# Questions of science: chatting with ChatGPT about complex systems

Nuno Crokidakis  Marcio Argollo de Menezes  Daniel O. Cajueiro  Instituto de
Física, Universidade Federal Fluminense, Niterói/RJ, Brazil  National
Institute of Science and Technology for Complex Systems (INCT-SC), Brazil.
Department of Economics, University of Brasilia, Campus Universitário Darcy
Ribeiro - FACE, Brasilia, DF 70910-900.  Machine Learning Laboratory in
Finance and Organizations (LAMFO), Universidade de Brasília (UnB), 70910-900,
Brasília, Brazil.

###### Abstract

We present an overview of the complex systems field using ChatGPT as a
representation of the community’s understanding. ChatGPT has learned language
patterns and styles from a large dataset of internet texts, allowing it to
provide answers that reflect common opinions, ideas, and language patterns
found in the community. Our exploration covers both teaching and learning, and
research topics. We recognize the value of ChatGPT as a source for the
community’s ideas.

###### keywords:

ChatGPT , Complex Systems , large language models.

††journal: a journal of physics

###### Contents

  1. 1 Introduction
  2. 2 Large language models and ChatGPT
  3. 3 Definition of a complex system
  4. 4 Teaching and learning
    1. 4.1 Topics to cover in a course
    2. 4.2 Introduction of a specific topic and the impact of it in the course
    3. 4.3 Sources
    4. 4.4 Same concept in different levels
      1. 4.4.1 Fractal
      2. 4.4.2 Phase transition
    5. 4.5 Connection between different concepts
      1. 4.5.1 Connection between chaos and fractals
      2. 4.5.2 Connection between fractals and complex networks
      3. 4.5.3 Connection between chaos and complex networks
      4. 4.5.4 Connection between time series and complex networks
      5. 4.5.5 Connections between chaos and machine learning
      6. 4.5.6 Connections between complex networks and machine learning
    6. 4.6 Learning activities
    7. 4.7 Practical laboratory experiments
    8. 4.8 Coding
  5. 5 Research
    1. 5.1 Overview of the field
      1. 5.1.1 Complex systems
      2. 5.1.2 Econophysics
      3. 5.1.3 Geographic complexity
      4. 5.1.4 Complexity in biological systems
    2. 5.2 Critiques
    3. 5.3 Chalenges
  6. 6 Final remarks
  7. 7 Acknowledgment

##  1 Introduction

ChatGPT and other transformers-based large language models (LLMs) have
demonstrated remarkable abilities to perform tasks such as completion tasks,
question answering, reading comprehension and translation that were once
thought to be exclusive to humans [1, 2, 3, 4, 5, 6, 7, 8].

We are currently in a great era for researchers and scientists studying and
developing in the field of complex systems. Half of the physics Nobel prize of
2021 was awarded to the physicist Giorgio Parisi for his contributions to the
theory of complex systems [9] and the other half to two meteorologists Syukuro
Manabe and Klaus Hasselmann to the modeling of the Earth’s climate [10].
Parisi has made significant contributions to the literature on complex
systems, including areas such as spin glass [11, 12, 13], stochastic resonance
[14], surface growth [15], multifractality [16], and bird flocking [17].
Manabe showed that there is a relation between the increase in global
temperature and the level of CO2 in the atmosphere [18]. Hasselmann developed
a model that establishes a connection between weather and climate, implying
that climate models can be reliable despite the unpredictability and
complexity of weather patterns [19, 20]. The diversity of research topics
among the award recipients highlights the multidisciplinary nature of the
field. In addition, the applied dimension of the contributions stresses the
importance of applied contributions to the field. In particular, it is worth
mentioning that the research of field of complex systems has developed
multiple frameworks, such as chaos theory [21], fractal theory [22], complex
networks [23, 24], and agent-based models [25], to model and solve important
practical problems in diverse fields.

OpenAI has just now introduced the ChatGPT, a large language model tool
trained, to aid humans in a variate of human tasks including question
answering, text edition and has been used as a great information provider [8,
26]. ChatGPT has been trained on a large dataset of internet texts, which
means that it has learned to mimic the patterns and styles of language used by
many different people online. When you talk to ChatGPT, it responds based on
what it has learned from that dataset, so its answers can be thought of as
representing the average opinions, ideas, and language patterns that are
commonly found on the internet.

In this paper, we overview different aspects of the field of complex systems
by means of chats with ChatGPT. We split this paper into two parts. The first
part explores issues related to teaching and learning, where ChatGPT may work
as a teacher assistant. The second part explores issues related to research,
where, in this case, ChatGPT works like a research assistant. Besides
exploring different concepts from different points of view in the field of
complex systems, we also recognize that ChatGPT can be a valuable resource for
teachers and scientists.

##  2 Large language models and ChatGPT

The precursors of the large language models that we know today are the
classical and sparse ones known as $n$-grams models [27, 28]. In Natural
Language Processing (NLP), $n$-gram models are probabilistic models that aim
to predict the next word in a text using a limited amount of previous context.
The value of $n$ indicates the number of words that are used to establish this
context. Mathematically speaking, the goal of an $n$-gram model is to estimate
the probability $P(t_{n}|t_{1},\cdots,t_{n-1})$ of a particular token (word),
$t_{n}$, occurring based on the history of words that have come before it,
specifically $t_{1}$ through $t_{n-1}$. We estimate these probabilities by
counting the number of times that a sequence of tokens arise in a large
corpus. We may consider, for instance, that we intend to estimate the
probability of the word “mountains” after the history “These mist covered”
that comes from the first sentence “These mist covered mountains are a home
now for me” of the song “Brothers in Arms” of the British band Dire Straits.
In order to estimate this probability, we have to count the number of times
the sequence of tokens $t_{1}$=“These”, $t_{2}$= “mist”, $t_{3}$= “covered”
arise before $t_{4}$=“mountains”. Thus, we may estimate
$P(t_{4}|t_{1},t_{2},t_{3})$ as
$\frac{P(t_{1},t_{2},t_{3},t_{4})}{P(t_{1},t_{2},t_{3})}$, where
$P(t_{1},t_{2},\cdots,t_{n})$ is the frequency of the $n$-gram
$t_{1},\cdots,t_{n}$ in the corpus.

Most popular language models today are based neural network models. Although
there are many different types of neural network models, all share some
ingredients with the Multi-Layer Perceptron (MLP)[29]. An MLP consists of
multiple layers of interconnected nodes, called “neurons”, where each neuron
receives input from the neurons in the previous layer and produces an output
that is sent to the neurons in the next layer. The input to the first layer is
typically a vector of features and the output of the last layer is the desired
output. Each neuron in an MLP computes a linear combination of its inputs and
applies a nonlinear function, called “activation function”, to the result. The
weights on the inputs determine the strength of the connections between
neurons. The nonlinear activation function is necessary to ensure a neural
network is able to approximate any continuous function [30]. We may describe
mathematically the outputs of the neurons of layer $l$ of MLP as

$y_{j}^{(l)}=h^{(l)}\left(\sum_{i=1}^{M_{l-1}}w_{ij}^{(l)}y_{i}^{(l-1)}+w_{0j}^{(l)}\right),\;j=1,\cdots,n_{l},$
(1)

where $n_{l}$ is the number of neurons of layer $l$, $w_{ij}$ is the weight
associated with the input $i$ of the (previous) layer $l-1$ and the neuron $j$
of layer $l$, $y_{i}^{(l-1)}$ is the $i$th output of the (previous) layer
$l-1$, $M_{l}$ is the number of neurons of layer $l$, and $h^{l}$ is the
activation function of layer $l$111Common types of activation functions are
tanh, sigmoide and Relu.. We call the term
$a_{j}^{(l)}=\sum_{i=1}^{M_{l-1}}w_{ij}^{(l)}y_{i}^{(l-1)}+w_{0j}^{(l)}$
activation of the neuron $j$222It is also denoted as
$a_{j}^{(l)}=\sum_{i=0}^{M_{l-1}}w_{ij}^{(l)}y_{i}^{(l-1)}$, where we
associate the bias (constant) $w_{0j}^{(l)}$ with the input
$y_{0}^{(l-1)}=1$.. The input of the first layer is $y_{i}^{0}=x_{i}$, which
is actually the input of the neural network with $M_{0}$ attributes. The
output of the neural network is $y_{i}^{nl}=y_{i}^{0}$, for $i=1,\cdots
M_{nl}$. Thus, the neural network has $M_{nl}=M_{o}$ outputs.

In order to estimate the weights (parameters) of a MLP, we need a dataset with
a pairs of inputs and outputs. Since the outputs provide a reference for what
the neural network should learn for each input, we call this kind of learning
as supervised learning. We use the gradient descent (ascent) to minimize
(maximize) the output square error (maximal likelihood associated to the
problem). Since we have only the errors (or the likelihoods) associated with
the last layer, we use the chain rule of calculus to evaluate the gradients
associated with the weights of the other layers. This is the so-called
backpropagation algorithm.

A crucial development towards the creation of large language models has been
the use of word embeddings to represent words. Word embeddings are vectors
that represent words in high dimensional Euclidean spaces [31]. The concept of
word embeddings originates from the idea that the meaning of a word can be
inferred from the context in which it appears. We may generate word embeddings
using neural network models that learn to predict a word based on its
surrounding context. One of the most common neural network models used to
generate word embeddings is the Continuous Bag-of-Words (CBOW) model [32]. The
CBOW model consists of three layers: an input layer, a hidden layer, and an
output layer. The input layer encodes the context words as one-hot vectors333A
one-hot vector is a vector that has the same dimensionality as the vocabulary
size and has exactly one element set to 1 and all other elements set to 0. The
position of the 1 corresponds to the index of the word in the vocabulary.,
while the output layer predicts the probability distribution over the
vocabulary for the target word. The hidden layer maps the input vectors to a
lower-dimensional representation, which is the word embedding. The input layer
has $C$ inputs (the size of the context used), the intermediate layer has size
$N$ (the size of the word embedding vectors) and the output layer has $N_{V}$
outputs. The size of the hidden layer, as well as the number of context words
used as input (denoted by $C$), are hyperparameters that we need to tune for
each application. Thus, we have to estimate $C$ context matrices of size
$N_{V}\times N$ and one output matrix of size $N\times N_{V}$. Note that the
problem we solve here is very similar to the one considered by the $n$-grams
models. Consider, for instance, the first sentence of the song “All by myself”
of Eric Carmen “When I was young I never needed anyone”. Suppose that we
consider a context of size 6. Thus, we use the tokens $t_{1}$=“I”
,$t_{2}$=“was”, $t_{3}$=“young ”, $t_{5}$=“never”, ,$t_{6}$=“needed” and
,$t_{7}$=“anyone” to predict ,$t_{4}$=“I”. To generate the word embeddings,
the model is trained on a large corpus of text in a supervised fashion. During
training, the weights of the network are updated to minimize the difference
between the predicted probability distribution and the actual distribution of
the target word. The actual word embedding vector that is supposedly to
represent a word is the average of the rows of the context matrices
corresponding to the word and the corresponding column of the output matrix.

Transformers form the foundation of ChatGPT and other contemporary large
language models. They are neural network models that consist of three
ingredients [33]: a layer that represent the words using word embedding
vectors, a position encoding, and a mechanism of attention. The position
encoding informs the position of each token in the input sequence and it is
added to the corresponding word embedding vectors. This is achieved through a
sinusoidal function that creates a unique encoding for each position in the
sequence. The self-attention mechanism allows the model to focus on different
parts of the input sequence depending on the task at hand. Self-attention
operates by representing the input sequence as a set of queries, keys, and
values444These concepts come from the field of information retrieval. The
query is a short piece of text, such as a keyword or phrase, that the user
enters into a search engine. The key is the piece of information associated
with the object that is being searched. The value is the object itself the
user is looking for., that are matrices that should be learned. The resulting
attention scores are then used to weight the values, producing an output that
is a weighted sum of the values, with the weights determined by the attention
scores. There is today a large list of transformers that have been used in
successful NLP tasks such as Google’s BERT [1], PEGASUS [2], T5 [3], and
Switch [4], Facebook’s BART [5], and Open AI’s Generative Pre-Training (GPT)
[6], GPT-2 [7], and GPT-3 [8].

ChatGPT is built on GPT-3 [8], that were trained using a large amount of
internet data. In general, we train language models to predict the next word,
but not to deal with questions of users. A language model may invent facts,
may present toxic behavior, or simply do not follow commands. In this case, we
say that a language model is poorly aligned. In order to deal with that Open
AI uses reinforcement learning from human feedback to align this model based
on three steps, based on the feedback of 40 people hired to label the data
[26]: (1) They collect a dataset of human-labeled comparisons between outputs
of the models; (2) They train a reward model on this dataset to predict the
output the humans prefer; (3) They use this reward model to fine tune the
model using a proximal policy optimization algorithm [34] that is type of
reinforcement learning algorithm555A reinforcement learning problem explores a
situation where the objective is to map states to actions in order to maximize
a numerical reward signal. A Policy gradient method is a reinforcement
learning algorithm that learns a parametric representation of the policy and
use gradient ascent to iteratively improve it. The proximal policy gradient
algorithm constrains the magnitude of the policy update at each iteration
reducing the risk of the policy diverging.  [35].

##  3 Definition of a complex system

The development of statistical mechanics by Maxwell, Boltzmann and Gibbs in
the second half of the $19^{th}$ century introduced probabilistic reasoning as
a new paradigm in Physics: given incomplete information regarding the initial
state of a system, one cannot predict to which state it will evolve.
Nevertheless, it is still possible to derive a probabilistic prediction of the
final state reached by the system [36], as long as the precise defintion of
the initial state is exchanged for an ensemble of initial states.

Following G. Parisi [36], we define a system as complex if its behavior
crucially depends on the details of the system. They are often composed of
many interacting elements with (emergent) properties that cannot be inferred
from the property of a single element [37, 38]. Since the behavior of a
complex system cannot be deduced without full knowledge of its basic elements
and their interactions, one again resorts to the concept of probabilistic
prediction to determine the probability distribution of behaviors given a set
of systems, each with a given set of elements and interactions among them
[36]. With such approach one can identify the most minimalist model which
reproduces features observed in many distinct models falling into the same
universality class [37].

We start this tour over the ideas, methods and contributions of the field of
complex systems asking ChatGPT for the definition of a complex system.
Although there is not a general accepted definition of a general complex
system, Figure 1 shows that ChatGPT provides a very good definition “A complex
system is a system that is made up of a large number of interconnected and
interdependent parts or components, where the behavior of the system as a
whole is not easily predictable from the behavior of the individual
components.”. It also highlights several ingredients presented in these
systems and the fields of application.

How do you define a complex system?  ChatGPT: A complex system is a system
that is made up of a large number of interconnected and interdependent parts
or components, where the behavior of the system as a whole is not easily
predictable from the behavior of the individual components. Complex systems
are characterized by non-linear interactions, emergence of new behaviors or
properties, and sensitivity to initial conditions and perturbations. Examples
of complex systems include the human brain, ecosystems, the economy, traffic
networks, and social systems. The study of complex systems involves
interdisciplinary approaches from various fields including physics, biology,
mathematics, computer science, and sociology, among others.

Figure 1: Definition of a complex system.

##  4 Teaching and learning

There are often numerous intriguing questions that arise when we seek to teach
or learn a new subject. In this section, we explore these questions by
utilizing ChatGPT as a proxy for the answers that can be found within the
complex systems community. Section 4.1 investigates which topics should we
choose for a course. Section 4.2 explores the impact of a given topic in a
course. Section 4.3 seeks for interesting sources about a subject. Section 4.4
asks how to present the same concept in different levels. Section 4.5 looks
for connections between different concepts. While Section 4.6 requests some
learning activities, Section 4.7 searches for practical laboratory
experiments. In Section 4.8, we focus on the issues related to coding.

###  4.1 Topics to cover in a course

Courses about complex systems are not very common regular courses in Physics
Departments, although there are good examples, such as the one at the Center
for the Study of Complex Systems at the University of
Michigan666https://lsa.umich.edu/cscs/graduate-students/graduate-courses.html,
the one at the Complexity Sciences Center at the University of California,
Davis777https://csc.ucdavis.edu/Courses.html, the one at the University of
Bologna888https://www.unibo.it/en/teaching/course-unit-catalogue/course-
unit/2022/433619, the one at the Abdus Salam International Centre for
Theoretical Physics999https://www.ictp.it/opportunity/international-master-
physics-complex-systems and the one at the Politecnico di
Torino101010http://www.pcs.polito.it/presentation.

Given the vast array of topics that could be deemed essential for a course on
complex systems, a key question arises: what should be selected for inclusion
in the curriculum? Thus, we ask this to ChatGPT considering a course for
third- year undergraduate physics students. Figure 2 shows that these topics
are aligned with relevant sources of the field [39, 40, 41, 42]. The topics it
suggests are Graph Theory and Network Analysis [43, 44, 45, 46, 47], Celular
Automata [48, 49, 50], Agent-Based Modeling [25, 51, 52], Chaos Theory [53,
54], Self-Organized criticality [55, 56, 57, 58, 59] and Complexity and
Information Theory [27, 60, 61, 62]. In the next question presented in Figure
3, we ask about a complex systems course for PhD students. The big differences
between this answer and the previous one are that the descriptions of the
topics are more advanced and ChatGPT also suggests the inclusion of machine
learning methods [63, 64, 65] that have become prominent in different
investigations of complex systems.

In Figure 4, we ask ChatGpt to present the necessary prerequisites for a
complex systems course for PhD students. Its answer covers basic topics of
mathematics, statistics, systems, computer programming and some topics that a
student of physics could have learned in an undergraduate course of complex
systems.

What are the recommended topics to cover in a complex systems course for
third-year undergraduate physics students? Considering the content’s level of
difficulty and relevance, what would be the most suitable sequence for these
topics?  ChatGPT: A complex systems course for third-year undergraduate
physics students should cover a range of topics that provide a comprehensive
understanding of complex systems and their behavior. The following topics are
recommended for such a course: Introduction to complex systems: This should
cover the basics of complex systems and their properties, including emergence,
self-organization, and non-linearity. Graph theory and network analysis: This
topic should introduce students to graph theory and its applications in
complex systems. It should cover topics such as random graphs, small-world
networks, scale-free networks, and community detection. Cellular automata:
This topic should introduce students to cellular automata and their
applications in modeling complex systems. It should cover topics such as the
Game of Life, the forest fire model, and the Ising model. Agent-based
modeling: This topic should introduce students to agent-based modeling and its
applications in complex systems. It should cover topics such as the Schelling
model, the Axelrod model, and the Swarm model. Chaos theory: This topic should
introduce students to chaos theory and its applications in complex systems. It
should cover topics such as the logistic map, the Lorenz system, and the
Mandelbrot set. Self-organized criticality: This topic should introduce
students to self-organized criticality and its applications in complex
systems. It should cover topics such as sandpile models, earthquakes, and
avalanches. Complexity and information theory: This topic should introduce
students to complexity and information theory and their applications in
complex systems. It should cover topics such as entropy, mutual information,
and the complexity-entropy causality plane. In terms of the sequence of
topics, it is recommended to start with an introduction to complex systems,
followed by graph theory and network analysis, and then move on to cellular
automata and agent-based modeling. Chaos theory and self-organized criticality
can be covered next, followed by complexity and information theory. However,
the sequence can be adjusted based on the instructor’s preferences and the
students’ background.

Figure 2: Topics of a course of complex systems for a third-year undergraduate
student of physics.

What if the course is for PhD students?  ChatGPT: If the complex systems
course is intended for PhD students, the recommended topics could be more
advanced and in-depth compared to a course designed for undergraduate
students. The following topics are recommended for such a course: Introduction
to complex systems: This should cover advanced topics related to complex
systems, including the philosophical and mathematical foundations of
complexity science. Network science: This topic should cover advanced topics
in network science, including community detection, network motifs, network
dynamics, and multilayer networks. Nonlinear dynamics: This topic should cover
advanced topics in nonlinear dynamics, including bifurcation theory, chaos
theory, and attractors. Information theory: This topic should cover advanced
topics in information theory, including measures of complexity, entropy, and
mutual information. Agent-based modeling: This topic should cover advanced
topics in agent-based modeling, including the design and implementation of
complex systems models, calibration and validation of models, and sensitivity
analysis. Machine learning and data analytics: This topic should cover
advanced topics in machine learning and data analytics, including clustering,
classification, regression, and deep learning, and their applications in
complex systems analysis. Complex systems applications: This topic should
cover advanced applications of complex systems in various domains, such as
social systems, biological systems, and ecological systems. In terms of the
sequence of topics, it is recommended to start with an introduction to complex
systems, followed by network science and nonlinear dynamics. Agent-based
modeling and information theory can be covered next, followed by machine
learning and data analytics. Finally, the course can conclude with advanced
applications of complex systems in various domains. However, the sequence can
be adjusted based on the instructor’s preferences and the students’
background. The level of difficulty and mathematical rigor of the course
should be appropriately adjusted to suit the PhD students’ training and
research interests.

Figure 3: Topics of a course of complex systems for a PhD student of physics.

What should be the prerequisites for this course? ChatGPT: The prerequisites
for a complex systems course depend on the course’s intended audience and the
level of difficulty of the topics covered. However, some general prerequisites
for a complex systems course are as follows: Mathematics: A strong background
in mathematics is essential for understanding complex systems. Students should
have a good understanding of calculus, linear algebra, differential equations,
probability theory, and statistics. Programming: Students should have
experience in programming, preferably in a language such as Python, MATLAB, or
R, which are commonly used in complex systems modeling and analysis. Physics:
A basic understanding of physics, including mechanics, thermodynamics, and
electromagnetism, is helpful in understanding the dynamics of complex systems.
Systems thinking: Students should have a basic understanding of systems
thinking, including feedback loops, system boundaries, and emergent behavior.
Complex systems concepts: Familiarity with the basic concepts of complex
systems, such as emergence, self-organization, and non-linearity, is helpful
in understanding the course’s content. Network theory: Familiarity with basic
network theory concepts, such as graph theory, is useful in understanding the
dynamics of complex networks. Statistics and data analysis: Familiarity with
basic statistical concepts, such as hypothesis testing and regression
analysis, and experience in data analysis are helpful in understanding complex
systems’ modeling and analysis. The specific prerequisites for a complex
systems course may vary depending on the course’s content and the instructor’s
preferences. It is recommended to consult the course syllabus or the
instructor for more detailed information on the prerequisites.

Figure 4: Prerequisites for a complex systems course in PhD level.

###  4.2 Introduction of a specific topic and the impact of it in the course

In Section 4.1, we ask about a recommendation for topics for a course on
complex systems and ChatGPT does not include, for instance, the topic on phase
transitions. Phase transitions are ubiquitous in physics [66], and specially
in complex systems [67]. So, now, in the present section, we ask about the
possibility of including it. In Figure 5, we show that ChatGPT agrees with the
inclusion of this topic and it explains that phase transitions can aid
students in understanding the collective behavior and in modeling critical
phenomena in a variety of systems. Then, in Figure 6, we ask about a specific
example of phase transitions we may consider in the course. ChatGPT suggests
that the percolation transition [68, 69, 70, 71] that illustrates several
fundamental concepts in the study of complex systems, including the idea of
criticality and the emergence of collective behavior.

What do you think about introducing the theory of phase transitions in an
introductory course of complex systems?  ChatGPT: Introducing the theory of
phase transitions in an introductory course on complex systems can be a
valuable addition to the course material. The theory of phase transitions is a
fundamental concept in the study of complex systems, and it has many
applications across different fields, including physics, chemistry, materials
science, and biology. Phase transitions refer to the abrupt changes in the
properties of a system as a result of small changes in external parameters
such as temperature, pressure, or magnetic field. The behavior of complex
systems can often be described by phase transitions, and understanding these
transitions can provide insights into the collective behavior of the system.
Moreover, phase transitions can be used to model critical phenomena in a
variety of systems, including social networks, ecological systems, and
financial markets. Therefore, introducing the theory of phase transitions in
an introductory course on complex systems can help students develop a deep
understanding of the behavior of complex systems and prepare them for further
study in related fields. Overall, I believe that introducing the theory of
phase transitions in an introductory course on complex systems can be a
valuable addition to the course material and can help students develop a
deeper understanding of the behavior of complex systems.

Figure 5: Inclusion of a topic of phase transitions in a course of complex
systems.

Do you suggest any specific example of phase transition we may use in a
complex systems course? Consider the sake of understanding and simplicity.
ChatGPT: Sure! One example of a phase transition in a complex system that is
often used in introductory courses is the percolation transition. Percolation
refers to the process by which a fluid or gas passes through a porous
material, such as soil or rock. In the context of complex systems, percolation
can also refer to the process by which a connected network of nodes or
clusters emerges in a system. The percolation transition occurs when there is
a critical probability of connections between nodes in the system, above which
a giant connected component emerges that spans the entire system. Below this
critical probability, the connected component is small and fragmented. This
transition is an example of a second-order phase transition, which means that
it is characterized by a continuous change in the system’s properties as the
probability of connections changes. The percolation transition can be used to
illustrate several fundamental concepts in the study of complex systems,
including the idea of criticality and the emergence of collective behavior. It
is a simple and intuitive example that can help students understand the
fundamental concepts of phase transitions in complex systems. Moreover,
percolation has applications in a wide range of fields, including physics,
biology, and computer science. For example, it can be used to model the spread
of diseases in a population, the flow of electrical current through a network,
or the behavior of fluid in porous materials. Overall, the percolation
transition is a useful example to introduce the theory of phase transitions in
an introductory course on complex systems. It is a simple and relatable
example that can help students understand the fundamental concepts of
criticality, emergence, and universality in complex systems.

Figure 6: Example about phase transitions that we may use in a course of
complex systems.

###  4.3 Sources

Another important dimension of teaching or self-learning is the choice of good
sources. We ask, in particular, in this section about possible sources for a
possible course in complex systems in a PhD level. ChatGPT answers this
question providing the following references: Networks, Crowds, and Markets:
Reasoning About a Highly Connected World [72], Nonlinear Dynamics and Chaos:
With Applications to Physics, Biology, Chemistry, and Engineering [54],
Introduction to the Theory of Complex Systems [42], Agent-Based and
Individual-Based Modeling: A Practical Introduction [73], Information Theory,
Inference, and Learning Algorithms [74], Machine Learning: A Probabilistic
Perspective [75] and Ecological Networks: Linking Structure to Dynamics in
Food Webs [76]. Figure 7 shows these sources with a small summary of their
content. These references are all well-known and all of them are wonderful.
However, it is worth mentioning that the field of complex systems is full of
great references that we would like to consider here. Complex Adaptive
Systems: An Introduction to Computational Models of Social Life [39] is one of
these books. This book is a helpful guide to help us to understand how to make
sense of the world with agent-based models. Another book is Complexity and
criticality [58]. This book is a very good introduction to the theme of
complexity and criticality, where a central theme of the book is scale
invariance.

What are some recommended books to cover the topics of the complex systems
course for PhD students?  ChatGPT: There are several excellent books on
complex systems that cover the topics recommended for a PhD-level course. Here
are some recommended books: ”Networks, Crowds, and Markets: Reasoning About a
Highly Connected World” by David Easley and Jon Kleinberg. This book provides
a comprehensive introduction to network science and its applications in
complex systems. ”Nonlinear Dynamics and Chaos: With Applications to Physics,
Biology, Chemistry, and Engineering” by Steven H. Strogatz. This book provides
a comprehensive introduction to nonlinear dynamics and chaos theory and their
applications in various fields. ”Introduction to the Theory of Complex
Systems” by Stefan Thurner, Rudolf Hanel, and Peter Klimek. This book provides
an advanced introduction to complex systems theory, including topics such as
self-organization, criticality, and complex networks. ”Agent-Based and
Individual-Based Modeling: A Practical Introduction” by Steven F. Railsback
and Volker Grimm. This book provides an in-depth introduction to agent-based
modeling and its applications in complex systems. ”Information Theory,
Inference, and Learning Algorithms” by David J. C. MacKay. This book provides
an advanced introduction to information theory and its applications in complex
systems. ”Machine Learning: A Probabilistic Perspective” by Kevin P. Murphy.
This book provides an in-depth introduction to machine learning and its
applications in complex systems analysis. ”Ecological Networks: Linking
Structure to Dynamics in Food Webs” by Mercedes Pascual and Jennifer A. Dunne.
This book provides an in-depth introduction to complex ecological networks and
their dynamics. These books provide a broad range of coverage of topics in
complex systems, including network science, nonlinear dynamics, agent-based
modeling, information theory, machine learning, and ecological networks.
Depending on the course’s content and the instructor’s preferences, some or
all of these books may be recommended for a PhD-level complex systems course.

Figure 7: Sources for a PhD course on complex systems.

###  4.4 Same concept in different levels

In this section, we explore the idea of presenting the same concept for
audiences of different levels. We consider two important examples that are
prominent in the field of complex systems, namely fractal and phase
transition.

####  4.4.1 Fractal

We consider here the presentation of the concept of fractals and fractal
dimension [22] for different audiences. Fractals are important for complex
systems modeling since many natural systems exhibit fractal-like patterns, and
fractal geometry (and dimension) can be used to describe and analyze the
behavior of these systems [77] Furthermore, they provide a way to describe the
system’s behavior at different levels of detail.

We ask for an introduction of these concepts in three different levels. Figure
8 shows the explanation of these concepts for a 10 year old kid, in which
ChatGPT uses an example to explain them. It describes the characteristics of
self-similarity of the snowflake without using this term in order to explain
the concept of fractal and it compares its irregular shape with the shapes of
lines, circles and squares to explain the notion of fractal dimension.

In the second case, we ask it for the same concept, but for teaching it to an
undergraduate student of physics. Figure 9 shows a more complete answer that
explores explicitly the concept of self-similarity [78]. It calls the
attention that fractal dimensions are not integer values and it also provides
the description of the box counting algorithm [79, 80, 81]. An interesting
characteristic of this answer is that ChatGPT also calls the attention for the
applications of these concepts.

In the last case, we ask for teaching this concept to a PhD student of
mathematics. As we show in Figure 10, this answer explicitly talks about the
scaling behavior of the fractal object and it provides the notion of
Hausdorff-Besicovitch dimension [82, 83, 84]. The applications are also
different from the last case. In the second case, the applications are more
applied physics focused and in the third case are more general.

Can you help me to explain the concept of fractal and fractal dimension to a
10 year old kid? ChatGPT: Sure, I’d be happy to help! Fractals are shapes that
repeat themselves at different scales. That means if you zoom in on a fractal,
you will see the same shape over and over again, getting smaller and smaller.
They often have patterns that repeat themselves in a self-similar way. A good
example of a fractal is a snowflake. If you look closely at a snowflake,
you’ll notice that it has a six-pointed pattern that repeats itself as you
zoom in. Each point has smaller points on it, and those points have even
smaller points on them, and so on. This is what makes a snowflake a fractal.
The concept of fractal dimension is a little trickier, but I’ll do my best to
explain it. Fractal dimension is a way of measuring how complicated a fractal
is. Unlike regular shapes like circles or squares, fractals have a dimension
that is not a whole number, but a fraction. For example, a line has a
dimension of 1 because it only has one dimension, length. A square has a
dimension of 2 because it has two dimensions, length and width. But a fractal
like a snowflake has a dimension that is between 1 and 2, because it has some
length, but also a lot of detail in the pattern that repeats itself. So when
we talk about the fractal dimension of a shape, we’re talking about how much
detail it has in its pattern, and how much it repeats itself at different
scales. I hope that helps! Let me know if you have any more questions.

Figure 8: The explanation of the concepts of fractal and fractal dimension to
a 10 year old kid.

How about an undergraduate student of physics? How would you explain this
concept? ChatGPT: For an undergraduate physics student, I can provide a more
technical explanation of fractals and fractal dimension. Fractals are
geometric shapes that exhibit self-similarity at different scales. This means
that if you zoom in on a fractal, you will see similar patterns repeating
themselves. For example, the Mandelbrot set is a famous fractal that looks
similar at all levels of magnification. Fractal dimension is a mathematical
concept used to describe the complexity of a fractal shape. Unlike traditional
dimensions like length, width, and height, fractal dimension is a non-integer
value that describes how much space a fractal shape occupies. This value is
typically between 1 and 2, but can be greater than 2 in certain cases. To
calculate the fractal dimension of a shape, mathematicians use a process
called box counting. They divide the shape into smaller and smaller boxes, and
count the number of boxes that contain a part of the shape. As the size of the
boxes gets smaller, the number of boxes needed to cover the shape increases.
The fractal dimension is then calculated by taking the logarithm of the number
of boxes needed, divided by the logarithm of the size of the boxes. Fractal
dimension is useful in physics because it can be used to describe the behavior
of complex systems. For example, the fractal dimension of a coastline can give
us insight into how the coast erodes over time, or the fractal dimension of a
fracturing rock can give us information about its mechanical properties. I
hope this explanation helps! Let me know if you have any further questions.

Figure 9: The explanation of the concepts of fractal and fractal dimension to
an undergraduate student of physics.

How about a phd student of mathematics? Can I give more details?  ChatGPT:
Yes, I can certainly provide more details for a PhD student of mathematics!
Fractals are objects that have a property called self-similarity, which means
that they look similar at different scales. This self-similarity can be exact
or approximate, and can occur in any dimension. Some well-known examples of
fractals include the Koch snowflake, the Sierpinski triangle, and the
Mandelbrot set. Fractal dimension is a way of measuring the scaling behavior
of a fractal object. Unlike traditional dimensions, which are integers that
describe the number of independent parameters required to specify a point in
space, fractal dimension is a real number that describes the degree of
irregularity or self-similarity of an object. There are different ways to
define fractal dimension, but one common method is through the Hausdorff-
Besicovitch dimension. This involves covering the fractal object with small
balls of radius r, and then counting the number of balls needed to cover the
object. The fractal dimension is then defined as the limit of
log(N(r))/log(1/r) as r approaches zero, where N(r) is the number of balls
needed to cover the object of radius r. Another way to define fractal
dimension is through the concept of box-counting dimension, which is similar
to the method described in the undergraduate explanation. In this method, the
fractal object is covered with boxes of size r, and the number of boxes needed
to cover the object is counted. The fractal dimension is then defined as the
limit of log(N(r))/log(1/r) as r approaches zero. Fractal dimension is an
important concept in mathematics because it allows us to measure the
complexity of fractal objects in a precise and rigorous way. It also has
applications in many fields, including geometry, topology, physics, and
computer science. I hope this explanation is helpful for a PhD student of
mathematics! Let me know if you have any further questions or if you would
like more details on any specific aspect of the concept.

Figure 10: The explanation of the concepts of fractal and fractal dimension to
a PhD student of mathematics.

####  4.4.2 Phase transition

As an another example, we can consider distinct explanations, in distinct
levels, of the concept of phase transition [66]. Phase transitions were
originally discussed in physics, in the context of liquids and magnetic
systems [66]. However, such phenomenon was after observed in a variety of
natural and artificial systems, like social, economic and biological systems.
Indeed, in such distinct complex systems phase transitions are ubiquitous
[85].

We ask ChatGPT to introduce the main idea of phase transitions to a 10 year
old kid. Figure 11 shows the explanation it produced. It uses as example the
phase transitions that occur in water (liquid/solid and liquid/gas) to
introduce the concept of form change. In other words, it discusses what
happens with a traditional substance, water, when we heat up or cold it down.

After that, we ask how ChatGPT can explain the same concept to an
undergraduate student of physics. Figure 12 shows the answer, that explores
the concepts of temperature and pressure to introduce the idea of phase
transition points [86, 87, 88]. It also develops more details of phase
transitions, like the standard classification of them in continuous or first-
order transitions [89], and explain the differences between them. Finally, it
calls our attention to the fact that phase transitions are very important in
several areas of physics.

Finally, our last question about the subject of phase transitions to ChatGPT
is how to teach this concept to a PhD student of physics. As discussed in
figure 13, ChatGPT improves the previous answer adding the concept of
criticality, which implies in the occurrence of power laws, scale invariance
and critical exponents. These last concepts are indeed very important in phase
transitions [90] and are useful to model a collection of applications [91,
92].

Can you help me to explain the concept of phase transitions to a 10 year old
kid? ChatGPT: Sure, I can help you explain the concept of phase transitions to
a 10-year-old kid! Phase transitions happen when a substance (like water)
changes from one form to another form. For example, when ice melts and becomes
liquid water, that’s a phase transition. When a substance changes from one
form to another form, it goes through a phase transition. Imagine you have a
bucket of ice. The ice is solid and very cold. But if you leave the bucket in
the sun, the ice will start to melt and become liquid water. As the ice melts,
it goes through a phase transition and becomes liquid. Another example is when
you boil water. When you heat up water in a pot, it starts to boil and turns
into steam. This is also a phase transition. Phase transitions can happen when
you heat up a substance, cool it down, or put pressure on it. And depending on
the substance and the conditions, it can go through different types of phase
transitions. But the important thing to remember is that it’s all about
changing from one form to another form!

Figure 11: The explanation of the concept of phase transitions to a 10 year
old kid.

How about an undergraduate student of physics? How would you explain this
concept? ChatGPT: For an undergraduate student of physics, I would explain
phase transitions as follows: Phase transitions are changes in the physical
properties of a substance when it undergoes a change in state, such as from a
solid to a liquid, or from a liquid to a gas. These changes occur at specific
temperatures or pressures known as phase transition points. At a phase
transition point, the substance undergoes a change in its internal structure
or order. For example, when water freezes and becomes ice, its molecules
become more ordered and form a crystalline structure. Conversely, when ice
melts and becomes liquid water, the molecules become more disordered and move
more freely. There are several types of phase transitions, including first-
order transitions and continuous transitions. First-order transitions involve
a change in the order of the substance, such as from a liquid to a solid, and
involve a change in the enthalpy (heat content) of the substance. Continuous
transitions, on the other hand, involve a change in a physical property of the
substance, such as its electrical conductivity or magnetic susceptibility, and
do not involve a change in enthalpy. Phase transitions are important in many
areas of physics, including materials science, condensed matter physics, and
statistical mechanics. Understanding the properties and behavior of phase
transitions is essential for predicting the behavior of materials under
different conditions, and for developing new materials with unique properties.

Figure 12: The explanation of the concept of phase transition to an
undergraduate student of physics.

How about a phd student of physics? Can I give more details?  ChatGPT: Sure, I
can provide more details for a PhD student of physics. In physics, phase
transitions refer to the changes in the macroscopic properties of a system
when it undergoes a change in its thermodynamic state. These changes can occur
as a result of temperature, pressure, or other external factors. At the phase
transition point, there is a sudden change in the behavior of the system, and
the properties of the system can be described by different mathematical
models. Phase transitions can be classified according to their order. First-
order phase transitions involve a change in the order of the system, such as
from a solid to a liquid, and are characterized by a latent heat of
transition. Continuous phase transitions, on the other hand, involve a change
in a physical property of the system, such as its electrical conductivity or
magnetic susceptibility, and are characterized by critical exponents. Phase
transitions are of great importance in condensed matter physics, statistical
physics, and materials science. They are used to understand the properties and
behavior of many systems, such as magnets, superconductors, and liquid
crystals. They also have practical applications in technology, such as in the
design of computer chips and the development of new materials. One of the key
concepts in the study of phase transitions is criticality. Criticality refers
to the behavior of a system at the critical point, where there is a sudden
change in the properties of the system. At the critical point, the system
exhibits scale invariance, which means that its properties are independent of
the length scale. This leads to the appearance of power-law behavior, which is
described by critical exponents. In summary, phase transitions are an
important concept in physics that describe the changes in the macroscopic
properties of a system when it undergoes a change in its thermodynamic state.
They have many applications in condensed matter physics, statistical physics,
and materials science, and are characterized by criticality and critical
exponents.

Figure 13: The explanation of the concepts of phase transition to a PhD
student of physics.

###  4.5 Connection between different concepts

One important ingredient of teaching and learning is the ability to make
connections between different fields. In this section, we investigate possible
connections between different concepts using ChatGPT as a proxy for the
complex systems and physics community. We discuss six examples: (1) Connection
between chaos and fractals; (2) Connection between fractals and complex
networks; (3) Connection between chaos and complex networks; (4) Connection
between times series and complex networks; (5) Connection between chaos and
machine learning. (6) Connection between complex networks and machine
learning.

####  4.5.1 Connection between chaos and fractals

In this subsection, we look into possible connections between the field of
chaos [21, 54] and fractals [22, 80, 81]. Figure 14 calls our attention to an
important connection between these areas. The connection is by means of the
concept of strange attractors111111This term was used in the first time by
[93]. [94]. An attractor is an invariant121212The property of invariance means
that the orbits starting in the attractor does not escape from it.,
closed131313A set that contains all limit points. and indecomposable141414The
set cannot be separated into smaller pieces. set of points, which attracts all
orbits of a dynamical system that start in some neighborhood (the domain of
attraction) [95, 96]. It is said to be strange if it has a non-integer fractal
dimension and several researchers devised method to calculate an approximate
value of this fractal dimension [97]. One interesting point is that there
actually needs an infinite number of dimensions to entirely characterize it
[98].

Are there important connections between the fields of chaos theory and
fractals?  ChatGPT: Yes, there are important connections between the fields of
chaos theory and fractals. Fractals are mathematical objects that exhibit
self-similarity at different scales. They are generated by iterative processes
and often have a highly irregular, fragmented appearance. Chaos theory, on the
other hand, studies the behavior of dynamical systems that are highly
sensitive to initial conditions. These systems exhibit unpredictable and
seemingly random behavior, even though they are deterministic and follow well-
defined rules. One of the important connections between chaos theory and
fractals is the concept of ”strange attractors”. In chaotic systems, the
behavior of the system may converge to a stable set of points, known as an
attractor. However, if the system is highly sensitive to initial conditions,
the attractor may take on a highly irregular, fractal shape. These strange
attractors can be visualized using fractal geometry and are a hallmark of
chaotic systems. Fractal geometry is also used to analyze the properties of
chaotic systems, such as their dimensionality. Fractal dimension is a measure
of the complexity of a geometric object, and it can be used to characterize
the irregular shapes of strange attractors and other features of chaotic
systems. In summary, the fields of chaos theory and fractals are closely
related and share many important concepts and techniques. The study of chaotic
systems often involves the use of fractal geometry to analyze and understand
their behavior, and the irregular shapes of strange attractors are a common
example of the connection between these fields.

Figure 14: Connection between chaos and fractals.

####  4.5.2 Connection between fractals and complex networks

In this subsection, we explore possible connections between the field of
fractals [22, 80, 81] and complex networks [43, 44, 45, 46, 47]. Besides the
fact that we can apply both fractals and complex networks to model complex
systems, Figure 14 presents the use of fractal dimensions to characterize the
self-similar behavior of complex networks. We may describe this connection
from two different strands of the literature. The first strand refers to the
literature that characterize “complex” networks151515We put complex between
quotes because at that time the term “complex network” was not actually coined
yet. embedded in Euclidean spaces, such as river networks, using fractal
dimensions [99]. An interesting point is that we may justify these scaling
properties by the theory that river networks have structures that spend
minimal energy in draining their watersheds [100]. The other strand uses
fractal dimension to characterize the self-similar structures in a large
number of complex networks [101, 102]. A compelling property of fractal
dimension associated with complex networks are that they can be used to
characterize several other attributes of the networks such as vulnerability
[103] and robustness [104].

Are there important connections between the fields of fractals and complex
networks?  ChatGPT: Yes, there are important connections between the fields of
fractals and complex networks. Fractals and complex networks both deal with
the study of complex structures and patterns in nature and in human-made
systems, and they can be used to model and understand a wide range of
phenomena. Fractals are geometric shapes that exhibit self-similarity, meaning
that they look similar at different scales. They are characterized by their
fractal dimension, which describes how the shape fills up space. Fractals are
found in many natural systems, such as coastlines, clouds, and snowflakes, and
they can also be used to model phenomena like turbulence and the behavior of
financial markets. Complex networks, on the other hand, are systems made up of
many interacting elements, such as social networks, the internet, and
biological systems. They are characterized by their topology, which describes
how the elements are connected to each other. Complex networks are studied
using tools from graph theory, and they can be used to model and understand
phenomena like the spread of diseases, the behavior of financial markets, and
the structure of the brain. There are several ways in which fractals and
complex networks are connected. For example: Fractal analysis can be used to
study the structure of complex networks. By measuring the fractal dimension of
different parts of a network, researchers can gain insights into its topology
and how it evolves over time. Complex networks can be modeled using fractals.
For example, researchers have used fractal geometry to model the structure of
river networks and the internet. Fractal and complex network models can be
combined to study complex systems. For example, researchers have used fractal
networks to model the spread of diseases and the behavior of financial
markets. Overall, the connections between fractals and complex networks are an
exciting area of research that has the potential to yield new insights into
the structure and behavior of complex systems in nature and in human-made
systems.

Figure 15: Connection between fractals and complex networks.

####  4.5.3 Connection between chaos and complex networks

In this subsection, we make connections between the fields of chaos [21, 54]
and complex networks [43, 44, 45, 46, 47]. Besides the joint usage of chaos
theory and complex networks to the study of pattern formation, Figure 16
points that chaos theory can be used to study important dynamic properties of
complex, such as the emergence of synchronization and stability [105, 106,
107, 108, 109].

However, there are other interesting attempts that connect these two fields.
The main idea is to build a complex network from a chaotic dynamical system.
Different methods have been used. We may for instance represent each cycle by
a node of the network [110] or we may represent regions of space by nodes
[111]. The advantage of modeling chaotic systems using complex networks is
that we may apply the tools developed to study complex networks [112] to
understand chaotic systems.

Are there important connections between the fields of chaos theory and complex
networks?  ChatGPT: Yes, there are important connections between the fields of
chaos theory and complex networks. Chaos theory is a branch of mathematics
that deals with nonlinear dynamical systems, which exhibit sensitive
dependence on initial conditions, meaning that small differences in initial
conditions can lead to vastly different outcomes over time. Complex networks,
on the other hand, refer to systems consisting of many interacting elements,
such as social networks, biological networks, and communication networks. The
study of complex networks has revealed that many real-world networks exhibit a
small-world property, meaning that they have both high local clustering and
short path lengths between nodes. This property is often attributed to the
presence of hubs, highly connected nodes that act as major conduits for
information flow within the network. Chaos theory has been used to study the
dynamics of complex networks, including the emergence of synchronization and
the stability of network dynamics. One important concept in this area is the
synchronization of chaotic systems, which occurs when two or more chaotic
systems with similar dynamics become coordinated. This phenomenon has been
observed in a variety of natural and engineered systems, including neural
networks, power grids, and chemical oscillators. Another area of research that
has benefited from the connection between chaos theory and complex networks is
the study of pattern formation. Many complex systems exhibit patterns that
emerge from the interaction of their constituent elements, and the dynamics of
these patterns can often be described using nonlinear differential equations.
Chaotic systems can exhibit a wide range of patterns, including spatiotemporal
chaos and fractal structures, and understanding the underlying mechanisms
behind these patterns can provide insights into the behavior of complex
systems. Overall, the connections between chaos theory and complex networks
provide a rich framework for studying the behavior of complex systems, and
have led to numerous insights into the dynamics of natural and engineered
systems.

Figure 16: Connection between chaos and complex networks.

####  4.5.4 Connection between time series and complex networks

In this subsection, we investigate possible connections between the field of
time series [113, 114, 115] and complex networks [43, 44, 45, 46, 47]. Figure
17 points out that we may use complex networks to analyze time series data
using tools such as the visibility graph [116] and recurrence network methods
[117]. As we mentioned in Section 4.5.3, the advantage of these pseudo-created
networks is to use techniques of the field of complex networks to model time
series. On the other hand, as Figure 17 also calls our attention, we may
naturally use times series tools for analyzing the evolution of complex
networks [118].

Are there important connections between the fields of complex networks and
time series analysis?  ChatGPT: Yes, there are important connections between
the field of complex networks and time series analysis. In fact, these two
fields are closely related and have been the subject of much research in
recent years. Complex networks represent systems consisting of interconnected
entities, such as social networks, transportation networks, and biological
networks. Time series analysis, on the other hand, deals with the study of
data collected over time, such as stock prices, weather patterns, and
physiological signals. One of the main connections between these two fields is
that complex networks can be used to model and analyze time series data. In
particular, researchers have developed a variety of techniques for
representing time series data as networks, such as the visibility graph and
recurrence network methods. These techniques involve mapping the time series
data onto a network, where the nodes represent the individual data points and
the edges represent relationships between them. Once the time series data has
been represented as a network, researchers can use techniques from network
science to analyze the data. For example, they can use measures such as degree
centrality, clustering coefficient, and community structure to identify
important features of the data and gain insights into its underlying dynamics.
Conversely, time series analysis can be used to study the dynamics of complex
networks. In particular, researchers have used time series analysis techniques
to analyze the temporal evolution of networks, such as how the connectivity
between nodes changes over time. Overall, the connections between complex
networks and time series analysis are important and have led to many new
insights and techniques for analyzing and understanding complex systems.

Figure 17: Connection between time series and complex networks.

####  4.5.5 Connections between chaos and machine learning

In this subsection, we explore possible connections between the field of chaos
[21, 54] and machine learning [75, 119, 120, 121]. Figure 18 points out that
we may use machine learning to model and forecast chaotic behavior [122, 123,
124]. However, it is not available in ChatGPT answer that chaos theory can
help us to understand the dynamical behavior of recurrent neural networks
behaviors [125, 126, 127].

Are there important connections between the fields of chaos and machine
learning?  ChatGPT: Yes, there are important connections between the fields of
chaos and machine learning. Chaos theory is concerned with the study of
complex, nonlinear systems that exhibit sensitive dependence on initial
conditions. These systems are characterized by their unpredictable behavior
over time, making it difficult to make accurate predictions about their future
behavior. Machine learning, on the other hand, is concerned with developing
algorithms and models that enable computers to learn from data and make
predictions or decisions based on that data. Machine learning algorithms are
used in a wide range of applications, including image recognition, natural
language processing, and predictive modeling. One of the key connections
between chaos and machine learning is that both fields deal with complex
systems that exhibit nonlinear behavior. Machine learning algorithms can be
used to model and analyze chaotic systems, enabling researchers to better
understand the dynamics of these systems and make more accurate predictions
about their behavior. Another connection between the two fields is the use of
neural networks. Neural networks are a class of machine learning algorithms
that are loosely inspired by the structure and function of the human brain.
They are capable of learning complex, nonlinear relationships between input
and output data and can be used to model chaotic systems. In summary, the
connections between the fields of chaos and machine learning are significant
and continue to be an area of active research. By leveraging the tools and
techniques from both fields, researchers can gain new insights into complex
systems and develop more accurate predictive models.

Figure 18: Connection between chaos and machine learning.

####  4.5.6 Connections between complex networks and machine learning

In this subsection, we look into the connections between complex networks [43,
44, 45, 46, 47] and machine learning [75, 119, 120, 121]. As we may find in
Figure 19, we may use machine learning algorithms to identify structures in a
large network such as communities and important nodes. For instance,
conventional clustering algorithms may be used to identify communities [128]
and reinforcement learning [129, 130, 131] may be used to identify important
nodes. On the other hand, supervised machine learning methods that depend on
attributes that come from the given network may be used for tasks as node
classification [132] or link prediction [133, 134]. There is today many
machine learning methods that can be addapted to solve diverse complex
networks problems [64].

Are there important connections between the fields of complex networks and
machine learning?  ChatGPT: Yes, there are important connections between the
fields of complex networks and machine learning. Complex networks are a branch
of network science that studies the behavior of complex systems represented as
graphs, where nodes represent entities and edges represent their relationships
or interactions. Machine learning, on the other hand, is a field of computer
science that focuses on the development of algorithms and models that enable
computers to learn from data and make predictions or decisions without being
explicitly programmed. Complex networks and machine learning are closely
related because many real-world systems can be represented as complex
networks, and machine learning techniques can be used to analyze and model the
behavior of these networks. For example, machine learning algorithms can be
used to identify patterns and structures in large-scale network data, such as
community structures, important nodes or links, and network dynamics. On the
other hand, complex network analysis can also provide useful insights and
features for machine learning tasks. For example, network-based features can
be used as input to machine learning algorithms for prediction tasks, such as
link prediction, node classification, and community detection. Additionally,
network analysis techniques can be used to preprocess data for machine
learning algorithms, such as data reduction or feature extraction. In summary,
the fields of complex networks and machine learning are highly interrelated,
and both have much to offer in terms of insights, models, and algorithms for
understanding and analyzing complex systems.

Figure 19: Connection between complex networks and machine learning.

###  4.6 Learning activities

Learning activities, such as hands-on and experiential learning experiences,
encourage active learning by engaging students. In this section, we explore
the possibility of ChatGPT suggesting an interesting learning activity to help
individuals learn an important concept in the complex systems field. Thus, we
ask ChatGPT a learning activity to teach children about consensus in opinion
dynamics [135] and it suggests a game that the objective is to reach a
consensus.

What is a learning activity that can be used to teach children about consensus
in opinion models?  ChatGPT: A great learning activity to teach children about
consensus in opinion models is to use a group decision-making game. Here’s an
example of a simple game you can use: Divide the children into small groups of
3-4. Give each group a problem to solve or a decision to make. For example,
you could ask them to choose a color for a new classroom poster or decide on a
game to play during recess. Ask each child in the group to write down their
individual choice on a piece of paper. Have the children share their choices
with each other, and encourage them to discuss why they made their choice. Ask
the group to work together to come up with a single choice that everyone can
agree on. Encourage them to listen to each other’s opinions and try to find
common ground. Once the group has reached a consensus, ask them to share their
final decision with the class. This activity will help children understand the
importance of listening to others and working together to reach a common goal.
It will also teach them about the concept of consensus and how it can be used
to make decisions as a group.

Figure 20: Learning activity to understand the concept of consensus.

###  4.7 Practical laboratory experiments

In addition to lectures and other learning activities, teachers may schedule
laboratory experiments as practical activities to help students learn new
concepts in the complex systems field.

First, we want to know how to design practical laboratory experiments in the
field of complex systems and, then, we ask for the suggestion of such
experiments. Figure 21 presents the basic steps for designing practical
laboratory experiments. It considers the identification of the concepts we
want to demonstrate, the correct choice of equipment and materials, a previous
test of the experiment, safety issues, interactiveness, data visualization and
simplicity.

Second, we want to know what specific experiments we should consider. Figure
22 shows some interesting suggestions, such as the bouncing ball experiment
[136, 137] to demonstrate sensitivity to initial conditions ubiquitous in
chaotic systems, a traffic flow simulation [138, 139, 140] to show the
emergent behavior of complex systems, a network analysis [112, 141], such as
social [23, 142] or transportation networks [143, 129], to investigate the
structure of a complex network, a pattern formation in a chemical reaction to
reveal the formation of complex patterns from simple chemical reactions [144,
145], and a swarm robotics [146, 147] experiment where robots work together to
realize a task to show how complex systems are formed from simple components.

How can I design practical laboratory experiments in the field of complex
systems that effectively demonstrate key concepts and principles?  ChatGPT:
Designing practical laboratory experiments in the field of complex systems
requires careful consideration of a variety of factors, such as the specific
principles and concepts that you want to demonstrate, the equipment and
materials available, and the level of experience of your audience. Here are
some general tips for designing effective laboratory experiments in this
field: Identify the key concepts and principles you want to demonstrate: Start
by clearly defining the concepts and principles you want to teach, and make
sure they are appropriate for the level of your audience. Complex systems can
encompass a wide range of topics, such as chaos theory, network dynamics, or
emergence, so it’s important to choose a few key concepts to focus on. Choose
appropriate equipment and materials: Based on the concepts you want to teach,
select the appropriate equipment and materials for your experiment. You may
need specialized software, sensors, or data collection tools to conduct your
experiment effectively. Test your experiment beforehand: Before conducting the
experiment in front of your audience, it’s essential to test it beforehand to
ensure that it works as intended. This will also give you an opportunity to
identify any potential problems and make adjustments as needed. Ensure safety:
When working with laboratory equipment, safety should always be a top
priority. Make sure you and your audience are familiar with any safety
protocols or precautions, and take steps to minimize risks. Make it
interactive: Complex systems can be abstract and difficult to visualize, so
it’s important to make your experiment as interactive as possible. Encourage
your audience to ask questions and participate in the experiment, and provide
opportunities for them to manipulate variables and see how the system
responds. Use data visualization: Complex systems often generate large amounts
of data, so it’s important to use data visualization tools to help your
audience understand the patterns and trends that emerge. This can include
graphs, charts, or other visual representations of the data. Keep it simple:
While complex systems can be fascinating, it’s important to keep your
experiment simple and focused on a few key concepts. Too many variables or too
much complexity can be overwhelming for your audience and may make it
difficult for them to understand the principles you’re trying to teach. By
following these tips, you can design practical laboratory experiments in the
field of complex systems that effectively demonstrate key concepts and
principles. Remember to always be mindful of safety and to tailor your
experiments to the level of your audience.

Figure 21: Design of laboratory experiments.

Can you suggest some practical experiments? ChatGPT: Sure, here are a few
examples of practical experiments that you could conduct in the field of
complex systems: Bouncing ball experiment: This experiment is a classic
example of chaos theory and demonstrates how small changes in initial
conditions can lead to vastly different outcomes over time. In this
experiment, you drop a ball onto a surface and measure the height of each
bounce. By making small changes to the initial conditions (such as the height
of the drop or the angle of the surface), you can observe how the behavior of
the system changes over time. Traffic flow simulation: This experiment
demonstrates how simple rules can lead to emergent behavior in complex
systems. In this experiment, you create a simulation of traffic flow using a
simple set of rules (such as maintaining a safe following distance and
adjusting speed based on traffic density). By changing the parameters of the
simulation, you can observe how the behavior of the system changes over time.
Network analysis: This experiment involves analyzing the structure and
behavior of networks (such as social networks or transportation networks) to
understand how information or resources flow through them. You could use
network analysis software to analyze the structure of a network and identify
key nodes or pathways. Pattern formation in chemical reactions: This
experiment demonstrates how complex patterns can emerge from simple chemical
reactions. In this experiment, you mix two or more chemicals and observe how
the reaction progresses over time. By changing the concentrations of the
chemicals or the temperature of the reaction, you can observe how different
patterns emerge. Swarm robotics: This experiment involves programming a group
of robots to work together to accomplish a task (such as moving objects or
exploring an environment). By observing how the robots interact and how their
behavior changes over time, you can gain insights into how complex systems can
be built from simple components. These are just a few examples of practical
experiments in the field of complex systems. Depending on your specific
interests and the resources available to you, there are many other experiments
you could design and conduct.

Figure 22: Suggestions of laboratory experiments.

###  4.8 Coding

An important dimension of teaching and learning complex systems theory is
coding. Figure 23 shows the Python libraries that may be useful for complex
systems’ researchers. It points to NumPy [148] for numerical tools, SciPy
[149] for statistical tools, NetworkX [150] for complex networks modelling,
Matplotlib [151] for plotting, Pandas [152] for struturing and analysing data,
SymPy [153] for symbolic tools, PyCX [154] for complex systems simulations and
Scikit-learn [155] for machine learning models. Although it provides a great
portfolio of libraries, we miss the Mesa library [156], that is a great
library for agent-based models simulation. Thus, we ask about the Mesa
library. Figure 24 shows the relevance of this library for complex systems
researchers. Finally, we ask ChatGPT to provide an example of use of the
NetworkX library. Figures 25, 26 and 27 provide a simples example of building,
characterizing and plotting a social network for self-learning161616It is
worth mentioning that the code it provides has a minor execution error, since
the social network it builds is not fully connected and we need this property
to evaluate the average shortest path. Basically, we have to include the pair
(’Frank’,’Alice’) in the list friendships before running, but we note that the
syntatix is perfect..

Which python libraries may be useful for researchers of the field of complex
systems? ChatGPT: Python has a rich set of libraries that can be useful for
researchers working in the field of complex systems. Here are some of the most
commonly used ones: NumPy: This library provides support for large, multi-
dimensional arrays and matrices, and offers a range of mathematical functions
to work with them. This is particularly useful for complex systems research
where large amounts of data need to be processed and analyzed. SciPy: This
library builds on top of NumPy and provides additional functions for
scientific and technical computing, such as optimization, integration, and
linear algebra. NetworkX: This library provides tools for the creation,
manipulation, and study of complex networks. It offers functions for
generating random graphs, computing network metrics, and visualizing network
structures. Matplotlib: This library provides a wide range of tools for
creating visualizations and plots, which can be used to illustrate complex
system behavior. Pandas: This library provides tools for data manipulation and
analysis. It offers data structures for efficient handling of large datasets
and functions for statistical analysis. SymPy: This library provides support
for symbolic mathematics, which is particularly useful for researchers working
in the field of mathematical modeling of complex systems. PyCX: This library
provides a set of tools for creating simulations of complex systems, which can
help researchers to explore and analyze the behavior of such systems. Scikit-
learn: This library provides a range of tools for machine learning, which can
be useful for analyzing and predicting the behavior of complex systems. There
are many more libraries available for Python, but these are some of the most
commonly used ones for researchers working in the field of complex systems.

Figure 23: List of Python libraries useful for complex systems’ researchers.

Don’t you think the mesa library can be useful? ChatGPT: Yes, the Mesa library
can be very useful for researchers working in the field of complex systems.
Mesa is a Python library for agent-based modeling (ABM), which is a powerful
technique for simulating and analyzing the behavior of complex systems that
are made up of individual agents that interact with one another. Mesa provides
a framework for creating, running, and analyzing agent-based models, and
includes a range of tools and utilities for building and visualizing these
models. Some of the key features of Mesa include: A simple and flexible API
for defining agents and their behaviors, which allows researchers to easily
implement and test new models. Built-in tools for creating visualizations and
animations of the models, which can be useful for understanding the dynamics
of complex systems. A range of utility functions for analyzing and visualizing
the data generated by the models, which can help researchers to identify
patterns and trends in the behavior of the agents. Support for parallel
execution, which allows the models to be run efficiently on multi-core
processors and distributed computing clusters. Some examples of applications
of the Mesa library include modeling the spread of diseases, analyzing the
behavior of financial markets, and simulating the interactions between animals
in an ecosystem. Overall, if you are working in the field of complex systems
and are interested in agent-based modeling, the Mesa library can be a valuable
tool for building and analyzing models of complex systems.

Figure 24: The relevance of Mesa library.

Can you provide an interesting example of the use of the library NetworkX?
ChatGPT: Sure! Here’s an example of how NetworkX can be used to analyze the
structure of a social network: Suppose we have data on the friendships between
a group of individuals, and we want to analyze the structure of the resulting
social network. We can represent the network as a graph, where each individual
is a node and each friendship is an edge. We can use NetworkX to create this
graph and analyze its properties. Here’s how we can do it: First, we need to
create the graph. We can use the Graph() function from NetworkX to create an
empty graph, and then add nodes and edges to it based on our data. Here’s some
code to do this:

    import networkx as nx

    # Create an empty graph
    G = nx.Graph()

    # Add nodes to the graph for each individual
    individuals = [’Alice’, ’Bob’, ’Charlie’, ’David’,
                   ’Eve’, ’Frank’]
    for i in individuals:
        G.add_node(i)

    # Add edges to the graph for each friendship
    friendships = [(’Alice’, ’Bob’), (’Alice’, ’Charlie’),
                    (’Bob’, ’Charlie’), (’David’, ’Eve’),
                    (’Eve’, ’Frank’)]
    for i, j in friendships:
        G.add_edge(i, j)

    %\end{minted}

Figure 25: Example with the NetworkX library: creation of the network.

Once we have created the graph, we can analyze its properties. For example, we
can compute the degree distribution of the nodes in the graph, which tells us
how many edges each node has. We can also compute the average shortest path
length between nodes, which tells us how connected the graph is. Here’s some
code to do this:

    # Compute the degree distribution
    degrees = [d for n, d in G.degree()]
    degree_counts = nx.degree_histogram(G)

    # Compute the average shortest path length
    avg_shortest_path = nx.average_shortest_path_length(G)

    # Print the results
    print(’Degree distribution:’, degree_counts)
    print(’Average shortest path length:’, avg_shortest_path)

    %\end{minted}

Figure 26: Example with the NetworkX library: characterization of the network.

Finally, we can visualize the graph using NetworkX’s built-in plotting
functions. We can use a spring layout to position the nodes, which will make
the graph look more natural. Here’s some code to do this:

    import matplotlib.pyplot as plt

    # Compute the layout using a spring layout algorithm
    pos = nx.spring_layout(G)

    # Plot the graph
    nx.draw_networkx_nodes(G, pos, node_color=’lightblue’)
    nx.draw_networkx_edges(G, pos, edge_color=’gray’)
    nx.draw_networkx_labels(G, pos)
    plt.axis(’off’)
    plt.show()

    %\end{minted}

This code will display a visualization of the social network, where each node
represents an individual and each edge represents a friendship. The layout of
the nodes is determined by a spring layout algorithm, which tries to position
the nodes so that they are evenly spaced and not too close together. We can
use this visualization to gain insights into the structure of the social
network and to identify key individuals or communities within it

Figure 27: Example with the NetworkX library: plot of the network.

##  5 Research

Researchers, specially those who are outsiders to a particular field, face
regularly the challenge of keeping up with the latest developments and
understanding the subject matter. In this section, we use ChatGPT as a proxy
for the answers that can be found within the physics and complex systems
community to spot the purpose, trends, critiques and challenges of the field.
Section 5.1 presents an overview of the complex systems field identifying the
purpose and trends of the field. Besides focusing only on the entire complex
systems field, we also explore the subfields of econophysics, geographic
complexity and complexity in biological systems. While in Section 5.2, we
investigate the critiques to the field, in Section 5.3, we discuss the
challenges of the field.

###  5.1 Overview of the field

We split this subsection into three subsections. While Subsection 5.1.1
explores the field of complex systems, Subsections 5.1.2 and 5.1.3 investigate
respectively the subfields of econophysics and geographic complexity.

####  5.1.1 Complex systems

We start asking ChatGPT the purpose of the field of complex systems. Figure 28
shows the basic relevant ingredients of the field [39, 40, 41, 42], which is
very connected with the definition presented in Section 3. It also calls the
attention that the field is a multidisciplinar one and it may be applied in
different disciplines. It concludes that another important function of the
field is to develop strategies for managing and controlling complex systems.

In the following, we ask ChatGPT about the current trends and developments of
the field. We may find in Figure 29 the answer to this question that
highlights the trends of interest in network science [43, 44, 45, 46, 47],
artificial intelligence and machine learning [63, 64, 65], agent-based
modelling [25, 51, 52], complexity economics [157, 158, 159, 160, 161] and
interdisciplinary collaboration [162, 163]. Note that the first three topics
are topics already suggested as topics for a course on complex systems in
Section 4.1. Complexity economics or Econophysics is a subfield of the field
of complex systems and we explore it in Section 5.1.2. Finally, it is
noteworthy that the trend of interdisciplinary collaboration in complex
systems is gaining momentum, as experts from various fields recognize the
importance of complex systems methods and seek to apply them in their own
work. In addition, collaboration from other fields is highly encouraged, as
the participation of experts from diverse fields can enhance the credibility
and reputation of research in the complex systems field.

What is the purpose of the complex systems field? ChatGPT: The purpose of the
complex systems field is to study and understand the behavior of complex
systems, which are systems made up of many interconnected and interdependent
components that exhibit non-linear, dynamic, and often unpredictable behavior.
The complex systems field seeks to develop theories, models, and methods to
analyze and predict the behavior of complex systems, such as the economy,
ecosystems, social networks, biological systems, and many others. The field
draws on a wide range of disciplines, including mathematics, physics, computer
science, biology, and social sciences, to develop a multidisciplinary approach
to understanding complex systems. One of the goals of the complex systems
field is to identify patterns and emergent properties that arise from the
interactions of the individual components of a complex system. These patterns
and properties may not be apparent when looking at the individual components
in isolation, but can be observed at the system-level. Another important
purpose of the complex systems field is to develop strategies for managing and
controlling complex systems, especially in the face of uncertainty and change.
This may involve identifying critical components or feedback loops within a
system that can be targeted for intervention, or developing adaptive
strategies that can respond to changes in the system over time.

Figure 28: The purpose of the complex systems field.

Which are the current trends and developments in the complex systems field?
ChatGPT: The complex systems field is a rapidly evolving and dynamic field,
and there are many current trends and developments that are shaping the
direction of research and applications in the field. Some of these trends and
developments include: Network Science: Network science is a subfield of
complex systems that focuses on the study of complex networks, which are
systems of interconnected entities. Network science is being applied to a wide
range of fields, including social networks, transportation networks,
biological networks, and many others. Artificial Intelligence and Machine
Learning: Advances in artificial intelligence and machine learning are
enabling researchers to develop more sophisticated models of complex systems,
and to better understand the behavior of these systems. AI and machine
learning techniques are also being used to develop more effective strategies
for managing and controlling complex systems. Agent-Based Modeling: Agent-
based modeling is a simulation technique that is used to study the behavior of
complex systems by modeling the behavior of individual agents within the
system. Agent-based modeling is being used to study a wide range of complex
systems, including ecosystems, transportation systems, and social networks.
Complexity Economics: Complexity economics is a relatively new field that
applies complex systems thinking to the study of economic systems. Complexity
economics seeks to develop a better understanding of the behavior of economic
systems, including how they evolve and how they respond to shocks and changes.
Interdisciplinary Collaboration: Collaboration between researchers from
different disciplines is becoming increasingly important in the complex
systems field. By bringing together experts from different fields, researchers
can develop more comprehensive and integrated approaches to studying complex
systems. Overall, the complex systems field is a rapidly evolving and dynamic
field that is making important contributions to our understanding of complex
systems and their behavior. As new tools and techniques continue to emerge, it
is likely that the field will continue to grow and develop in new and exciting
ways.

Figure 29: Current trends and developments of the complex systems field.

####  5.1.2 Econophysics

In this section, we explore the field of econophysics that is a subfield of
the field of complex systems. We start by asking ChatGPT about the purpose of
Econophysics. We show the answer to this question in Figure 30, where it
asserts (with a very complete answer) that econophysics is that uses tools of
complex systems to investigate economic behavior [157, 158, 159, 160, 161,
164].

Next, we ask ChatGPT for the possible trends in Econophysics. Figure 31
stresses the trends of interest in network theory [165, 166, 167], machine
learning [168, 169], agent-based modelling [170, 171], nonlinear-dynamics
[172, 173] and complexity economics [174, 175].

We know that the study of persistence in financial time series is largely
investigated [176, 177, 178, 179, 180]. Thus, we ask if this topic cannot be
considered a trend anymore. In Figure 32 mentions that although this is not a
new trend anymore, it agrees that it is an active and important topic since it
has implications for risk management and the behavior of financial markets.
Furthermore, it presents some of the techniques used to run these
investigations such as the Hurst exponent [181, 80] and Detrended Fluctuation
Analysis (DFA) [182, 183].

In the following, we ask about a possible impact of the field of econophysics
in mainstream economics. Figure 33 presents the answer to this question that
defends that measuring this impact is difficult, since the field is still
relatively new and its ideas are not yet widely accepted in the mainstream.
Anyway, it suggests three criteria, namely the amount of citations that
econophysics papers receive from mainstream economics papers, the amount of
collaboration between the two fields and the extent econophysics models are
adopted by mainstream economics. We have to agree that these criteria are
really good suggestions.

Thus, we ask ChatGPT the reason that the ideas of econophysics are not widely
accepted in mainstream economics. Figure 34 shows the answer to this question
that provides four different reasons: (1) Different approaches (methods from
physics versus assumptions about individual agent behavior); (2) Lack of
empirical validation171717This is the classical critique to agent-based
models. However, it is worth mentioning that in the last few years, some
researchers have proposed solutions to this issue [184, 185, 186].; (3)
Resistance to change; (4) Lack of familiarity with the methods used in
econophysics. We agree with all of them. In particular, due to the difficulty
to run large scale experiments in economics, the field is very conservative
and, therefore, resistant to change. However, we want to highlight that maybe
the first item is the most relevant. Although this is not explicit in the
first item, the primary objective of mainstream economics is to model the
choice behavior of agents and many of the econophysics models do not follow
this objective [187, 188].

In spite of the resistance of mainstream economics to the field of
econophysics, it is worth mentioning that some works published in high impact
economic journals could be by members of the community of econophysics. Among
them we may cite Self-organized criticality and economic fluctuations [189],
The granular origins of aggregate fluctuations [190] and The dynamics of
inequality [191].

What is the purpose of econophysics?  ChatGPT: Econophysics is a field of
study that applies concepts and tools from physics to economic systems. Its
purpose is to use methods from physics, such as statistical mechanics, network
theory, and complex systems theory, to model and analyze complex economic
phenomena. The goal of econophysics is to develop a deeper understanding of
economic systems, including financial markets, income distributions, and
macroeconomic dynamics, by studying them as complex systems. By doing so,
econophysicists hope to uncover underlying patterns and structures that may be
difficult to detect using traditional economic approaches. Econophysics also
aims to provide insights into the behavior of economic systems during times of
crisis, such as stock market crashes or economic recessions, and to develop
models that can help policymakers make more informed decisions.

Figure 30: The purpose of the econophysics field.

Which are the current trends and developments in econophysics? ChatGPT: There
are several current trends and developments in the field of econophysics, some
of which are: Network theory: Econophysicists are increasingly using network
theory to study economic systems. This involves modeling economic agents as
nodes in a network, and the connections between them as edges. By analyzing
the properties of these networks, researchers can gain insights into the
structure of economic systems, including financial markets. Machine learning:
Machine learning techniques are being increasingly used in econophysics to
analyze large and complex data sets. This includes techniques such as
clustering, regression, and deep learning. Agent-based modeling: Agent-based
modeling is a technique used in econophysics to simulate the behavior of
economic agents, such as investors or firms, in a complex system. This allows
researchers to explore the emergent properties of the system and make
predictions about its behavior. Nonlinear dynamics: Nonlinear dynamics is a
branch of mathematics that deals with the behavior of systems that are
sensitive to initial conditions. Econophysicists are using this approach to
study the dynamics of financial markets, including the emergence of bubbles
and crashes. Complexity economics: Complexity economics is a new approach that
seeks to understand economic systems as complex adaptive systems. This
involves studying the interactions between economic agents and the emergence
of patterns and structures that are not easily predicted from individual
behavior. Overall, these trends and developments in econophysics are helping
to provide new insights into the behavior of economic systems and inform
policy decisions.

Figure 31: Current trends and developments of the econophysics field.

How about the topic that investigates persistence in financial time series?
Cannot it be considered a trend anymore? ChatGPT: The investigation of
persistence in financial time series is still a relevant and active area of
research in econophysics. Persistence refers to the tendency of financial
variables, such as asset prices or volatility, to exhibit long-term
correlations over time. This is an important phenomenon to understand because
it can have implications for risk management and the behavior of financial
markets. Econophysicists have developed several techniques to analyze
persistence in financial time series, such as detrended fluctuation analysis
(DFA) and the Hurst exponent. These techniques can help to identify whether a
time series exhibits long-term correlations and to quantify the degree of
persistence. While the investigation of persistence in financial time series
is not a new trend in econophysics, it remains an important topic of study.
There is ongoing research aimed at understanding the causes of persistence,
its effects on financial markets, and how it can be modeled and predicted.
Therefore, this topic can still be considered an active area of research in
the field of econophysics.

Figure 32: The relevance of the topic that investigates persistence in
financial time series.

Is it possible to measure the impact of econophysics in mainstream economics?
ChatGPT: Measuring the impact of econophysics on mainstream economics can be
challenging, as the field is still relatively new and its ideas are not yet
widely accepted in the mainstream. However, there are some ways to assess its
impact: Citation analysis: One way to measure the impact of econophysics on
mainstream economics is to analyze the number of citations that econophysics
papers receive from mainstream economics journals. While this is not a perfect
measure, it can give an indication of the level of attention and recognition
that econophysics is receiving. Collaboration and interdisciplinary research:
Another way to measure the impact of econophysics is to look at the level of
collaboration and interdisciplinary research between econophysicists and
mainstream economists. If there are more joint research projects or
conferences, it could be an indication of a growing interest in econophysics
from mainstream economics. Adoption of econophysics models: A third way to
measure the impact of econophysics is to look at the extent to which
econophysics models and concepts are being adopted in mainstream economic
research. For example, if mainstream economists are using network theory or
agent-based models in their research, it could be an indication that
econophysics is having an impact. Overall, while it may be difficult to
measure the impact of econophysics on mainstream economics, there are some
indications that its ideas and models are gaining attention and recognition in
the field.

Figure 33: Impact of the field of econophysics in mainstream economics.

Why the ideas of econophysics are not widely accepted in mainstream economics?
ChatGPT: The ideas of econophysics are not widely accepted in mainstream
economics for several reasons: Different approaches: Econophysics and
mainstream economics have different approaches to studying economic systems.
Econophysics uses methods from physics and complex systems theory to model and
analyze economic phenomena, while mainstream economics typically uses
mathematical models based on assumptions about individual behavior. Lack of
empirical validation: Some mainstream economists are skeptical of econophysics
because many of its models have not yet been empirically validated. While
econophysics models have shown promise in explaining certain phenomena, such
as stock market behavior, there is still much work to be done to validate
these models and demonstrate their usefulness. Resistance to change: Like any
field, mainstream economics has its established ways of doing things, and it
can be difficult for new ideas and approaches to gain acceptance. Econophysics
is a relatively new field, and it may take time for its ideas to be accepted
and integrated into mainstream economics. Complexity and interdisciplinary
nature: Econophysics is an interdisciplinary field that draws on ideas and
methods from physics, mathematics, and computer science, among other
disciplines. This can make it difficult for economists who are not familiar
with these fields to understand and appreciate econophysics models and ideas.
Overall, while there are several reasons why the ideas of econophysics are not
widely accepted in mainstream economics, there is growing interest in the
field and the potential for it to provide new insights into economic
phenomena.

Figure 34: The reasons that econophysics’ ideas are widelly accepted in
economics.

####  5.1.3 Geographic complexity

In this section, we explore another subfield of the complex systems field
known as geographic complexity [192, 193, 194, 195]. As in the case of
econophysics considered in Section 5.1.2, we start by asking ChatGPT about the
purpose of this field. Figure 35 shows the answer to this question that says
it is an interdisciplinary field with the objective of investigating how
physical and human systems interact with each other across various scales of
space and time. There other interesting ingredients in the answer to this
question, such as the types of phenomena the field intends to model and the
methodological approaches used in the field. The field uses methods from
geographic information science [196], spatial analysis [197], complexity
theory [198], and simulation modeling to investigate phenomena such as climate
change [199], land-use change [200, 201], natural disasters [202], economic
development [203, 204], and urbanization [205, 206, 207, 208].

In the following, we ask about the current trends and developments of this
field. Figure 36 shows the answer to this question that points the following
trends: big data and machine learning [209, 210], integration of physical and
social systems [211], multi-scale analysis [212, 213], collaborative modeling
[214] and Open science and reproducibility [215, 216]. The trend of utilizing
big data and machine learning techniques is widespread due to the increased
availability of large datasets and advancements in data analysis methods. The
trend of open science and reproducibility is also a widespread phenomenon in
empirical sciences caused by the reproducibility crisis. In addition, it
fosters transparency and efficiency that saves time and resources. The trend
of collaborative modeling is associated with the same reasons of the trend
found in the complex systems literature. The trend of big data seems to be
continuous trend in this literature that explores geographic phenomena at
different scales.

What is the purpose of the field known as geographic complexity?  ChatGPT:
Geographic complexity is an interdisciplinary field that seeks to understand
the complex and dynamic relationships between physical and human systems at
different spatial and temporal scales. The purpose of this field is to provide
insights into the complex nature of geographical phenomena, such as climate
change, land-use change, natural disasters, economic development, and
urbanization, among others. Geographic complexity draws on various theoretical
and methodological approaches, including geographic information science,
spatial analysis, complexity theory, and simulation modeling, to understand
the interplay between natural and human systems. It also seeks to provide
practical solutions to real-world problems related to spatial planning,
resource management, and sustainable development. Overall, the purpose of
geographic complexity is to provide a more comprehensive and nuanced
understanding of the complex spatial dynamics that shape our world, and to
develop strategies that promote more sustainable and equitable outcomes for
people and the planet.

Figure 35: The purpose of the geographic complexity field.

Which are the current trends and developments in the geographic complexity
field? ChatGPT: There are several current trends and developments in the
geographic complexity field. Here are a few examples: Big data and machine
learning: The availability of large and complex datasets, combined with
advances in machine learning algorithms, is revolutionizing the way
geographers analyze and model complex spatial phenomena. This has led to the
development of new tools and techniques for spatial analysis, such as deep
learning, network analysis, and agent-based modeling. Integration of physical
and social systems: One of the key challenges in the geographic complexity
field is to understand the interplay between physical and social systems.
Recent developments in this area have focused on developing more integrated
models that can capture the feedback loops and dynamics between these systems,
such as land-use change and urbanization. Multi-scale analysis: Many
geographic phenomena operate at different spatial and temporal scales, and
understanding how they interact requires a multi-scale approach. Advances in
GIS and remote sensing technologies have made it possible to collect and
analyze data at increasingly fine scales, while also providing new tools for
integrating data across scales. Collaborative modeling: Collaborative modeling
involves engaging stakeholders and experts from different fields to develop
models and scenarios that reflect their diverse perspectives and goals. This
approach has gained popularity in recent years as a way to develop more robust
and effective strategies for spatial planning and resource management. Open
science and reproducibility: The geographic complexity field has increasingly
embraced open science principles, which promote the sharing of data, code, and
methods to promote transparency and reproducibility in research. This has led
to the development of new tools and platforms for sharing and collaborating on
spatial data and models, such as GitHub and OpenGeoHub. Overall, the
geographic complexity field is rapidly evolving, driven by advances in
technology, data, and theory. These developments are leading to new insights
and approaches for understanding and addressing complex spatial problems, and
are likely to have significant impacts on a wide range of fields, from urban
planning to environmental science.

Figure 36: Current trends and developments of the geographic complexity field.

####  5.1.4 Complexity in biological systems

In this section, we explore the subfield of complexity in biological systems
[217, 218]. As in the case of Sections 5.1.2 and 5.1.3, we start by asking
ChatGPT about the purpose of this field. Figure 37 shows the answer to this
question that says that this field investigates how complex biological systems
emerge from the interactions of simpler components. Life, the emergence of
“order from order” [219, 220, 221], is undoubtedly a complex process [217].
From organisms to cells, tissues, animals, societies and ecological
environments, information organizes in a hierarchical way with interactions
involving a wide range of scales, both in space and time [222, 220]. Examples
include the mutation accumulation theory of aging [223, 224, 225], neural
networks, the networks of metabolic reactions inside a cell [226] and
regulatory genes promoting proteins involved in physiological responses [227],
the latter belonging to a new field called ”Systems Biology” [221]

In the following, we ask about the current trends and developments of this
field. Figure 38 shows the answer to this question that points the following
trends Multi-omics analysis [228, 229], Single-cell analysis [230], Machine
learning and artificial intelligence [231, 232], Network medicine [233, 234]
and Synthetic biology [235, 236].

What is the purpose of the field that studies complexity in biological
systems?  ChatGPT: What is the purpose of the field that studies complexity in
biological systems? The purpose of the field that studies complexity in
biological systems is to understand how complex biological systems, such as
cells, organisms, and ecosystems, emerge from the interactions of simpler
components, such as molecules, cells, and organisms. This field, called
”systems biology,” seeks to understand the underlying principles that govern
the behavior of biological systems, including how they adapt to changing
environments, communicate with each other, and evolve over time. Systems
biology combines experimental and theoretical approaches to study biological
systems as a whole, rather than focusing on individual components in
isolation. By analyzing large datasets and

Figure 37: The purpose of the field that studies complexity in biological
systems.

Which are the current trends and developments in the field that studies
complexity in biological systems? ChatGPT: The field that studies complexity
in biological systems, or systems biology, is constantly evolving and there
are several current trends and developments. Here are a few examples: Multi-
omics analysis: One of the most important trends in systems biology is the
integration of multiple ”omics” technologies, such as genomics,
transcriptomics, proteomics, and metabolomics, to gain a more comprehensive
understanding of biological systems. By combining data from these different
omics platforms, researchers can identify novel regulatory networks and
interactions that would be difficult to detect with a single approach. Single-
cell analysis: Advances in single-cell technologies, such as single-cell RNA
sequencing, are enabling researchers to study biological systems at an
unprecedented level of detail. By analyzing gene expression and other
molecular features of individual cells, systems biologists can gain insights
into cellular heterogeneity, cell signaling, and the dynamics of biological
systems. Machine learning and artificial intelligence: With the increasing
availability of large-scale data sets, machine learning and artificial
intelligence (AI) methods are becoming increasingly important in systems
biology. These approaches can help identify patterns and relationships in
complex data sets, and can be used to build predictive models of biological
systems. Network medicine: Network medicine is a relatively new field that
aims to integrate systems biology approaches with clinical data to develop
more personalized approaches to diagnosis, treatment, and prevention of
disease. By analyzing the interactions between genes, proteins, and other
molecules in disease networks, network medicine researchers can identify novel
drug targets and biomarkers for disease. Synthetic biology: Synthetic biology
involves designing and constructing novel biological systems or modifying
existing ones for specific applications. Systems biology approaches are
critical for understanding and optimizing synthetic biological systems, and
advances in synthetic biology are providing new tools for studying and
manipulating biological systems.

Figure 38: Current trends and developments of the field that studies
complexity in biological systems.

###  5.2 Critiques

In this subsection, as a proxy to the community, we turn to ChatGPT to explore
the criticisms that the complex systems field has faced, following our earlier
attempt to provide an overview of the field in Section 5.1.1. Figure 39 shows
a very relevant spectrum of the critiques received by the field: lack of
rigorous mathematical foundations, lack of experimental validation,
overreliance on reductionism, Lack of clarity on what constitutes a complex
system and lack of progress in solving real-world problems. All these
critiques deserve special attention, since they are recurrent in the field.

It is unfair to say that the field lacks mathematical rigor, since the lack of
rigor is not widespread in the field. While there are some areas of the field
that depend strongly on simulations, there are other areas that the
mathematical rigor dominates. Among the areas with more rigor, we may cite the
fields of chaos theory and information theory, with an intermediate level of
rigor, we may cite the fields of complex networks and cellular automatas, and
the field that depends mostly on simulations are the agent-based models (and
less on mathematics). However, even if we consider the agent-based models,
there are many attempts to become them more rigorous. For instance, while
Challet and other physicists introduced the minority game181818A minority game
is a simple computational model for the collective behavior of agents where
they compete through adaptation for scarce resources. It has been applied to
investigate relevant issues, such as the dynamics of financial markets [237,
238], the effects of information in transportation networks [239, 240], the
concept of intelligence [241], the dynamics of auctions [242] and strategies
in deregulated markets [243].  using computer simulations [244, 245] to study
the El-Farol Bar problem [246], there is today a mathematical basis for this
model [247, 248]. Furthermore, in general in the field of agent based models,
the works [249] and [250] use the statistics of experiment design to make the
results of agent-based models more credible.

The lack of experimental validation is not also widespread in the field
either. In particular, we should mention the list of practical laboratory
experiments presented in Section 4.7 and many others that may be conceived
following the same lines. Furthermore, a big fraction of the literature of
complex systems depend on data and its consequent validation. Furthermore,
even in the case of the agent based models, as we mention in Section 5.1.2,
there are today some attempts to their validation.

While there is no universal agreement on what constitutes a complex system,
current accepted definitions are highly consistent with each other. A useful
resource for exploring these definitions is the Journal of Physics: complexity
[251], whose editors have independently provided a range of self-concordant
definitions that align closely with the one presented in Section 3.

The lack of progress in solving real-world problems in our opinion is the less
relevant critique, since models that come from the field of complex systems
have dealt with many real world problems in the fields of traffic management
[252, 253], climate modeling [254], epidemiology [255, 256], power grid
management [257, 258], ecological modeling [76, 259], information diffusion
[260, 261, 262, 263, 264, 265], political science [266, 267, 268, 269, 270]
and marketing [271, 272]. While many of these approaches depend on the
structure of a complex network to model the interaction of the actors of the
system, we may also find specific applications of fractals, such as cancer
diagnosis and therapeutic management [273] and the characterization of soil
structure [274].

What are the criticisms about the area of complex systems in physics from the
physicists of other areas?  ChatGPT: There are several criticisms of complex
systems in physics from physicists of other areas, including: Lack of rigorous
mathematical foundations: Some physicists criticize complex systems research
for being too reliant on computer simulations and not having a solid
mathematical foundation. They argue that without a firm mathematical basis,
the results of complex systems research cannot be trusted. Lack of
experimental validation: Another criticism is that much of the research in
complex systems has not been experimentally validated. Critics argue that
without empirical evidence, the findings of complex systems research are
little more than speculation. Overreliance on reductionism: Critics argue that
complex systems research is too reductionist, meaning it attempts to explain
complex phenomena by breaking them down into simpler parts. They argue that
this approach is limited and fails to capture the true complexity of the
systems being studied. Lack of clarity on what constitutes a complex system:
Some physicists argue that there is no clear definition of what constitutes a
complex system. They argue that this lack of clarity makes it difficult to
apply complex systems research to real-world problems. Lack of progress in
solving real-world problems: Critics argue that despite decades of research in
complex systems, little progress has been made in using this approach to solve
real-world problems. They argue that the field is too theoretical and lacks
practical applications. It should be noted that not all physicists hold these
criticisms, and many see the study of complex systems as a valuable and
promising area of research. However, these criticisms do exist and are worth
considering in any discussion of complex systems research.

Figure 39: Critiques to the field of complex systems.

###  5.3 Chalenges

In this subsection, we use ChatGPT to explore the challenges faced by the
complex systems field. This is in addition to our earlier overview of the
field in Subsection 5.1.1 and our examination of criticisms of the field in
Subsection 5.2.

Thus, Figure 40 presents the potential challenges of the field of complex
systems in the next few years. We may summarize the challenges in five topics:
dealing with big data, developing more accurate models, interdisciplinary
collaboration, ethical issues and bridging theory and practice. While these
topics will undoubtedly be important for complex systems research in the years
to come, it is difficult to determine their relative significance.

Dealing with big data is actually a present issue and it is also possibly a
challenge in the future. We have never produced and recorded so much data as
we do today. There are actually many specific challenges associated with the
management of big data, such as data security and privacy, data integration,
data quality, and cost of storing [275, 276, 277].

In a world with increasing complexity, the need for more accurate models is a
fundamental issue. Furthermore, the development of more accurate models can be
attributed to the advancements in machine and deep learning literature, which
have pushed the boundaries of predictability.

Interdisciplinary collaboration is a trend today as we see in Subsection 5.1.1
and it is also a challenge in the next few years. Interdisciplinary
collaboration may be an important step for the acceptance of complex systems
methods in other fields as we mention in Section 5.1.2.

Ethical issue is an important dimension in all fields that deal with data and
it has been considered seriously by researchers that develop artificial
inteligence and machine learning models [278, 279]. In addition, using the
predictions of these models for public policy implementation requires caution,
as it may have unintended negative effects on individuals or groups.
Therefore, it is crucial to consider the potential consequences before making
any decisions. In addition to the concerns surrounding the use of private data
mentioned above, if we use the predictions of these models for the
implementation of public policies [280], we need to be careful about the
consequences of these uses in order to not prejudice anyone.

We may associate the challenging of Bridging theory and practice with the
critique to the field that complex systems do not solve real world problems as
presented in Section 5.2. As we explore there, although we do not believe this
is true, we also believe that in the next few years, the complex systems field
will provide more solutions for real world problems.

What are the key challenges that researchers and practitioners in the field of
complex systems are likely to face in the next few years?  ChatGPT: As a
language model, I cannot predict the future with certainty, but based on
current trends and challenges in the field of complex systems, some of the key
challenges that researchers and practitioners are likely to face in the next
few years are: Dealing with big data: Complex systems are often characterized
by large amounts of data generated from various sources such as social media,
sensors, and simulations. Analyzing and understanding these large datasets can
be challenging, and new techniques and tools will be required to handle and
make sense of these data. Developing more accurate models: Complex systems are
inherently unpredictable, and developing accurate models that capture the
dynamics and behavior of these systems is a significant challenge. Improving
model accuracy and developing new modeling techniques will be essential for
making predictions and decisions based on complex systems. Interdisciplinary
collaboration: Complex systems are multidisciplinary, and understanding them
requires knowledge from various fields such as physics, mathematics, computer
science, biology, and social sciences. Collaborating across disciplines can be
challenging, and developing effective communication and collaboration
mechanisms will be necessary to advance the field. Addressing ethical issues:
Complex systems can have far-reaching effects on society, and as such, there
is a need to address ethical issues such as data privacy, fairness, and
accountability. Researchers and practitioners will need to develop ethical
frameworks to ensure that the development and deployment of complex systems
are guided by ethical principles. Bridging theory and practice: While the
field of complex systems has made significant advances in theory and modeling,
there is a need to bridge the gap between theory and practice. Developing
practical applications of complex systems theory and models will be essential
for addressing real-world problems such as climate change, pandemics, and
social inequalities. Overall, the challenges facing researchers and
practitioners in the field of complex systems are complex and multifaceted.
However, by developing new techniques, collaborating across disciplines,
addressing ethical issues, and bridging theory and practice, the field can
continue to advance and make significant contributions to society.

Figure 40: Challenges of the field of complex systems.

##  6 Final remarks

In this study, we have used ChatGPT as a representation of the community’s
view on the complex systems field. Through our questioning of ChatGPT, we have
generated an overview of the field that encompasses both teaching and research
topics. During these discussions, we supplemented the “community” perspective
with our own insights on the field. Our contributions have not only added new
ingredients to the “community”s ideas, but but also provided sources to
support the community’s views and also our own views.

ChatGPT can be a helpful tool for understanding the community’s ideas on
teaching and research because of its strong summarization skills. However, we
should use it carefully because if the data it uses to learn about the
community has biases, ChatGPT may also learn and reproduce those biases.
Furthermore, the popularity bias may also affect ChatGPT’s answers paying
attention to the most popular issues instead of necessarily the most important
ones.

##  7 Acknowledgment

DOC (grant number 302629/2019-0) and NC (grant number 310893/2020-8) thank
CNPq for partial financial support. NC also thanks FAPERJ (grant number
203.217/2017). We want to reinforce that ChatGPT has written its all answers.

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