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@anxosan
Anxo Sánchez
Grupo Interdisciplinar de Sistemas Complejos
Departamento de Matemáticas
Institute UC3M-BS for Finan...
@anxosan
¿What are complex systems?
@anxosan
¿What are complex systems?
❖ Many interacting components / agents
@anxosan
¿What are complex systems?
❖ Many interacting components / agents
❖ Emergent collective behavior
@anxosan
¿What are complex systems?
❖ Many interacting components / agents
❖ Emergent collective behavior
❖ Examples:
❖ Wa...
@anxosan
¿What are complex systems?
❖ Many interacting components / agents
❖ Emergent collective behavior
❖ Examples:
❖ Wa...
@anxosan
(Physics Nobel Laureate) Phil Anderson, 1972
“More is different” (emergence)
@anxosan
When there are MORE 

than one simple agent (e.g. molecule)
(Physics Nobel Laureate) Phil Anderson, 1972
“More is...
@anxosan
When there are MORE 

than one simple agent (e.g. molecule)
those agents may self-organize in collective objects ...
@anxosan
When there are MORE 

than one simple agent (e.g. molecule)
those agents may self-organize in collective objects ...
@anxosan
When there are MORE 

than one simple agent (e.g. molecule)
those agents may self-organize in collective objects ...
@anxosan
MICRO: the relevant elementary agents        
INTER: their basic, simple interactions        
MACRO: the emerging...
@anxosan
MICRO: the relevant elementary agents        
INTER: their basic, simple interactions        
MACRO: the emerging...
@anxosan
Intrinsically (3x) interdisciplinary:
❖ MICRO belongs to one science
❖ MACRO to another science
❖ Mechanisms: a t...
@anxosan
Intrinsically (3x) interdisciplinary:
❖ MICRO belongs to one science
❖ MACRO to another science
❖ Mechanisms: a t...
@anxosan
“More is different” (fluctuations)
• Role of fluctuations: many, but not infinite,

agents / interactions
@anxosan
“More is different” (fluctuations)
• Role of fluctuations: many, but not infinite,

agents / interactions
@anxosan
“More is different” (fluctuations)
• Role of fluctuations: many, but not infinite,

agents / interactions
@anxosan
“More is different” (fluctuaciones)
• Role of fluctuations: many, but not infinite,

agents / interactions
• Nonl...
@anxosan
“More is different” (fluctuaciones)
• Role of fluctuations: many, but not infinite,

agents / interactions
• Nonl...
@anxosan
“More is different” (fluctuaciones)
• Role of fluctuations: many, but not infinite,

agents / interactions
• Nonl...
@anxosan
“More is different” (fluctuaciones)
• Role of fluctuations: many, but not infinite,

agents / interactions
• Nonl...
@anxosan
“More is different” (fluctuaciones)
• Role of fluctuations: many, but not infinite,

agents / interactions
• Nonl...
@anxosan
950C
1Kg
1cm
Water level vs. temperature
“More is different” (phase transition)
@anxosan
950C
1Kg
1cm
970C
1cm
1Kg
Water level vs. temperature
“More is different” (phase transition)
@anxosan
950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
Water level vs. temperature
“More is different” (phase transition)
@anxosan
950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
? Extrapolation?
Water level vs. temperature
“More is different” (phase transi...
@anxosan
950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
1010C
Macroscopic linear
extrapolation
breaks down!
? Extrapolation?
Water lev...
@anxosan
950C
1Kg
1cm
970C
1cm
1Kg
990C
1Kg
1010C
Macroscopic linear
extrapolation
breaks down!
? Extrapolation?
(a single...
@anxosan
“More is different” (phase transition)
Water level:
economic index
@anxosan
95 97 99
“More is different” (phase transition)
Water level:
economic index
@anxosan
95 97 99 101
Crash = result of
collective behavior of
individual traders
“More is different” (phase transition)
W...
@anxosan
Statistical Mechanics 

Phase Transition
@anxosan
Statistical Mechanics 

Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
A...
@anxosan
Statistical Mechanics 

Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
B...
@anxosan
Statistical Mechanics 

Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
B...
@anxosan
Statistical Mechanics 

Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
B...
@anxosan
Statistical Mechanics 

Phase Transition
Atoms,Molecules
Drops,Bubbles
Complexity
MICRO
MACRO More is
different
B...
“More is different” (frontier science)
Chemicals
Ion channels
Neurons
Brain
Thoughts
Economy, Culture, Social groups
“More is different” (frontier science)
Chemicals
Ion channels
Neurons
Brain
Thoughts
Economy, Culture, Social groups
“More is different” (frontier science)
Conce...
Chemicals
Ion channels
Neurons
Brain
Thoughts
Economy, Culture, Social groups
It helps to bridge them by addressing 

with...
@anxosan
Santa Fe Institute for Complex Systems
“... a private, non-profit, multidisciplinary research and education cente...
@anxosan
Santa Fe Institute for Complex Systems
“... a private, non-profit, multidisciplinary research and education cente...
@anxosan
Santa Fe Institute for Complex Systems
“... a private, non-profit, multidisciplinary research and education cente...
@anxosan
Santa Fe Institute for Complex Systems
“... a private, non-profit, multidisciplinary research and education cente...
@anxosan
Institutions in Spain
@anxosan
Asociación para el estudio de Sistemas Complejos
Sociotecnológicos (COMSOTEC)
@anxosan
Complexitat.cat
@anxosan
Asociación Madrileña de Ciencias de la Complejidad
(ComplejiMad)
@anxosan
Foundations
@anxosan
Foundations
❖ Statistical Mechanics (Boltzmann, Gibbs, 1900)
@anxosan
Foundations
❖ Statistical Mechanics (Boltzmann, Gibbs, 1900)
❖ Nonlinear Science
❖ Chaos (Lorenz, 1963)
❖ Coheren...
@anxosan
Foundations
❖ Statistical Mechanics (Boltzmann, Gibbs, 1900)
❖ Nonlinear Science
❖ Chaos (Lorenz, 1963)
❖ Coheren...
@anxosan
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear physics
❖Nanotechnology, quantum compu...
@anxosan
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear physics
❖Nanotechnology, quantum compu...
@anxosan
J. Muñoz, R. Cuerno & M. Castro (2006)
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear...
@anxosan
J. Muñoz, R. Cuerno & M. Castro (2006)
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear...
@anxosan
J. Muñoz, R. Cuerno & M. Castro (2006)
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear...
@anxosan
J. Muñoz, R. Cuerno & M. Castro (2006)
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear...
@anxosan
J. Muñoz, R. Cuerno & M. Castro (2006)
Cross-disciplinary (frontier) science
❖ Physics
❖Statistical and nonlinear...
@anxosan
❖Economy
❖Micro vs macro, financial markets, management, …
Cross-disciplinary (frontier) science
@anxosan
❖Economy
❖Micro vs macro, financial markets, management, …
Cross-disciplinary (frontier) science
@anxosan
E. Moro, Leganés (2006)
❖Economy
❖Micro vs macro, financial markets, management, …
Cross-disciplinary (frontier) ...
@anxosan
E. Moro, Leganés (2006)
❖Economy
❖Micro vs macro, financial markets, management, …
Cross-disciplinary (frontier) ...
@anxosan
E. Moro, Leganés (2006) P. Richmond, Dublin (2006)
❖Economy
❖Micro vs macro, financial markets, management, …
Cro...
@anxosan
❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…
Cross-disciplinary (frontier) science
@anxosan
❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…
Cross-disciplinary (frontier) science
@anxosan
❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…
Cross-disciplinary (frontier) science
@anxosan
M. San Miguel, Palma de Mallorca (2005)
❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…
Cros...
@anxosan
M. San Miguel, Palma de Mallorca (2005)
❖Sociology
❖Norms and institutions, cultural dynamics, cooperation,…
Cros...
@anxosan
A. Arenas, Tarragona (2002)M. San Miguel, Palma de Mallorca (2005)
❖Sociology
❖Norms and institutions, cultural d...
@anxosan
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …
Cross-disciplinary (frontier) science
@anxosan
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …
Cross-disciplinary (frontier) science
@anxosan
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …
Cross-disciplinary (frontier) science
@anxosan
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …
Cross-disciplinary (frontier) science
@anxosan
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, …
Cross-disciplinary (frontier) science
@anxosan
Cross-disciplinary (frontier) science
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, ……
@anxosan
Cross-disciplinary (frontier) science
❖ Biology
❖ Ecology, inmune system, genetic networks, biofilms, ……
@anxosan
Innovation between Science and Technology
@anxosan
Innovation between Science and Technology
❖ Bottom-up approximation
@anxosan
Innovation between Science and Technology
❖ Bottom-up approximation
❖ Robust, self-organized systems
@anxosan
Innovation between Science and Technology
❖ Bottom-up approximation
❖ Robust, self-organized systems
❖ Control of...
@anxosan
Innovation between Science and Technology
❖ Bottom-up approximation
❖ Robust, self-organized systems
❖ Control of...
@anxosan
Innovation between Science and Technology
❖ Bottom-up approximation
❖ Robust, self-organized systems
❖ Control of...
@anxosan
Mathematics
@anxosan
Mathematics
❖ Graph theory (Complex networks)
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Mathematics
❖ Graph theory (Complex networks)
❖ Stochastic processes and statistics (Disorder, noise)
❖ Functiona...
@anxosan
Applications
@anxosan
Applications
@anxosan
Applications
@anxosan
Applications
@anxosan
Applications
@anxosan
Applications
@anxosan
The Sicomoro course
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
The Sicomoro course
❖ Intro (this lecture, not over yet, examples coming)
❖ Sociophysics
❖ Econophysics (Bartolo ...
@anxosan
Examples
@anxosan
Examples
❖ Traffic
• Question 1: How are jams formed in highways?
• Question 2: How are jams formed in cities?
@anxosan
Examples
❖ Traffic
• Question 1: How are jams formed in highways?
• Question 2: How are jams formed in cities?
❖ ...
@anxosan
• Important problem
Examples: Traffic
@anxosan
• Important problem
❖ 82% travellers and 53% commercial transport (Germany)
❖10% asphalt land (Netherlands)
❖ Bil...
@anxosan
• Important problem
❖ 82% travellers and 53% commercial transport (Germany)
❖10% asphalt land (Netherlands)
❖ Bil...
@anxosan
• Important problem
❖ 82% travellers and 53% commercial transport (Germany)
❖10% asphalt land (Netherlands)
❖ Bil...
@anxosan
Simulation work needed:
❖ Discrete models suited to simulation and amenable to analytics
(at least to some extent...
@anxosan
1. Acceleration: if possible, increase speed by 1; vmax=5
7.5
2. Braking: slow down to the fastest possible speed...
@anxosan
1. Acceleration: if possible, increase speed by 1; vmax=5
7.5
2. Braking: slow down to the fastest possible speed...
@anxosan
1. Acceleration: if possible, increase speed by 1; vmax=5
7.5
2. Braking: slow down to the fastest possible speed...
@anxosan
3. Randomization: with probability p, brake (no apparent cause)
4. Motion
Parallel updating (important)
1D Nagel-...
@anxosan
3. Randomization: with probability p, brake (no apparent cause)
4. Motion
Parallel updating (important)
1D Nagel-...
@anxosan
1D Nagel-Schreckenberg model (1992)
@anxosan
Empirical findings:
“Fundamental diagram”
1D Nagel-Schreckenberg model (1992)
@anxosan
1D Nagel-Schreckenberg model (1992)
Empirical findings:
Phantom jams
@anxosan
1D Nagel-Schreckenberg model (1992)
Simulation results:
Good agreement
@anxosan
What can be inferred?
@anxosan
• The NaSch is a good (stylized) description of highway 

traffic
• (Can be extended to more complicated 

situat...
@anxosan
What can be inferred?
@anxosan
BML Automata (Biham,
Middleton & Levine, 1992)
• Traffic lights
even instants
odd instants
• No overlap
• Paralle...
@anxosan
Aleatoriedad: Probabilidad g de giro, g < 0.5 (BML g=0)
g
1-g
City traffic: 2D models
CMMS Automata (Cuesta,
Mart...
@anxosan
City traffic: 2D models
Main result:
Phase diagram
@anxosan
• The phase transition picture applies to traffic as well as 

to molecules
• Phase diagram similar to water; g s...
@anxosan
Generalization: particle flow
@anxosan
Generalization: particle flow
@anxosan
Examples: Opinion formation
Question: How can the minority win?
@anxosan
Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms init...
@anxosan
Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms init...
@anxosan
Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms init...
@anxosan
Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms init...
@anxosan
Serge Galam has proposed a mechanism, social inertia,
that leads to a democratic rejection of social reforms init...
@anxosan
S. Galam: Le Monde, 26 February, 2005
Examples: Opinion formation
@anxosan
S. Galam: Le Monde, 26 February, 2005
Examples: Opinion formation
@anxosan
Binary opinion, either yellow or blue, about reform
Galam´s model
@anxosan
Binary opinion, either yellow or blue, about reform
For Against
Galam´s model
@anxosan
Binary opinion, either yellow or blue, about reform
For Against
Initially, there is a blue minority
Galam´s model
@anxosan
Social life: Discussion in groups
(e.g., at work, at the bar, at the church,…)
Galam´s model
@anxosan
Social life: Discussion in groups
(e.g., at work, at the bar, at the church,…)
Example,k
=16
M, maximum
cell size...
@anxosan
Galam´s model
Interaction: Majority convinces minority in a cell
@anxosan
6
10
Galam´s model
Interaction: Majority convinces minority in a cell
@anxosan
Every agent
becomes yellow
6
10
Galam´s model
Interaction: Majority convinces minority in a cell
@anxosan
Galam´s model
Social inertia: Ties resolved in favor of blue
@anxosan
8
8
Galam´s model
Social inertia: Ties resolved in favor of blue
@anxosan
8
8
Galam´s model
Social inertia: Ties resolved in favor of blue
@anxosan
Galam´s model
Evolution: Random reshuffling in cells
@anxosan
Galam´s model
Evolution: Random reshuffling in cells
@anxosan
Galam´s model
Evolution: Random reshuffling in cells
@anxosan
Phase diagram: Initial minority vs max cell size
p: initial minority population
Threshold
line
Eur. Phys. J. B 39...
@anxosan
What can be inferred?
@anxosan
What can be inferred?
• There is a threshold value pc<½ such that for p>pc 

the minority becomes a majority
• Fo...
@anxosan
By way of conclusion
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
❖ Relevant to real life problems of different nature

(...
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
❖ Relevant to real life problems of different nature

(...
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
❖ Relevant to real life problems of different nature

(...
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
❖ Relevant to real life problems of different nature

(...
@anxosan
By way of conclusion
❖ Complexity Science is here to stay
❖ Relevant to real life problems of different nature

(...
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Introduction to Complex Systems

Presentación utilizada por Anxo Sanchez (@anxosan) en la primera sesión del Curso de Introducción a los Sistemas Complejos de la Fundacion Sicomoro y ComplejiMad

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Introduction to Complex Systems

  1. 1. @anxosan Anxo Sánchez Grupo Interdisciplinar de Sistemas Complejos Departamento de Matemáticas Institute UC3M-BS for Financial Big Data (IFiBiD) Universidad Carlos III de Madrid Instituto de Biocomputación y Física de Sistemas Complejos (BIFI) Universidad de Zaragoza Introduction to complex systems
  2. 2. @anxosan ¿What are complex systems?
  3. 3. @anxosan ¿What are complex systems? ❖ Many interacting components / agents
  4. 4. @anxosan ¿What are complex systems? ❖ Many interacting components / agents ❖ Emergent collective behavior
  5. 5. @anxosan ¿What are complex systems? ❖ Many interacting components / agents ❖ Emergent collective behavior ❖ Examples: ❖ Water (molecules vs phases) (more later) ❖ Proteins (aminoacids vs molecule / structure) ❖ Brain (neurons vs intelligence) ❖ Society (people vs institutions /norms) ❖ Biosphere (species vs ecosystem) (more later)
  6. 6. @anxosan ¿What are complex systems? ❖ Many interacting components / agents ❖ Emergent collective behavior ❖ Examples: ❖ Water (molecules vs phases) (more later) ❖ Proteins (aminoacids vs molecule / structure) ❖ Brain (neurons vs intelligence) ❖ Society (people vs institutions /norms) ❖ Biosphere (species vs ecosystem) (more later) ❖ Frontier between sciences or subjects
  7. 7. @anxosan (Physics Nobel Laureate) Phil Anderson, 1972 “More is different” (emergence)
  8. 8. @anxosan When there are MORE 
 than one simple agent (e.g. molecule) (Physics Nobel Laureate) Phil Anderson, 1972 “More is different” (emergence)
  9. 9. @anxosan When there are MORE 
 than one simple agent (e.g. molecule) those agents may self-organize in collective objects (e.g. cells) (Physics Nobel Laureate) Phil Anderson, 1972 “More is different” (emergence)
  10. 10. @anxosan When there are MORE 
 than one simple agent (e.g. molecule) those agents may self-organize in collective objects (e.g. cells) which have emergent behavior (e.g. life) (Physics Nobel Laureate) Phil Anderson, 1972 “More is different” (emergence)
  11. 11. @anxosan When there are MORE 
 than one simple agent (e.g. molecule) those agents may self-organize in collective objects (e.g. cells) which have emergent behavior (e.g. life) that 
 IS DIFFERENT 
 from the behavior of the simple agent (e.g. chemical reactions) (Physics Nobel Laureate) Phil Anderson, 1972 “More is different” (emergence)
  12. 12. @anxosan MICRO: the relevant elementary agents         INTER: their basic, simple interactions         MACRO: the emerging collective objects “More is different” (emergence) Complex Systems Paradigm:
  13. 13. @anxosan MICRO: the relevant elementary agents         INTER: their basic, simple interactions         MACRO: the emerging collective objects orders, transactions “More is different” (emergence) Complex Systems Paradigm: traders herds,crashes,booms Economy:
  14. 14. @anxosan Intrinsically (3x) interdisciplinary: ❖ MICRO belongs to one science ❖ MACRO to another science ❖ Mechanisms: a third science “More is different” (emergence) Complex Systems Paradigm:
  15. 15. @anxosan Intrinsically (3x) interdisciplinary: ❖ MICRO belongs to one science ❖ MACRO to another science ❖ Mechanisms: a third science Decision making, psychology Financial economics Statistical mechanics, Physics
 Mathematics “More is different” (emergence) Complex Systems Paradigm: Economy:
  16. 16. @anxosan “More is different” (fluctuations) • Role of fluctuations: many, but not infinite,
 agents / interactions
  17. 17. @anxosan “More is different” (fluctuations) • Role of fluctuations: many, but not infinite,
 agents / interactions
  18. 18. @anxosan “More is different” (fluctuations) • Role of fluctuations: many, but not infinite,
 agents / interactions
  19. 19. @anxosan “More is different” (fluctuaciones) • Role of fluctuations: many, but not infinite,
 agents / interactions • Nonlinear systems with instabilities
  20. 20. @anxosan “More is different” (fluctuaciones) • Role of fluctuations: many, but not infinite,
 agents / interactions • Nonlinear systems with instabilities • External influences: noise, disorder
  21. 21. @anxosan “More is different” (fluctuaciones) • Role of fluctuations: many, but not infinite,
 agents / interactions • Nonlinear systems with instabilities • External influences: noise, disorder • Creative effects, e.g., stochastic resonance
  22. 22. @anxosan “More is different” (fluctuaciones) • Role of fluctuations: many, but not infinite,
 agents / interactions • Nonlinear systems with instabilities • External influences: noise, disorder • Creative effects, e.g., stochastic resonance
  23. 23. @anxosan “More is different” (fluctuaciones) • Role of fluctuations: many, but not infinite,
 agents / interactions • Nonlinear systems with instabilities • External influences: noise, disorder • Creative effects, e.g., stochastic resonance
  24. 24. @anxosan 950C 1Kg 1cm Water level vs. temperature “More is different” (phase transition)
  25. 25. @anxosan 950C 1Kg 1cm 970C 1cm 1Kg Water level vs. temperature “More is different” (phase transition)
  26. 26. @anxosan 950C 1Kg 1cm 970C 1cm 1Kg 990C 1Kg Water level vs. temperature “More is different” (phase transition)
  27. 27. @anxosan 950C 1Kg 1cm 970C 1cm 1Kg 990C 1Kg ? Extrapolation? Water level vs. temperature “More is different” (phase transition)
  28. 28. @anxosan 950C 1Kg 1cm 970C 1cm 1Kg 990C 1Kg 1010C Macroscopic linear extrapolation breaks down! ? Extrapolation? Water level vs. temperature “More is different” (phase transition)
  29. 29. @anxosan 950C 1Kg 1cm 970C 1cm 1Kg 990C 1Kg 1010C Macroscopic linear extrapolation breaks down! ? Extrapolation? (a single molecule does not boil)Water level vs. temperature “More is different” (phase transition)
  30. 30. @anxosan “More is different” (phase transition) Water level: economic index
  31. 31. @anxosan 95 97 99 “More is different” (phase transition) Water level: economic index
  32. 32. @anxosan 95 97 99 101 Crash = result of collective behavior of individual traders “More is different” (phase transition) Water level: economic index
  33. 33. @anxosan Statistical Mechanics 
 Phase Transition
  34. 34. @anxosan Statistical Mechanics 
 Phase Transition Atoms,Molecules Drops,Bubbles Complexity MICRO MACRO More is different Anderson abstractization
  35. 35. @anxosan Statistical Mechanics 
 Phase Transition Atoms,Molecules Drops,Bubbles Complexity MICRO MACRO More is different Biology Social Science Brain Science Economics and Finance Business
 AdministrationICT Semiotics and Ontology Anderson abstractization
  36. 36. @anxosan Statistical Mechanics 
 Phase Transition Atoms,Molecules Drops,Bubbles Complexity MICRO MACRO More is different Biology Social Science Brain Science Economics and Finance Business
 AdministrationICT Semiotics and Ontology Chemicals E-pages Neurons Words people Customers Traders Cells,life Meaning Social groups WWW Cognition, perception Markets Herds, Crashes Anderson abstractization
  37. 37. @anxosan Statistical Mechanics 
 Phase Transition Atoms,Molecules Drops,Bubbles Complexity MICRO MACRO More is different Biology Social Science Brain Science Economics and Finance Business
 AdministrationICT Semiotics and Ontology Chemicals E-pages Neurons Words people Customers Traders Cells,life Meaning Social groups WWW Cognition, perception Markets Herds, Crashes Anderson abstractization
  38. 38. @anxosan Statistical Mechanics 
 Phase Transition Atoms,Molecules Drops,Bubbles Complexity MICRO MACRO More is different Biology Social Science Brain Science Economics and Finance Business
 AdministrationICT Semiotics and Ontology Chemicals E-pages Neurons Words people Customers Traders Cells,life Meaning Social groups WWW Cognition, perception Markets Herds, Crashes Anderson abstractization
  39. 39. “More is different” (frontier science)
  40. 40. Chemicals Ion channels Neurons Brain Thoughts Economy, Culture, Social groups “More is different” (frontier science)
  41. 41. Chemicals Ion channels Neurons Brain Thoughts Economy, Culture, Social groups “More is different” (frontier science) Conceptual boundary between disciplines
  42. 42. Chemicals Ion channels Neurons Brain Thoughts Economy, Culture, Social groups It helps to bridge them by addressing 
 within a common conceptual framework
 the fundamental problems 
 of one of them 
 in terms of the collective phenomena of another. “More is different” (frontier science) Conceptual boundary between disciplines
  43. 43. @anxosan Santa Fe Institute for Complex Systems “... a private, non-profit, multidisciplinary research and education center”
  44. 44. @anxosan Santa Fe Institute for Complex Systems “... a private, non-profit, multidisciplinary research and education center” “Since its founding in 1984, the Santa Fe Institute (SFI) has devoted itself to fostering a multidisciplinary scientific research community pursuing frontier science. SFI seeks to catalyze new research activities and serve as an "institute without walls.” Topics • Physics and Computation of Complex Systems • Human Behavior, Institutions and Social Systems • Living Systems: Emergence, Hierarchy and Dynamics
  45. 45. @anxosan Santa Fe Institute for Complex Systems “... a private, non-profit, multidisciplinary research and education center”
  46. 46. @anxosan Santa Fe Institute for Complex Systems “... a private, non-profit, multidisciplinary research and education center” Research projects • A theory of invention and innovation • Theory of embodied intelligence • Biology, behavior, and disease • Social networks, big data, and physics-powered inference • Information, thermodynamics, and the evolution of complexity in 
 biological systems • Neighborhoods, slums, & human development • Emergence of complex societies • Hidden laws in biological and social systems • Evolution of complexity on earth • Cities, scaling, & sustainability
  47. 47. @anxosan Institutions in Spain
  48. 48. @anxosan Asociación para el estudio de Sistemas Complejos Sociotecnológicos (COMSOTEC)
  49. 49. @anxosan Complexitat.cat
  50. 50. @anxosan Asociación Madrileña de Ciencias de la Complejidad (ComplejiMad)
  51. 51. @anxosan Foundations
  52. 52. @anxosan Foundations ❖ Statistical Mechanics (Boltzmann, Gibbs, 1900)
  53. 53. @anxosan Foundations ❖ Statistical Mechanics (Boltzmann, Gibbs, 1900) ❖ Nonlinear Science ❖ Chaos (Lorenz, 1963) ❖ Coherents Structures (Fermi, Pasta y Ulam, 1955) ❖ Patterns (Bénard, 1900; Belusov, 1951; Winfree, 1967) ❖ Evolutionary Dynamics (Maynard-Smith, 1974)
  54. 54. @anxosan Foundations ❖ Statistical Mechanics (Boltzmann, Gibbs, 1900) ❖ Nonlinear Science ❖ Chaos (Lorenz, 1963) ❖ Coherents Structures (Fermi, Pasta y Ulam, 1955) ❖ Patterns (Bénard, 1900; Belusov, 1951; Winfree, 1967) ❖ Evolutionary Dynamics (Maynard-Smith, 1974) ❖ Computation (1990)
  55. 55. @anxosan Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  56. 56. @anxosan Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  57. 57. @anxosan J. Muñoz, R. Cuerno & M. Castro (2006) Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  58. 58. @anxosan J. Muñoz, R. Cuerno & M. Castro (2006) Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  59. 59. @anxosan J. Muñoz, R. Cuerno & M. Castro (2006) Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  60. 60. @anxosan J. Muñoz, R. Cuerno & M. Castro (2006) Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  61. 61. @anxosan J. Muñoz, R. Cuerno & M. Castro (2006) Cross-disciplinary (frontier) science ❖ Physics ❖Statistical and nonlinear physics ❖Nanotechnology, quantum computation, astronomy,…
  62. 62. @anxosan ❖Economy ❖Micro vs macro, financial markets, management, … Cross-disciplinary (frontier) science
  63. 63. @anxosan ❖Economy ❖Micro vs macro, financial markets, management, … Cross-disciplinary (frontier) science
  64. 64. @anxosan E. Moro, Leganés (2006) ❖Economy ❖Micro vs macro, financial markets, management, … Cross-disciplinary (frontier) science
  65. 65. @anxosan E. Moro, Leganés (2006) ❖Economy ❖Micro vs macro, financial markets, management, … Cross-disciplinary (frontier) science
  66. 66. @anxosan E. Moro, Leganés (2006) P. Richmond, Dublin (2006) ❖Economy ❖Micro vs macro, financial markets, management, … Cross-disciplinary (frontier) science
  67. 67. @anxosan ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  68. 68. @anxosan ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  69. 69. @anxosan ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  70. 70. @anxosan M. San Miguel, Palma de Mallorca (2005) ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  71. 71. @anxosan M. San Miguel, Palma de Mallorca (2005) ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  72. 72. @anxosan A. Arenas, Tarragona (2002)M. San Miguel, Palma de Mallorca (2005) ❖Sociology ❖Norms and institutions, cultural dynamics, cooperation,… Cross-disciplinary (frontier) science
  73. 73. @anxosan ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, … Cross-disciplinary (frontier) science
  74. 74. @anxosan ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, … Cross-disciplinary (frontier) science
  75. 75. @anxosan ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, … Cross-disciplinary (frontier) science
  76. 76. @anxosan ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, … Cross-disciplinary (frontier) science
  77. 77. @anxosan ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, … Cross-disciplinary (frontier) science
  78. 78. @anxosan Cross-disciplinary (frontier) science ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, ……
  79. 79. @anxosan Cross-disciplinary (frontier) science ❖ Biology ❖ Ecology, inmune system, genetic networks, biofilms, ……
  80. 80. @anxosan Innovation between Science and Technology
  81. 81. @anxosan Innovation between Science and Technology ❖ Bottom-up approximation
  82. 82. @anxosan Innovation between Science and Technology ❖ Bottom-up approximation ❖ Robust, self-organized systems
  83. 83. @anxosan Innovation between Science and Technology ❖ Bottom-up approximation ❖ Robust, self-organized systems ❖ Control of complex systems ❖ Internet ❖ Agent-based software ❖ Design of organization ❖ Risks: spam, SIDA/SARS/Avian flu ❖ New drugs / genetic therapy
  84. 84. @anxosan Innovation between Science and Technology ❖ Bottom-up approximation ❖ Robust, self-organized systems ❖ Control of complex systems ❖ Internet ❖ Agent-based software ❖ Design of organization ❖ Risks: spam, SIDA/SARS/Avian flu ❖ New drugs / genetic therapy ❖ Basis for new technologies (ICT, transport, …)
  85. 85. @anxosan Innovation between Science and Technology ❖ Bottom-up approximation ❖ Robust, self-organized systems ❖ Control of complex systems ❖ Internet ❖ Agent-based software ❖ Design of organization ❖ Risks: spam, SIDA/SARS/Avian flu ❖ New drugs / genetic therapy ❖ Basis for new technologies (ICT, transport, …) ❖ ¿A paradigm shift?
  86. 86. @anxosan Mathematics
  87. 87. @anxosan Mathematics ❖ Graph theory (Complex networks)
  88. 88. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise)
  89. 89. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions)
  90. 90. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory
  91. 91. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory ❖ Evolutionary dynamics and game theory
  92. 92. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory ❖ Evolutionary dynamics and game theory ❖ A “discrete analysis”
  93. 93. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory ❖ Evolutionary dynamics and game theory ❖ A “discrete analysis” ❖ New simulation techniques (Agents, OOP)
  94. 94. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory ❖ Evolutionary dynamics and game theory ❖ A “discrete analysis” ❖ New simulation techniques (Agents, OOP) ❖ Computational complexity
  95. 95. @anxosan Mathematics ❖ Graph theory (Complex networks) ❖ Stochastic processes and statistics (Disorder, noise) ❖ Functional analysis (Phase transitions) ❖ Control theory and signal theory ❖ Evolutionary dynamics and game theory ❖ A “discrete analysis” ❖ New simulation techniques (Agents, OOP) ❖ Computational complexity ❖ Data mining, data analysis
  96. 96. @anxosan Applications
  97. 97. @anxosan Applications
  98. 98. @anxosan Applications
  99. 99. @anxosan Applications
  100. 100. @anxosan Applications
  101. 101. @anxosan Applications
  102. 102. @anxosan The Sicomoro course
  103. 103. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming)
  104. 104. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics
  105. 105. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3)
  106. 106. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3) ❖ Fractals and scale invariance (Bartolo Luque, May 3)
  107. 107. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3) ❖ Fractals and scale invariance (Bartolo Luque, May 3) ❖ The game of evolution (José A. Cuesta, May 9)
  108. 108. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3) ❖ Fractals and scale invariance (Bartolo Luque, May 3) ❖ The game of evolution (José A. Cuesta, May 9) ❖ Genes and human genealogies (Susanna Manrubia, May 9)
  109. 109. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3) ❖ Fractals and scale invariance (Bartolo Luque, May 3) ❖ The game of evolution (José A. Cuesta, May 9) ❖ Genes and human genealogies (Susanna Manrubia, May 9) ❖ Complex networks (Javier Galeano, May 17)
  110. 110. @anxosan The Sicomoro course ❖ Intro (this lecture, not over yet, examples coming) ❖ Sociophysics ❖ Econophysics (Bartolo Luque, May 3) ❖ Fractals and scale invariance (Bartolo Luque, May 3) ❖ The game of evolution (José A. Cuesta, May 9) ❖ Genes and human genealogies (Susanna Manrubia, May 9) ❖ Complex networks (Javier Galeano, May 17) ❖ Complexity in biology (Ester Lázaro, May 17)
  111. 111. @anxosan Examples
  112. 112. @anxosan Examples ❖ Traffic • Question 1: How are jams formed in highways? • Question 2: How are jams formed in cities?
  113. 113. @anxosan Examples ❖ Traffic • Question 1: How are jams formed in highways? • Question 2: How are jams formed in cities? ❖ Opinion formation • Question: How can a minority win?
  114. 114. @anxosan • Important problem Examples: Traffic
  115. 115. @anxosan • Important problem ❖ 82% travellers and 53% commercial transport (Germany) ❖10% asphalt land (Netherlands) ❖ Billions of lost hours (Spain) Examples: Traffic
  116. 116. @anxosan • Important problem ❖ 82% travellers and 53% commercial transport (Germany) ❖10% asphalt land (Netherlands) ❖ Billions of lost hours (Spain) ❖ Billions of euros in gas (Europe)
 • Difficult solution ❖ Wrong traditional answer (new roads don’t fix it) Examples: Traffic
  117. 117. @anxosan • Important problem ❖ 82% travellers and 53% commercial transport (Germany) ❖10% asphalt land (Netherlands) ❖ Billions of lost hours (Spain) ❖ Billions of euros in gas (Europe)
 • Difficult solution ❖ Wrong traditional answer (new roads don’t fix it) Wider applicability: other transports, pedestrians, internet,… Examples: Traffic
  118. 118. @anxosan Simulation work needed: ❖ Discrete models suited to simulation and amenable to analytics (at least to some extent) ❖ Need for predictions: realistic models impossible ❖ Controlled experiments and identification of relevant parameters ❖ Monitor global variables Examples: Traffic
  119. 119. @anxosan 1. Acceleration: if possible, increase speed by 1; vmax=5 7.5 2. Braking: slow down to the fastest possible speed 1D Nagel-Schreckenberg model (1992)
  120. 120. @anxosan 1. Acceleration: if possible, increase speed by 1; vmax=5 7.5 2. Braking: slow down to the fastest possible speed 1D Nagel-Schreckenberg model (1992)
  121. 121. @anxosan 1. Acceleration: if possible, increase speed by 1; vmax=5 7.5 2. Braking: slow down to the fastest possible speed 1D Nagel-Schreckenberg model (1992)
  122. 122. @anxosan 3. Randomization: with probability p, brake (no apparent cause) 4. Motion Parallel updating (important) 1D Nagel-Schreckenberg model (1992)
  123. 123. @anxosan 3. Randomization: with probability p, brake (no apparent cause) 4. Motion Parallel updating (important) 1D Nagel-Schreckenberg model (1992)
  124. 124. @anxosan 1D Nagel-Schreckenberg model (1992)
  125. 125. @anxosan Empirical findings: “Fundamental diagram” 1D Nagel-Schreckenberg model (1992)
  126. 126. @anxosan 1D Nagel-Schreckenberg model (1992) Empirical findings: Phantom jams
  127. 127. @anxosan 1D Nagel-Schreckenberg model (1992) Simulation results: Good agreement
  128. 128. @anxosan What can be inferred?
  129. 129. @anxosan • The NaSch is a good (stylized) description of highway 
 traffic • (Can be extended to more complicated 
 situations/geometries) • Averages are not very relevant • New interesting magnitudes to monitor: “throughput” vs “volatility” What can be inferred?
  130. 130. @anxosan What can be inferred?
  131. 131. @anxosan BML Automata (Biham, Middleton & Levine, 1992) • Traffic lights even instants odd instants • No overlap • Parallel update • Periodic B.C. City traffic: 2D models
  132. 132. @anxosan Aleatoriedad: Probabilidad g de giro, g < 0.5 (BML g=0) g 1-g City traffic: 2D models CMMS Automata (Cuesta, Martínez, Molera & Sánchez, 1993)
  133. 133. @anxosan City traffic: 2D models Main result: Phase diagram
  134. 134. @anxosan • The phase transition picture applies to traffic as well as 
 to molecules • Phase diagram similar to water; g similar to temperature • Additional info by analytical means (low density limit, 
 other approaches) • Note interaction through excluded volume What can be inferred?
  135. 135. @anxosan Generalization: particle flow
  136. 136. @anxosan Generalization: particle flow
  137. 137. @anxosan Examples: Opinion formation Question: How can the minority win?
  138. 138. @anxosan Serge Galam has proposed a mechanism, social inertia, that leads to a democratic rejection of social reforms initially favored by the majority Examples: Opinion formation Question: How can the minority win?
  139. 139. @anxosan Serge Galam has proposed a mechanism, social inertia, that leads to a democratic rejection of social reforms initially favored by the majority Social inertia: Ties favor the “no” option Examples: Opinion formation Question: How can the minority win?
  140. 140. @anxosan Serge Galam has proposed a mechanism, social inertia, that leads to a democratic rejection of social reforms initially favored by the majority Social inertia: Ties favor the “no” option Examples: Opinion formation Question: How can the minority win? ! Conservative reaction to the risk of change ! Keep the social “statu quo”
  141. 141. @anxosan Serge Galam has proposed a mechanism, social inertia, that leads to a democratic rejection of social reforms initially favored by the majority Social inertia: Ties favor the “no” option Examples: Opinion formation Question: How can the minority win? ! Conservative reaction to the risk of change ! Keep the social “statu quo” (Taken from Maxi San Miguel, IFISC, Mallorca)
  142. 142. @anxosan Serge Galam has proposed a mechanism, social inertia, that leads to a democratic rejection of social reforms initially favored by the majority Social inertia: Ties favor the “no” option Examples: Opinion formation Question: How can the minority win? ! Conservative reaction to the risk of change ! Keep the social “statu quo” (Taken from Maxi San Miguel, IFISC, Mallorca)
  143. 143. @anxosan S. Galam: Le Monde, 26 February, 2005 Examples: Opinion formation
  144. 144. @anxosan S. Galam: Le Monde, 26 February, 2005 Examples: Opinion formation
  145. 145. @anxosan Binary opinion, either yellow or blue, about reform Galam´s model
  146. 146. @anxosan Binary opinion, either yellow or blue, about reform For Against Galam´s model
  147. 147. @anxosan Binary opinion, either yellow or blue, about reform For Against Initially, there is a blue minority Galam´s model
  148. 148. @anxosan Social life: Discussion in groups (e.g., at work, at the bar, at the church,…) Galam´s model
  149. 149. @anxosan Social life: Discussion in groups (e.g., at work, at the bar, at the church,…) Example,k =16 M, maximum cell size Cells defined by their size k Galam´s model
  150. 150. @anxosan Galam´s model Interaction: Majority convinces minority in a cell
  151. 151. @anxosan 6 10 Galam´s model Interaction: Majority convinces minority in a cell
  152. 152. @anxosan Every agent becomes yellow 6 10 Galam´s model Interaction: Majority convinces minority in a cell
  153. 153. @anxosan Galam´s model Social inertia: Ties resolved in favor of blue
  154. 154. @anxosan 8 8 Galam´s model Social inertia: Ties resolved in favor of blue
  155. 155. @anxosan 8 8 Galam´s model Social inertia: Ties resolved in favor of blue
  156. 156. @anxosan Galam´s model Evolution: Random reshuffling in cells
  157. 157. @anxosan Galam´s model Evolution: Random reshuffling in cells
  158. 158. @anxosan Galam´s model Evolution: Random reshuffling in cells
  159. 159. @anxosan Phase diagram: Initial minority vs max cell size p: initial minority population Threshold line Eur. Phys. J. B 39, 535 (2004) Galam´s model M: max cell size
  160. 160. @anxosan What can be inferred?
  161. 161. @anxosan What can be inferred? • There is a threshold value pc<½ such that for p>pc 
 the minority becomes a majority • For the effect to happen far from ½ M needs to be 
 small • Time to consensus T ~ ln N • Note that this is a proposal for a mechanism but 
 not a proof that this mechanism is the correct one • Models can be used to verify or falsify intuitions in
 social problems
  162. 162. @anxosan By way of conclusion
  163. 163. @anxosan By way of conclusion ❖ Complexity Science is here to stay
  164. 164. @anxosan By way of conclusion ❖ Complexity Science is here to stay ❖ Relevant to real life problems of different nature
 (cf. Examples)
  165. 165. @anxosan By way of conclusion ❖ Complexity Science is here to stay ❖ Relevant to real life problems of different nature
 (cf. Examples) ❖ Key concept: Emergence
  166. 166. @anxosan By way of conclusion ❖ Complexity Science is here to stay ❖ Relevant to real life problems of different nature
 (cf. Examples) ❖ Key concept: Emergence ❖ Involves many sciences, but strongly based on Mathematics and Computation
  167. 167. @anxosan By way of conclusion ❖ Complexity Science is here to stay ❖ Relevant to real life problems of different nature
 (cf. Examples) ❖ Key concept: Emergence ❖ Involves many sciences, but strongly based on Mathematics and Computation ❖ Requires working and thinking about frontiers
  168. 168. @anxosan By way of conclusion ❖ Complexity Science is here to stay ❖ Relevant to real life problems of different nature
 (cf. Examples) ❖ Key concept: Emergence ❖ Involves many sciences, but strongly based on Mathematics and Computation ❖ Requires working and thinking about frontiers ❖ Key for future R+D+i

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