# networks

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Chaotic Dynamics on
Large Networks
J. C. Sprott
Department of Physics

Presented at the
Chaotic Modeling and Simulation
International Conference
in Chania, Crete, Greece
on June 3, 2008
What is a complex system?
   Complex ≠ complicated
   Not real and imaginary parts
   Not very well defined
   Contains many interacting parts
   Interactions are nonlinear
   Contains feedback loops (+ and -)
   Cause and effect intermingled
   Driven out of equilibrium
   Evolves in time (not static)
   Usually chaotic (perhaps weakly)
A Physicist’s Neuron

N                             N
xout  tanh  a j x j
inputs                         j1

tanh x

x
A General Model
(artificial neural network)

1
N neurons
3
2                    4

N
xi  bi xi  tanh  aij x j

j 1
j i

“Universal approximator,” N  ∞
Solutions are bounded
Examples of Networks
System      Agents    Interaction     State        Source
Brain      Neurons    Synapses     Firing rate   Metabolism

Food Web     Species    Feeding      Population     Sunlight

Market                  actions

Political   Voters    Information     Party       The Press
System                              affiliation

Other examples: War, religion, epidemics, organizations, …
Political System
Information
a1               Political “state”
from others                                       N
Voter     x  bx  tanh  a j x j

a2                                      j 1
a3             aj = ±1/√N, 0

tanh x             Democrat

x
Republican
Types of Dynamics

1.   Static      Equilibrium
2.   Periodic Limit Cycle (or Torus)
“Stuck in a rut”
3.   Chaotic Strange Attractor
Arguably the most “healthy”
Especially if only weakly so
Route to Chaos at Large N (=317)
317
dxi / dt  bxi  tanh  aij x j
j1
400 Random networks
Fully connected

“Quasi-periodic route to chaos”
Typical Signals for Typical Network
Average Signal from all Neurons
All +1                 N = 317
b = 1/4

All −1
Simulated Elections
100% Democrat               N = 317
b = 1/4

100% Republican
Strange Attractors
N = 10
b = 1/4
Competition vs. Cooperation
317
dxi / dt  bxi  tanh  aij x j
j1
500 Random networks
Fully connected
b = 1/4

Competition

Cooperation
Bidirectionality
317
dxi / dt  bxi  tanh  aij x j
j1
250 Random networks
Fully connected
b = 1/4

Reciprocity

Opposition
Connectivity
317
dxi / dt  bxi  tanh  aij x j
j1
250 Random networks
N = 317, b = 1/4

Dilute                                     Fully connected

1%
Network Size
N
dxi / dt  bxi  tanh  aij x j
j1
750 Random networks
Fully connected
b = 1/4

N = 317
What is the Smallest Chaotic Net?
   dx1/dt = – bx1 + tanh(x4 – x2)

   dx2/dt = – bx2 + tanh(x1 + x4)

   dx3/dt = – bx3 + tanh(x1 + x2 – x4)

   dx4/dt = – bx4 + tanh(x3 – x2)

Strange                     2-torus
Attractor
Circulant Networks
dxi /dt = −bxi + Σ ajxi+j
Fully Connected Circulant Network
N 1
dxi / dt  bxi  tanh  a j xi j
j1

N = 317
Diluted Circulant Network
dxi / dt  bxi  tanh(xi42  xi126  xi254 )

N = 317
Near-Neighbor Circulant Network
dxi / dt  bxi  tanh(xi1  xi2  xi3  xi4  xi5  xi6)

N = 317
Summary of High-N Dynamics
   Chaos is generic for sufficiently-connected networks

   Sparse, circulant networks can also be chaotic (but

the parameters must be carefully tuned)

   Quasiperiodic route to chaos is usual

   Symmetry-breaking, self-organization, pattern

formation, and spatio-temporal chaos occur

   Maximum attractor dimension is of order N/2

   Attractor is sensitive to parameter perturbations, but

dynamics are not
References
   A paper on this topic is scheduled to
appear soon in the journal Chaos

   http://sprott.physics.wisc.edu/
lectures/networks.ppt (this talk)

   http://sprott.physics.wisc.edu/chaostsa/
(my chaos textbook)

   sprott@physics.wisc.edu (contact me)

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