Network modules Flashcards
(40 cards)
complex systems
open systems consisting of many sub-systems that interact nonlinearly
emergent behaviour as result of complex interactions
examples of complex systems
flocks of birds -> organized behaviour (emergent behaviour)
brain -> neuronal connections -> higher cognitive functions, emotions, consciousness (emergent computations at neural level)
polarization -> humans converging to show the same behaviour
ant nests, laser, weather, spiral waves, chemical patterns, swarms, traffic etc
cascading transitions
radicalization of individuals embedded in social transitions (conspiracy theories)
also in eco-systems -> when one thing collapses others also follow
chaos theory
seemingly random behavior in deterministic nonlinear system
small changes in initial conditions lead to large changes in the future (butterfly effect)
tiny perturbation in the system can explode and make system unpredictable
Verhulst equation - predicting growth of rabbit population on an island
r * xt * (K- xt) / K
r - population growth parameter
x - number of rabits in particular point of time
K - maximal capacity of an island
What are characteristics of K in Verhaulst equation?
it illustrates maximal capacity of the system
at first, it doesn’t affect the equation (so the growth is initially exponential)
however, the higher x, the higher impact of K decreasing the growth speed
xt
population (state variable) at time step t
r
growth rate parameter -> how aggrssively population grows
fix point
stable state (can be perturbed a bit by adding rabbits, but still comes back to normal)
in the graph -> to what value the function converge?
unstable fix point
in principle can be achieved, but any perturbation can throw it off balance
example: 0 rabbits
adding 2 rabbits to the system changes everything
how to simulate population growth in Python?
1) define model (x,r,k)
-> r * x * (k-x)/k
2) define r, k, x values
3) for loop to calculate population growth over np.empty x array
how changing values of r affects the system?
smaller values of r -> exponential function
medium values of r -> at first function exponential, then it stabilizes
the higher r values, the more perturbations in the system
system starts overshoting and undershoting at first between 2 points, then between larger number of points
How does increasing r relates to bifurcation model?
With the increasing r, there seems to be more perturbations in the system. It relates to the bifurcation diagram, as change in only one parameter -> r, causes major consequences for the system behaviour. With increasing r, previously stable system becomes more and more unstable, and oscillations increase from 2 points to higher number of points
What happens at very high r value?
perturbations become highly unpredictable, function starts to resemble deterministic chaos (random behaviour in the system which looks like noise)
What is bifurcation?
in dynamical system, bifurcation happens when small change in parameter value causes sudden change to system’s behaviour
characteristic feature of bifurcation diagrams
period-doubling cascade -> when system’s behavior changes from oscillating between one point, to two, then four, then eight etc. leading up to chaos
butterfly effect in the model
even if your model is valid, butterfly in the system can lead to unpredictable consequences
this is what happens in weather predictions!
chaos or noise in the brain?
EEG, hearbeat -> trying to predict chaos
importantly, chaotic brain is healthy brain
more regular EEG patterns -> unhealthy brain
catastrophes in the systme
Behavior of complex systems is determined by number and type of stable and unstable equilibrium points (attractor landscape)
These landscapes are invariant (only quantitative changes) within certain ranges of control parameter values
At bifurcation points the landscapes change qualitatively (type and or number of attractors change)
- tipping points! → you change one specific thing → and then everything changes
what is cusp catastrophe?
- card → 2 stable states → folded to the right and left (technically the card could be straigth, but it always tips to one side) → you need to over-press the card
- force → causes the tipping to other side
- there is also delay in jumping from one state to another
- when you exerct vertical pressure → card needs to go either direction
hysteresis
DELAY in phase transition
example:
although temperature for water freezing is 0 deg, due to delay water freezes at 4 deg
how hysteresis effect can be illustrated in psychological example?
polarizing topic -> such as abortion
people who are not interested are easily influencable… (normal distribution of attitudes)
but people highly involved show hysteresis -> they have very stable opinion -> if changed -> will alter drastically (binomial distribution of attitudes)
self-organization of the system
sponataneous global order based on local rules
“no single bird in control’’
reflective model
Spearman model
single latent variable (g) causing other variables
