdos was a “Kevin Bacon” long before Kevin Bacon himself!|

25. APL is small in random networks [Slide copied from Jari_Chennai2010.pdf]

26. [Slide copied from Jari_Chennai2010.pdf]

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

29. 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

30. 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 Source: Leskovec & Faloutsos Networks of the Real-world (1) Florence families Karate club network Collaboration network Friendship network

31. Biological networks metabolic networks food web neural networks gene regulatory networks Language networks Semantic networks Software networks … Source: Leskovec & Faloutsos Networks of the Real-world (2) Semantic network Yeast protein interactions Language network Software network

32. 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 hence we might want to create them)

33. Examples of interestingproperties of small world networks

34. Synchronizing fireflies … • <go to NetLogo> • Synchronization speed depends on small-world properties of the network  Network characteristics important for “integrating local nodes”

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

36. 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?

37. 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)

38. 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?

39. 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

40. And now the third characteristic

41. And if we consider all three…

42. … then we find this: Wang & Chen (2003) Complex networks: Small-world, Scale-free and beyond

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

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

45. The scale-free

46. Scale-free networks are: • Robust to random problems/mistakes ... • ... but vulnerable to selectively targeted attacks

47. 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

48. Some related issues

49. “The tipping point” (Watts*) • Consider a network in whicheach node determineswhether or nottoadopt, based on what his direct connections do. • Nodes have different thresholdstoadopt (randomlydistributed) • Question: when do you get cascades of adoption? • Answer: twophasetransitions or tipping points: • in sparsenetworks no cascades, as networks get more denseyou get cascades suddenly • as networks get more heterogenous, a suddenjump in the likelihood of cascades • as networksget 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

50. “Find the influentials” (or not?)