Physics

VITTORIO LORETO
FRANCESCA TRIA

A - Knowledge and understanding
OF 1) To possess a basic knowledge of complexity science, i.e. the collective properties that emerge with a large number of interacting components (atoms, particles or bacteria in a physical or biological context, or people, machines or businesses in a socio-economic context).
OF 2) Understanding the mechanisms underlying the emergence of complex macroscopic properties from knowledge of microscopic mechanisms.
OF 3) Mastering the basic toolbox of a complexity scientist: information theory, network theory, scale invariance and critical phenomena, properties of dynamical systems, agent models.
B - Application skills
OF 4) Knowing how to devise simple models for complex phenomenologies.
OF 5) Being able to tackle complex problems analytically or computationally, translating research questions into concrete solution and verification actions.
OF 6) Being able to apply the techniques and methods learnt also outside the areas covered in the course.
OF 7) Integrating the knowledge acquired in order to formalise problems and obtain results and predictions of increasing accuracy.
C - Autonomy of judgment
OF 8) Being able to analyse phenomena, also through the acquisition of data and evidence, that fall within the scope of complexity and identify their essential elements.
OF 9) Being able to synthesise phenomenologies in order to be able to distill relevant and relevant questions.
OF 10) Being able to identify interesting new research directions.
D - Communication skills
OF 11) Being able to communicate complex issues in a simple way, focusing on the essential elements and revealing cause-effect relationships as far as possible.
OF 12) Being able to organise a coherent, profound yet comprehensible presentation.
OF 13) Knowing how to express one's thoughts in a way that stimulates group work and interaction with colleagues.
E - Ability to learn
OF 14) Have the ability to consult reference texts and articles.
OF 15) Being able to assess the relevance of results, their place in the scientific panorama of reference and their potential importance for the research topics of interest.
OF 16) Being able to design and develop a research project, identifying the main objectives and the possible paths to reach them.

A - Knowledge and understanding
OF 1) To possess a basic knowledge of complexity science, i.e. the collective properties that emerge with a large number of interacting components (atoms, particles or bacteria in a physical or biological context, or people, machines or businesses in a socio-economic context).
OF 2) Understanding the mechanisms underlying the emergence of complex macroscopic properties from knowledge of microscopic mechanisms.
OF 3) Mastering the basic toolbox of a complexity scientist: information theory, network theory, scale invariance and critical phenomena, properties of dynamical systems, agent models.
B - Application skills
OF 4) Knowing how to devise simple models for complex phenomenologies.
OF 5) Being able to tackle complex problems analytically or computationally, translating research questions into concrete solution and verification actions.
OF 6) Being able to apply the techniques and methods learnt also outside the areas covered in the course.
OF 7) Integrating the knowledge acquired in order to formalise problems and obtain results and predictions of increasing accuracy.
C - Autonomy of judgment
OF 8) Being able to analyse phenomena, also through the acquisition of data and evidence, that fall within the scope of complexity and identify their essential elements.
OF 9) Being able to synthesise phenomenologies in order to be able to distill relevant and relevant questions.
OF 10) Being able to identify interesting new research directions.
D - Communication skills
OF 11) Being able to communicate complex issues in a simple way, focusing on the essential elements and revealing cause-effect relationships as far as possible.
OF 12) Being able to organise a coherent, profound yet comprehensible presentation.
OF 13) Knowing how to express one's thoughts in a way that stimulates group work and interaction with colleagues.
E - Ability to learn
OF 14) Have the ability to consult reference texts and articles.
OF 15) Being able to assess the relevance of results, their place in the scientific panorama of reference and their potential importance for the research topics of interest.
OF 16) Being able to design and develop a research project, identifying the main objectives and the possible paths to reach them.

a) It is essential to have a good grounding in statistical physics, calculus of probability and physics of dynamic systems.
b) Good analytical and computational knowledge is important.
c) Notions of data science and a strong inclination towards theoretical modelling are useful.

Introduction to Complex Systems: Hierarchical structure of nature, reductionism and complexity. Complex and complicated. Introduction to the notion of entropy and complexity in Thermodynamics and Statistical Physics, Dynamic Systems Theory, Information Theory and Algorithmic Complexity Theory. Maximum entropy principle and maximum likelihood principle. Power laws: Power law distributions, properties and representations. Zipf's law, Heaps' law and its relation with Zipf's law. Taylor's law. Generative models. Critical self-organisation and fractal growth: physical models of self-organised fractal growth; Sandpile model and Forest Fire; Hints about mean-field theories and renormalization group approaches. Graphs and complex networks: Examples of complex networks in different areas; Historical perspective; Basics of graph theory; Generative models;. Dynamics on networks. Innovation dynamics: Motivation and definition; The Hoppe model. Discussion on the adjacent possible principle. Urn model with triggering and its applications. Seminars on specific research topics related to complex systems: Social dynamics (e.g. consensus dynamics in linguistics, dynamics of opinions, cultural evolution); Information dynamics (infosphere, recommendation systems, misinformation and echo chambers, etc.); Urban dynamics (from the microscopic level of mobility to the macroscopic level of modelling socio-economic interactions); Econophysics and Economic Complexity; Neural networks and Artificial Intelligence.

The reference texts and teaching materials will be indicated by the teachers during the course and reported on the course web page.

All details can be found on the course webpage:

https://sites.google.com/site/sistemicomplessisapienza/course-of-physics-of-complex-systems?authuser=0

Attendance at all classes is strongly recommended.

The final examination will consist of an oral test in two consecutive stages. (i) the presentation of a
paper on an advanced topic chosen by the student and agreed with one of the lecturers. (ii) a series of open questions on the fundamental topics of the course. In the assessment of the examination, the determination of the final mark takes into account the following elements:

1. Competence in the core topics of the course: 60%.
2. Preparation and presentation of the thesis: 40%

In order to pass the examination, a mark of at least 18/30 is required.
In order to obtain a mark of 30/30 with distinction, the student must demonstrate that he/she has acquired an excellent knowledge of all the topics covered in the course, is able to link them in a logical and coherent manner and has shown initiative and ability in the preparation of the thesis. He/she also has to show independence of judgement, initiative and critical capacity in general.

-Information theory: Derivation of the Shannon entropy for a discrete probability distribution and related properties. Block entropy, differential entropy and Shannon entropy for a stochastic process. Asymptotic equipartition property (AEP) (Shannon-McMillan-Breiman theorem). Typical sequences. Cross entropy and relative entropy (Kullback-Leibler divergence). Positivity of the Kullback-Leibler divergence: Jensen inequality. Mutual Information. Discussion on the entropy of a language (e.g., English language): block entropy on n-grams and its empirical estimate, Shannon game and proportional gambling. Discussion on symbols codes: non-singular codes, uniquely decodable codes, prefix-free codes, and complete codes. Kraft’s equality and inequality. Bounds on the average length of optimal symbols codes. Huffman code. Algorithmic complexity (Kolmogorov complexity) and its relation with the Shannon entropy. Universal codes and Lempel-Ziv code (LZ77). Demonstration of its optimality. Kac’s lemma on the average recurrence time.

-Inference: Maximum entropy principle. General formulation and examples: probability distributions of microcanonical and canonical ensemble through maximum entropy principle. Exponential, Gaussian and power law distributions through the maximum entropy principle. Inferring the parameters by imposing the constraints. Extracting random variables from a probability distribution. A special case of the Gaussian distribution: the Box-Muller algorithm. Numerical methods: acceptance-rejection method, Markov Chain Monte Carlo Method. Master equation and detailed balance. Young’s argument for convergence to the equilibrium distribution in the case of detailed balance. Metropolis algorithm and Gibbs sampling (heat bath). Maximum likelihood principle. Equivalence of the maximum likelihood principle with the minimization of the Kullback-Leibler divergence between the experimental and the theoretical probability distribution. Bayesian inference. Inference of the success probability of a Bernoulli process.

-Power laws: scale invariance, probability distribution and moments computation. Log-log scale for power laws' representation. Logarithmic binning. Representation of a power law distribution through the cumulative function. Frequency-rank distribution and Zipf’s law. Multiplicative processes and the log-normal distribution. Power law distribution through a combination of exponentials. Yule-Simon model. Solution through continuous time approximation. Heaps’ law and its relation with Zipf’s law.

-Graph theory: Graphs representations. Diameter and clustering coefficient of a graph. Random graphs. Diameter in random graphs. Neighbour's degree distribution in random graphs. Clustering coefficient in a random graph. The small world property and the Watts and Strogatz model. Scale-free graphs. The Barabasi-Albert model. Configuration model. Clustering coefficient and diameter in scale-free uncorrelated networks.

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