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) ( ) 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: 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 Enter: Stanley Milgram ( ) Remember him? 8

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 ” 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 - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A The small world phenomenon (cont.) “Small world project” has been testing this assertion (not anymore, see to, 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… !

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TU/e - 0ZM05/0EM15/0A The Kevin Bacon experiment – Tjaden ( +/- 1996) Actors = actors Ties = “has played in a movie with”

TU/e - 0ZM05/0EM15/0A The Kevin Bacon game Can be played at: 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 - 0ZM05/0EM15/0A A search for high Kevin Bacon numbers… 3 2

TU/e - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A “Elvis has left the building …”

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TU/e - 0ZM05/0EM15/0A We find small average path lengths in all kinds of places… 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

TU/e - 0ZM05/0EM15/0A150 How weird is that? 22

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 Kevin Bacon himself!|

TU/e - 0ZM05/0EM15/0A150 APL is small in random networks 24 [Slide copied from Jari_Chennai2010.pdf]

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

TU/e - 0ZM05/0EM15/0A150 But let’s move on to the second network characteristic … 26

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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 network Friendship 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 protein interactions 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 … then we find this: 32 Wang & Chen (2003) Complex networks: Small-world, Scale-free and beyond

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TU/e - 0ZM05/0EM15/0A 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)

TU/e - 0ZM05/0EM15/0A150 Examples of interesting properties of small world networks 35

TU/e - 0ZM05/0EM15/0A150 Synchronizing fireflies … Synchronization speed depends on small-world properties of the network  Network characteristics important for “integrating local nodes” 36

TU/e - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A 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 - 0ZM05/0EM15/0A Scale-free networks are: Robust to random problems/mistakes but vulnerable to selectively targeted attacks

TU/e - 0ZM05/0EM15/0A Another BIG question: How do scale free networks arise? Potential answer: Perhaps through “preferential attachment” (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 - 0ZM05/0EM15/0A “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,

TU/e - 0ZM05/0EM15/0A Malcolm Gladwell (journalist/writer: wrote “Blink” and “The tipping point” Duncan Watts (scientist, Yahoo, Microsoft Research)

TU/e - 0ZM05/0EM15/0A150 Hmm... We will see that you do not always end up with small worlds! 50

TU/e - 0ZM05/0EM15/0A150 The bigger picture 51

TU/e - 0ZM05/0EM15/0A150 The general approach … understand how 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 - 0ZM05/0EM15/0A More material on the website...

TU/e - 0ZM05/0EM15/0A To Do: Read and comprehend the papers on small world networks, scale-free networks (see website). Think about applications of these results