Complex Systems Test 1

most people view things as physical vs bio vs social, complex system thinkers view based on interaction (it is cross disiplinary)

S^D = ans

A reductionist would break the system down into its smallest individuals and their interactions, and say that once we understand these, we know everything about the system.

Non-linearity
Heterogeneity
Emergence
Multi-scale dynaics
Far from equilibrium
strong transitions
memory
feedback loops

small changes in the neighbourhood or rule can have a large effect

diverity across components in the system e.g. attribute, behaviours or states

the whole is more than the sum of its parts

often lack a characteristic scale

predicitive vs explanatior and exploratory
realism vs instrumentalism
reductionism vs holism

micro known to produce macro unknown
micro unknown to reproduce macro known

Independent varable - usually time
structure - rules of the system
data - serves as an input to the rule
inputs
outputs - how we choose to capture the outcome of the system

depends on the choices made
can make predictions using validated theories
explore and develope new explanations of empircally observed phenomena

evenness
exposure
clustering
concentration
centralisation

how close the local environment looks like global environment

amount on contact with other groups

mean similarity ratio

1/N * sum over all agents #similar neighbours/#total neighbours

simple description
connected
evolving
self organising
emergent
ubiquitous

0.1 0.1 0.1
0.1 -0.8 0.1
0.1 0.1 0.1
diffusion rate of 0.1
9 point stencil = much more random diffusion pattern

1. irregular dynamics can be understood if it is chaotic
2. complex systems of componts that behave as a connected network e.g. networks in organisations
3. complex systems of evolving systems e.g. spread of infectious diseases
4. self organisation - critical transitions and tipping points e.g. a power grid
5. simple rules lead to emergent behaviour e.g vieiwng music as a time series

euclidean geom measures size at one particular scale characterisitic of the shape, if the shape is scale-free or involves all scales then Euclidean measurement fails

perform an operation that leaves a shape undetectable different in shape and form
geometcial properties have not changed

provide a systematic way to capture the "roughness" or "complexity" of some objects

self similarity (zoom similarity)
infinite-complexity
fractal dimension (quantifies the roughness)

classic/strict - exact self similarity, infinite scaling range
statistical - self similarity is approx and extends only a limited range of scales

strict fractal
start with a line and remove the middle third
repeat for each line - keep repeating

start with a triangle and remove middle third
create 3 triangles and repeat
keep repeating

D = - log(#copies)/log(scale factor)
#copies = how many times as big

fractal

- recursion, doing same thing over and over again
- growth process will frequently lead to fractals
- fractals are hyper-efficent in their construction e.g. plants maximise exposure to sunlight

initiator and generator
iterated function system
lindenmayer systems

initiator = starting object for the construction
generator = how the initiaor changes e.g. divide into 4 congruent replicas and remove the centre replica

created by a set of contratcive functions
each initiation adds more points and refines the shape
e.g. Sierpinski triangle
define triangle as 3 points A,B and C
f1 = x+A/2
f2 = x+b/2
f3 = x+c/2
IFS = f1Uf2Uf3

a formal string rewriting system
it is a grammar defines by the tuple G = (V,a,R)
V = alphabet made up of variables and constants
a = starting state
R = rules that define how the variables are replaced
e.g. Cantor Set
V = {F,f}
a = F
R = F --> FfF, f --> fff where F = draw forwards, f = move forward but don't draw
one step = FfF
second step = FfFfffFfF

contains smaller copies or subsets of itself

E.g. cantor set
after n iterations we have 2^n line segments, as n approach infinity, 2^n --> inifintiy hence infinite number of pieces in the cantor set
after n iterations the length of each segment = (2L/3)^n, as n-->infinity, (2L/3)^n --> 0

perimetre = 3(3/2)^n as n-->infin, perim --> infin
area = (3/4)^nA, as n approach infi, A approaches infin

takes into account not having exact replicas
D = lim e-->0 logN(e)/log(1/e)
N(e) = # balls with length scale e needed to cover the set X

N(e) = C/e^D
can estimate D by plotting e or 1/e and determining the slope
D = logN(e)-logC/-loge

takes into account the density of the points (how much coastline in the box) whereas box count equautes all occupies boxes equally

how does the universe evolve
rune large numerical simulation
create a cosmic web --> most of it is dark matter
use slime to test routes between galazies, looked back at paths and found evidence of dark matter

spatio-thermal dynamics
typically modelled using PDEs

Interaction between activator and inhibitor morphogens leads to the formation of spatial patters of pigmentations either throuhg the instability of homogeneous states or the formation of spatial gradients

interaction of two morphogens that allows for the autonomous generation of spatial patterns during development via a type of symmetry breaking

activator
inhibitor
reaction
diffusion (random walks)
regulation

slow moving
tells the cells to do something

fast moving
tells the cells to slow down activator production

with many particles performing random walks, the particles will slowly spread outwards at a certain rate

discretised world
space = grid the comain and bin particles in the grid cells, focus on the particle concentration (U and V)
time = select a sufficently small time step to integrate the system forward

slow down the rate of diffusion
if d = 0.1, means only 10% of the central cells concentration will concentrate to the spread at a given time

include diags

diff^2ij = U(i+1,j) + U(i-1,j) +U(i,j+1) + U(i,j-1) - 4U(i,j)
U= concentration of A

if rarer if concentration of B (V) is low then make proportion to UV^2
U --> U - UV^2
V --> V + UV^2
assuming collision, reaction occurs with rate r (let r=1)

not discretised world
dU/dt = Du*diff^2U + f(1-U)-UV^2
dV/dt = Dv*diff^2V +(k+f)V+UV^2

U(x,t+changet) = U + changet(dU/dt)
repeat for V

ecosystems sometimes spatially organise - turning like patterns
patterns seem to be indicator of two spanning homogeneous spatial states, often one with life, one without
bifurcation drop = can't get back to sustainable
similarity in pattern does not equal similarity in process

spatially distributed dynamical systems where both space and time are discrete
worlds are built ona regular and homogeneous spatial grid with one automate resideing at each lattice state
one of the most common modelling tool
able to form large networks of simple components and create complex dynamics from very simple rules

world
network
distance
state
state transition function/ update rule

dimension and size
initial conditions
boundary conditions

finite number of states available
often binary --> two state CA
the configuration = the state of all the cells

World = 1D square grid lattice with periodic boundaries
states = binary
initialisation = either random or single black cell in the middle of row of white cells
neighbourhood = central & right and &left
update rule = 2^8 (#choices^#possible neighbour config) 256 possible rule combos

B = 1, W = 0
WWW WWB WBW WBB BWW BWB BBW BBB
calc right to left starting at 2^0

1. cells die out or become stagnant
2. oscillations between small stable structures
3. cells seem ot come alive and do unusual things

# total neighbours = (2r+1)^D
#possible local configurations = K^k^n = K^K^(2r+1)^D
r = radius of neighbourhood
d = dimension of the space
gets very large very quickly

cares about the possible configurations of the entire world
L = length of space in each dimension
for reg domain, total # cells = L^D
each cell must be one of k states so K^L^D possible configurations
blows up much less quickly

class 1 = tends to a stable uniform state e.g. all 1s or 0s
class 2 = repeated patterns
class 3 = appear to generate randomess via intricate rested repeats and periodic patterns
class 4 = sits at the boarder of chaos, can propagate info, memory etc.

if every configuration has a unique predecessor then the CA is reversible and the state transiiton functions are bijective
there are no non-trivial reversible elementary CA
game of life is not reversible (contains orphans)

map a local condition x to its successor y

maps a global condition x to its successor y

collection of parts and complex structure/arrangement

consciousness determined by computational structure of our brains

survual is grounded in continuation of our mental life

flexible
easy adoption
adjustment for individial level properties
fairly intutitive
agents can really be defined as anything

locally ordered collective motion

discrete entities that usually have some kind of attributes
they are active or self propelled and move with infinite energy

world - dimensions, obstacles, boundary conditions
agents
possible states of the agents
the initial state
the interaction network (typically locally)
rules for updating agent states

specific problem solved by the ABM
availability of data and applying to model
method of model validation

1. initialise domain = square domain with periodic boundary conditions, agentvelocity = a random direcition with constant speed, agentposition = random inside of domain
2. initialise N agents and their attribute
position = a tuple represeting the x and y coord of agent
velocity = a tuple representing the agents current velocity v defined by the heading O and a constant 'speed'
3. determine each agents neighbours
4. average the velocities of each agents' neighbours including self
5. update agents attributes

characterise how ordered the microscopic state of the system is
for vicsek = polarised order parametre = average normalised velocity - ranges from 0 (no order) to 1 (coherent motion)
tells us something about the group behaviour

the state of the system depends on its history

understand a physical system - cannot conduct experiments in the physical system
an exercise in approximation

more realistic model, age specific
health burden outcomes
use what find in model to do economic analysis
we know how to use and build these models