Complexity Science: Modeling Complex Systems - Complexity …kti.tugraz.at/.../complexityscience/lectures/modelingcs.pdf · 2018-11-23 · We need tools and modeling approaches available

Complexity Science: Modeling Complex SystemsComplexity Science (VU) (706.723)

Elisabeth Lex

ISDS, TU Graz

November 22, 2018

Elisabeth Lex (ISDS, TU Graz) Complexity Science November 22, 2018 1 / 44

Repetition

Repetition

Basic concepts of complexity science: Self-organisation, emergence,non-linearity

Logistic model to study population growth

Measuring complexity:

Fractal dimension for fractal like systemsEntropy and statistical complexity if we focus on message exchangeComputing resources AIC, effective complexity, logical depthSystem properties complexity as sizeEvolution of the system thermodynamic depth, degree of hierarchy


Introduction

Modeling Systems

Agent-based Modeling

Cellular Automata


Introduction

What is Agent-based Modeling (ABM)?

Approach to modeling systems that consist of autonomous,interacting agents

Dynamic processes of agent interaction simulated repeatedly over time

Thus, an ABM is a model in which agents interact repeatedly


Introduction

Why do we need Agent-based Modeling?

We live in an increasingly complex world

Our systems are increasingly complex and interdependent: e.gelectrical infrastructures, telecommunication networks, transportationnetworks, social systems, social networks

Some systems have always been too complex to model realistically:e.g models for economic markets

We need tools and modeling approaches available that help usanalyze complex systems

Lots of empirical data available and computational power

Example use cases for ABM: modeling agent behaviour in the stockmarket, supply chains, consumer markets, spreading of epidemics,understanding social systems,..


Agents

Agent-based models

Agents are autonomous and model intelligent behavior with a simpleset of rules

The agents are situated in space (e.g. a grid or a network)

The agents interact with each other locally (i.e., they are social)

The agents have only a partial local information

There are often different types of agents following different set of rules

The rules may be deterministic or probabilistic

There are often random elements in the world


Agents

Agents

Agents are self-contained: identifiable, discrete, has set ofcharacteristics / attributes, behaviours and decision-making capability

Agents can have memory - then, they can learn and adapt theirbehaviour (dynamic agent attribute)

Examples for agents: people, groups, organizations, insects, swarms,robots, biological entities,..


Agents

Agent-based models

Argent-based models are used to simulate actions and interactions ofautonomous agents and to assess their effects on system as a whole

Understanding relations between individual decisions and systembehavior

Micromotives vs. Macrobehavior (Schelling’s book)

They are always computational, i.e., simulations

They are intuitive for implementation, experiments, interpretation


Agents

Advantages of Agent-based models

ABMs are extensible

ABM are interpretable: one can use them to transparently planreal-world concepts

Holistic modeling approach: can be used to answer multiple questions(“many question models”)

Typically, individual-level behavior better understood than aggregate(population) level. One can start with sth that is better understoodto understand macro behavior

ABMs help tackle complexity and well suited to model behavior


Agents

Tools to build ABM

Agent-based modeling and simulation toolkits: Repast (Java), Swarm(Objective C, Java), NetLogo, StarLogo, MASON, AnyLogix

Generall tools: e.g. MATLAB, spreadsheets, ABM with programminglanguages (Python, Java,...)


Agents

Cellular Automata

E.F. Codd and John von Neuman were the first ones to propose CAsin 1950s

Became popular when John Conway introduced Life Game

Book by Stephen Wolfram: A New Kind of Science

A type of agent based models

Dynamical system that is discrete in both space and time


Agents

Cellular Automata

Basis: collection of cells arranged in a grid, i.e. spatial structure

Each cell c has a state s (out of a finite number of possible states):e.g. 1 or 0, one or off, alive or dead

Each cell has a neighborhood: typically a list of adjacent cells

Set of rules uniformly applied to the contents of each cell at eachiteration of the automaton

structure of the cellular automaton evolves through a number of timesteps based on application of rules and contents of cells and theirneighbors


Agents

Example for Cellular Automaton

Screenshot fromhttps://natureofcode.com/book/chapter-7-cellular-automata/


https://natureofcode.com/book/chapter-7-cellular-automata/

Life

Conways’s Game of Life (1/3)

Developed by mathematician John Conway in 1970

Early example of emergent complexity

That means: based on which rules on chooses, one gets differentoutcomes with interesting properties

Implemented as 2-dimensional infinite grid partitioned into cells

Basis idea:

Life is played on grid of square cellsEach cell is either alive or deadAn alive cell is colored, a dead one notEach cell in the grid has a neighborhood consisting of the eight cells inevery direction including diagonals

Rules:

An agent stays alive if 2 or 3 neighbors are alive, otherwise it dies (as ifby loneliness or overcrowding if more neighbors are alive)New agent is born if exactly 3 neighbors are alive


Life

Life (2/3)

Let’s apply the rules:

To apply one step of the rules, we count the number of live neighborsfor each cell

The number of live neighbors is always based on the cells before therule was applied. In other words, we must first find all of the cellsthat change before changing any of them

A dead cell with exactly three live neighbors becomes a live cell (birth)

A live cell with two or three live neighbors stays alive (survival)

In all other cases, a cell dies or remains dead (overcrowding orloneliness)


Life

Example of Life

(a) Initial stage (b) After 40 times

Figure: Life simulation: (a) initial random layout of cells in the On state, (b) afterall cells updated 40 times


Life

Netlogo Example of the Life Model

http://ccl.northwestern.edu/netlogo/

Go to File/Model Library/Computer Science/Cellular Automata/Life


Life

Why is Game of Life interesting?

Rules are simple and use only local information as each cell’s state isbased on its current state and the state of its immediate neighbors

Resulting patterns of Life depend on initial conditions - eachsimulation gives different patterns of On and Off cells

Patterns can emerge in systems that are completely described bysimple, deterministic rules based on only local information

Based on simple rules of behavior and nature of agent interactions,systems can show collective intelligence, even without existence of acentral authority


Fractals

Elementary Cellular Automaton (Wolfram, 2002)1

Simplest grid: 1-dimensional line of cells

Simplest set of states s: 0 or 1

Simplest neighborhood in 1 dimensions for any given cell: cell plus itsneighbor on the left and its neighbor on the right

Cellular automaton lives over a period of time t

Init: st=0

Question: How can we compute the states for cells at st+1?

A cell’s new state at st+1 is a function of all states in the cell’sneighborhood at previous time step st−1

1https://www.wolframscience.com/nks/Elisabeth Lex (ISDS, TU Graz) Complexity Science November 22, 2018 19 / 44

Fractals

Elementary Cellular Automaton (Wolfram, 2002)1

Simplest grid: 1-dimensional line of cells

Simplest set of states s: 0 or 1

Simplest neighborhood in 1 dimensions for any given cell: cell plus itsneighbor on the left and its neighbor on the right

Cellular automaton lives over a period of time t

Init: st=0

Question: How can we compute the states for cells at st+1?

A cell’s new state at st+1 is a function of all states in the cell’sneighborhood at previous time step st−1

1https://www.wolframscience.com/nks/Elisabeth Lex (ISDS, TU Graz) Complexity Science November 22, 2018 19 / 44

Fractals

Example: Rule 90 Elementary Cellular Automaton (1/3)

We assume an infinite grid of cells

Each cell can have a state of 0 or 1

Initially, some cells in a row are set to 1, the others are 0

The state of a cell in the subsequent row is determined by the state of3 cells: the state of the cell directly above and the two cellsdiagonally above on each side, combined via an exclusive OR function


Fractals


How many state configurations can we have for these 3 cells?

23 = 8: 000, 001, 010, 011, 100, 101, 110, 111

States of three consecutive cells correspond to a 3-bit binary number

The cellular automaton is determined by what bit we assign to eachof the 8 possible 3-bit states, i.e. the automaton corresponds to an8-bit number

We apply the Rule 90 elementary cellular automaton

90 in binary is 01011010, i.e., we assign these bits to the 3-bit state:


Fractals


How many state configurations can we have for these 3 cells?23 = 8: 000, 001, 010, 011, 100, 101, 110, 111






Fractals


How many state configurations can we have for these 3 cells?23 = 8: 000, 001, 010, 011, 100, 101, 110, 111






Fractals



Fractals


Result: Sierpinski triangle. What can you observe?

Selfsimilarity, fractal patternCode:https://natureofcode.com/book/chapter-7-cellular-automata/



Fractals


Result: Sierpinski triangle. What can you observe?Selfsimilarity, fractal patternCode:https://natureofcode.com/book/chapter-7-cellular-automata/



The Schelling Model

Applying CAs to study social phenomena: The SchellingModel

T. Schelling (1971): A small preference for a specific kind ofneighbors lead to total segregation

Placed pennies and dimes on a chess board and moved them aroundaccording to various rulesInterpreted board as a city, each square represents a housePennies and dimes represented agents, e.g. two racesNeighborhood of an agent was the squares adjacent to the square inwhich the agent residedRules determined whether an agent was happy in its current locationIf unhappy it could move to another location or exit the board entirely


The Schelling Model

Applying CAs to study social phenomena: The SchellingModel

Result:

Board became segregated even if the agents did not prefer segregation

Board became segregated if an initially integrated board hadhappiness rules that expressed mild preference for neighbors of theirown type

I.e., model shows how global patterns (spatial segregation) canemerge from local preferences

A simple interaction mechanism leads to segregation

Segregation achieved even if no one explicitly aims for it - i.e. nocentral control


The Schelling Model

How does the Schelling Model work? (1/2)

Assume a population of individuals (aka agents) of type X or OTypes represent immutable characteristics (e.g., age)Two populations are initially placed into random locations of aneighborhood gridAfter placing all agents, each cell is either occupied by an agent oremptyThe neighbor relationships among the cells can be represented verysimply as a graph: cells are the nodes, edges are inserted between twocells that are neighbors on the grid


The Schelling Model

How does the Schelling Model work? (2/2)

Now, determine if each agent is satisfied with its current location

Agent is satisfied if is surrounded by at least t of its own type ofneighboring agents

Threshold t applies to all agents in the model (in reality everyonemight have a different threshold they are satisfied with)

The higher t, the higher the likelihood that agents will not besatisfied with their current location

Example:

For example, if t = 3, agent X is satisfied if at least 3 of its neighborsare also XIf fewer than 3 are X, then the agent is not satisfied, and it will want tochange its location in the gridAny algorithm can be used to choose new location (e.g., randomselection, nearest available location, 1 row at a time)


The Schelling Model

Example

(a) Initial stage (b) After one round

Figure: Left image: all dissatisfied agents have an asterisk next to them. Rightimage: shows new configuration after all dissatisfied agents have been moved tounoccupied cells (1 row at a time) where they are satisfied. May cause otheragents to become unsatisfied, then new round of movement begins


The Schelling Model

Netlogo Example of the Schelling Model


Go to File/Model Library/Social Science/Segregation


The Schelling Model

Observations from Schelling’s Model

Spatial segregation takes place even though no individual agentactively wants it

Segregation doesn’t happen due to built-in model agents that arewilling to be in the minority

Ideally, all agents are carefully arranged in an integrated pattern

However, from random start hard for agents to find such integratedpatterns

At more general level, Schelling model is an example of how fixedcharacteristics (e.g., ethnicity) can become highly correlated withmutable characteristics

E.g. decision where to live, which over time conforms to similarities inagents immutable types, producing segregation


Forest Fire

CAs: Simulation of forest fires

We model a forest as a grid of cells

A cell is either occupied by a tree or empty

The fire starts on the left edge of the forest

It spreads to the neighboring trees in all four directions

North, south, east, west

Fire can not skip an empty cell

There is no wind


Forest Fire

Netlogo Example of the Forest Fire model


Go to File/Model Library/Earth Science/Fire


Forest Fire

Questions

With density around 50% how much of the forest burns

With different initial settings do the same tree burn?

Each point that represents a tree burning is born and then dies

It never moves whatsoever

The fire is made of burning trees that do not move

But the fire itself moves!

Local vs. global level

Emergence of properties at a global level that do not exist on thelocal level


Forest Fire

Phase transition

Often there is a very small margin for parameters and the networkstructure where the system goes quickly from one state into another

This is called phase transition

We can observe a phase transition around 59%

Reaching the other edge of the grid


Ising

Simulate physical processes: The Ising model

The model originally comes from physics

It models the magnetization of a material

The cells are organized in a grid

Each cell has a spin si: it is represented by +1 or -1

The cells can flip their spin

The energy of a cell is calculated from its four neighbors (north,south, east, west) as Ei =

∑j sisj

The total energy is E =∑

iEi

The system always tries to reach the state of the minimal energy withsome randomness, which increases with temperature


Ising

Netlogo Example of the Ising model


Go to File/Model Library/Chemistry & Physics/Ising


Ising

Questions

What happens when the temperature is low?

The cells will align their spins

What happens when the temperature is high?

The alignment is not likely anymore

There is a specific temperature, which separates those two modes:2

ln(1+√2)

on an infinite grid

Phase transition

Ising model can be used to model also other types of processes - anyideas?

Opinion dynamics, consensus reaching, etc.


Ising

Questions






ln(1+√2)

on an infinite grid

Phase transition




Ising

Questions






ln(1+√2)

on an infinite grid

Phase transition




Ising

Questions






ln(1+√2)

on an infinite grid

Phase transition




Wealth Distribution

How does the wealth distribution work?

We have population living on a grid of cells

Each cell has an amount of grain and an grain capacity

People collect grain from the cells and eat (some of) the grain tosurvive

How much grain each person accumulates is her wealth

Initially, a roughly equal distribution

Each person attempts to move to a cell with more grain (if free)

People have a life expectancy and can die and can also die if theyhave no grain

If a person dies an offspring is born with a random amount of grain(no inheritance)


Wealth Distribution

Netlogo Example of the Wealth Distribution model


Go to File/Model Library/Social Science/Wealth Distribution


Wealth Distribution

Questions

What kind of wealth distribution do we expect to see?

A power-law distribution! Why?

Because agents are heterogeneous

They have different visions, metabolism, life expectancy, and so on

Those agents who gain an initial advantage will keep that advantage

Preferential attachment


Wealth Distribution

Questions








Wealth Distribution

Questions








Wealth Distribution

Summary

Agent-based modeling to model complex systems and to studyemergent phenomena, e.g. from animal behavior, social sciences,ecology, ...

Special case of ABMs: Cellular Automata

Examples for ABM: Life, Schelling, Forest Fire, Ising Model, Wealthdistribution


Wealth Distribution

Take away

We can model and understand real-world phenomena by constructingmodels that exhibit complex emergent behavior resulting from local,simplified agent interaction.


Wealth Distribution

How would you build an ABM?

Pro tip: Take an established model and see whether you can buildupon it

Requires model literacy!


Wealth Distribution

How would you build an ABM?

Pro tip: Take an established model and see whether you can buildupon it

Requires model literacy!


Wealth Distribution

Questions?


Complexity Science: Modeling Complex Systems - Complexity …kti.tugraz.at/.../complexityscience/lectures/modelingcs.pdf · 2018-11-23 · We need tools and modeling approaches available

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