Networks and sciences: The story of the small-world

Transcription

1 Networks and sciences: The story of the small-world Hugues Bersini IRIDIA ULB 2013 Networks and sciences 1

2 The story begins with Stanley Milgram ( ) In 1960, the famous experience of the submission to authority En 1967, the as famous experience of the six degrees of separation Stanley Milgram 2013 Networks and sciences 2

3 2013 Networks and sciences 3

4 The story moves on with Watts and Strogatz (1998) 2013 Networks and sciences 4

5 But networks are far from homogeneous!! With aggregates Random Scale-Free 2013 Networks and sciences 5

6 Poisson distribution Power-law distribution Exponential Network Scale-free Network 2013 Networks and sciences 6

7 The story ends up with Babarasi (in the years 2000) 2013 Networks and sciences 7

8 Networks are «scale-free»: P(k) ~k - connecteurs 2013 Networks and sciences 8

9 The kingdom of hubs Ron Carter Terry Clark Kenny Burrel Mon réseau de musiciens de Jazz 2013 Networks and sciences 9

10 Trophic Network 2013 Networks and sciences 10

11 How topology and hubs influence the global behaviour of networks The small-world effect very small distance among nodes Epidemic propagation (epidemiology or viral marketting) Study of robustness: target (on hubs) or random attacks 2013 Networks and sciences 11

12 Why scale-free? The rule of preferential attachment Rich get richer 2013 Networks and sciences 12

13 Research at IRIDIA on networks 2013 Networks and sciences 13

14 Network Dynamics Homogeneous units a i (t) (the same temporal evolution the same differential equations) da i /dt = F(a j, W ij,i) A given topology in the connectivity matrix: W ij Entries I which perturb the dynamics and to which the network gives meaning ( attractors) A very large family of concerned biological networks Idiotypic immune network Hopfield network Coupled Map Lattice Boolean network Ecological network (Lokta-Volterra) 2013 Genetic network Networks and sciences 14

15 Dynamics Chemistry Ecosystem: prey/predator Brain, heart 2013 Networks and sciences 15

16 Topology influences the dynamics of immune network 2013 Networks and sciences 16

17 Ab concentrations Ab concentrations Ab conc Frustrated chaos in biological networks time [d] clone case time [d] time [d] clone open chain clone closed chain Networks and sciences 17

18 Network of chemical reactions Dynamics = kinetics Metadynamics = appearance and disappearance of molecules 2013 Networks and sciences 18

19 Immune Networks 2013 Networks and sciences 19

20 Neural Networks 2013 Networks and sciences 20

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22 Elementary dynamics:propagation Epidemic propagation 2013 Networks and sciences 22

23 A new route to cooperation 2013 Networks and sciences 23

24 The prisoner's dilemma P1/P2 Cooperate Compete Cooperate (1,1) (-2,3) Compete (3,-2) (-1,-1) The winning strategy for both players is to compete. But doing so, they miss the cooperating one which is collectively better. The common good is subverted by individual rationality and self-interest Networks and sciences 24

25 But is competitive behaviour and collective distress avoidable? So far the prisoner's dilemma is lacking some crucial quality that real world situations have. 1) Iterated version: play several moves and cumulate your reward over these moves. 2) Distribute spatially the players (CA): each cell just cooperates with its immediate neighbours and adapts the local best strategy. Cluster of nice individuals emerge and can prosper in hostile environments -> EVOLUTIONARY GAME THEORY 2013 Networks and sciences 25

26 The spatial cellular automata simulation Largely inspired by Nowak s work on spatial prisoner dilemma A cellular automata in which every cell contains one agent (specialist or generalist) In all cells, asynchronously, an agent will subsequently: interact with its neighbors (Moore neighborhood) to consume them. Sum the payoff according to the payoff matrix replicate Adopt the identity of the fittest neighbor For a given number of iteration steps 2013 Networks and sciences 26

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28 Nowak s cooperators vs defectors 2013 Networks and sciences 28

29 Setting the stage k 6 =5 k 1 =3 k 2 =4 Stochastic replicator dynamics: Vertex x plays k x times per generation and accumulates payoff f x. Choose a random neighbor y with payoff f y. Replace strategy m x by m y with probability: k 5 =2 k 4 =3 k 3 =3 f p max 0, x f y k (T S) 2013 Networks and sciences 29

30 Games on graphs Conclusions: The more heterogeneous, the more cooperative. Cs benefit most from heterogeneity Networks and sciences 30

31 Plastic networks: parametrically and structurally: Network Metadynamics Various dynamical changes, that Varela called: dynamics and metadynamics - modification of connexions - addition of connexions - addition of new nodes - suppression of existing nodes The organisation is maintained independently of the constituants This is the case for neuro, immuno, chemical, sociological networks, PC networks 2013 Networks and sciences 31

32 A key interdependency 2013 Networks and sciences 32

33 Springer Link : With networks? Crawler How? Python script HTTP Requests What? DOI References Date 2013 Networks and sciences 33

34 Results 2013 Networks and sciences 34

35 Book analysis 2013 Networks and sciences 35

36 Book analysis Clustering coefficient and Degree 2013 Networks and sciences 36

37 Systemic contagion in financial systems 2013 Networks and sciences 37

38 Financial Network Intraday network structure of payment activity in the Canadian Large Value Transfer System (LVTS) Liasons Dangereuses: Increasing connectivity, risk sharing and systemic risk (by Stefano Battiston, Domenico Delli Gatti, Mauro Gallegati, Bruce C. Greenwald and Joseph E. Stiglitz) 2013 Networks and sciences 38

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42 Conclusions: networks are everywhere 2013 Networks and sciences 42