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Networks and sciences:
The story of the “small-world” Hugues Bersini
IRIDIA – ULB
2013 Networks and sciences 1
The story begins with Stanley
Milgram (1933-1984)
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
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The story moves on with Watts and
Strogatz (1998)
2013 Networks and sciences 4
But networks are far from homogeneous!!
Random
Scale-Free
With aggregates
2013 Networks and sciences 5
Poisson distribution
Exponential Network
Power-law distribution
Scale-free Network 2013 Networks and sciences 6
The story ends up with Babarasi (in the years 2000)
2013 Networks and sciences 7
Networks are « scale-free »:
P(k) ~k-
connecteurs
2013 Networks and sciences 8
The kingdom of hubs
Ron Carter
Terry Clark
Kenny Burrel
Mon réseau de musiciens
de Jazz
2013 Networks and sciences 9
Trophic Network
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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
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Why scale-free ?
The rule of preferential attachment
Rich get richer
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Research at IRIDIA on networks
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Network Dynamics
• Homogeneous units ai (t) (the same temporal evolution – the same differential equations)
– dai/dt = F(aj, Wij,I)
• A given topology in the connectivity matrix: Wij
• 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)
– Genetic network
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Dynamics
Chemistry Ecosystem:
prey/predator Brain, heart
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Topology influences the dynamics of
immune network
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Frustrated chaos in biological networks
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Network of chemical reactions
Dynamics = kinetics
Metadynamics = appearance
and disappearance of molecules
2013 Networks and sciences 18
Immune Networks
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Neural Networks
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Elementary dynamics:propagation
Epidemic propagation
2013 Networks and sciences 22
A new route to cooperation
2013 Networks and sciences 23
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.
2013 Networks and sciences 24
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
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
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Nowak’s cooperators vs defectors
2013 Networks and sciences 28
Setting the stage
• Stochastic replicator dynamics:
– Vertex x plays kx times per
generation and accumulates payoff
fx.
– Choose a random neighbor y
with payoff fy.
– Replace strategy mx by my with
probability:
k4=3
k1=3
k2=4 k
2=4
k3=3 k
5=2
k6=5
p max 0,fx fy
k(T S)
2013 Networks and sciences 29
Games on graphs
• Conclusions:
– The more heterogeneous, the more cooperative.
– Cs benefit most from heterogeneity.
2013 Networks and sciences 30
Plastic networks: parametrically and
structurally: Network Metadynamics
• Various dynamical changes, that Varela called: dynamics and metadynamics
• This is the case for neuro, immuno, chemical, sociological networks, PC networks
- modification of connexions
- addition of connexions
- addition of new nodes
- suppression of existing nodes
The organisation is maintained
independently of the constituants
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A key interdependency
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Springer Link : With networks ?
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• Crawler
• How ? – Python script
– HTTP Requests
• What ? – DOI
– References
– Date
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Results
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Book analysis
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Book analysis
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• Clustering coefficient and Degree
2013 Networks and sciences
Systemic contagion in financial
systems
2013 Networks and sciences 37
Liasons Dangereuses:
Increasing connectivity,
risk sharing and systemic risk
(by Stefano Battiston,
Domenico Delli Gatti,
Mauro Gallegati,
Bruce C. Greenwald and Joseph E. Stiglitz)
Intraday network structure
of payment activity in
the Canadian
Large Value Transfer System (LVTS)
Financial Network
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Conclusions: networks are everywhere
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