TIMETABLE:

Wednesday 16.00 -19.00 - Aula Careri

Thursday 16.00-19.00- Aula Careri

Lectures:

01/10/2025  (3 h)

Introduction to complex systems. 

Accompanying slides on the Prolusion [PDF]

02/10/2025  (3 h)

(1) Introduction to probability density functions (pdf). Observables and non-normalizable distributions. Distribution of a dependent variable and relation with the generation of random numbers. The sum of two random variables. Characteristic functions and Convolution theorem. Cumulants for the sum and the average of i.i.d random variables. (2) Moments for the sum and the average of i.i.d. random variables. Derivation of the Central Limit Theorem for distributions with well-defined moments. (3) Introduction to power-laws: normalisation and representations. Generation of random variables with a generic probability distribution: the case of power-laws. Criterion of Maximum Likelihood. Application to the estimation of the exponent of a power-law.

08/10/2025  (3 h)

(1) Frequency-rank plot and its relation with the pdf.  Again, on the criterion of Maximum Likelihood. Estimation of the exponent of a power-law and the error on the estimate. Estimate of the maximum value of a sample drawn from a power-law pdf. (2)  Probability distribution functions for extreme events drawn from a power-law distribution. Fréchet distribution. Cauchy-Lorentz distribution. Calculation of the first and second moments. Computation of the characteristic function and proof that a sum of Cauchy-distributed random variables is again Cauchy-distributed. (3) Lévy's distribution: calculation of the moments. Stable distributions and alpha-stable distributions. Conditions on the exponents of a power-law pdf for the existence of the moments. 

09/10/2025  (3 h)

(1) Central Limit Theorem for long-tailed distributions with well-defined variance. Central Limit Theorem for long-tailed distributions without a well-defined variance: the case of the Cauchy distribution as an alpha-stable distribution. (2) Benford's law. Empirical observations, scaling arguments and explanation based on multiplicative processes. More on multiplicative processes and log-normal distributions.  Zipf's law: empirical observations, normalisation depending on the value of the exponent. (3) Relationship between the Zipf's exponent and the exponent of the usual probability distribution function. Heaps' law: empirical observations. Taylor's law. Example of its derivation for a Poissonian process. 

15/10/2025  (3 h)

(1) First notions on scaling: Galileo (1638) and examples from biology and cities. Scale-invariance and power-laws. Mechanisms leading to the emergence of power-laws: combination of exponentials: Monkey typing, (2) inverse quantities. Yule-Simon process: calculation of the probability distribution of frequencies and (3) the frequency rank distribution. 

16/10/2025  (3 h)

Again on the Yule-Simon model. Solution with the quenched version of the model: at each time step, m species are reinforced, and one brand-new species is introduced. First-return times in a one-dimensional Random Walk. Introduction to critical phenomena. Example of percolation: Phenomenology and Real Space Renormalisation group in 1-D and for a 2-D triangular lattice. Self-Organised Criticality (SOC). Introduction to the overall phenomenology. Criticality vs. Self-Organised Criticality: The role of time-scale separations in SOC is to be tuned at the critical point. Sandpile model.

Accompanying slides about Power-laws and Scale Invariance [POWER_LAWS.pdf]

22/10/2025  (3 h)

(1) Self-Organised Criticality (SOC). Introduction to the overall phenomenology. Sandpile model. Definition of the main quantities. Real Space Renormalisation Group for the Sandpile model and computation of the critical exponents. (2) Introduction to network science. Phenomenology in several domains: technological networks, information networks, and social networks. (3) Basic notions of graph theory: undirected and directed networks, unweighted and weighted networks, single and multiple edges, self-edges, cycles, Directed Acyclic Graphs (DAG). Adjacency matrix. Bipartite networks and Incidence matrix. Projections of bipartite networks. Trees. Degree, in-degree, out-degree, mean degree, density. 

23/10/2025  (3 h)

(1) Walks, paths and loops and their computation in terms of the adjacency matrix. Laplacian matrix and its properties. (2) Eigenvalues and eigenvectors. Random walks on networks. Centrality measures: degree, eigenvector centrality, (3) Katz centrality. PageRank centrality. Betweenness centrality. 

29/10/2025  (3 h)

(1) Transitivity and clustering coefficient. Assortativity. Random graphs G(n,m) and G(n,p). (2) G(n,p) and its properties: diameter, size of the giant component. Configuration model. Knn assortativity and its interpretation in the configuration model. (3) Friendship paradox and excess degree distribution. Models of network formation. Preferential attachment. Barabasi-Albert model: definition.

30/10/2025  (3 h)

(1) Barabasi-Albert model: definition and calculation of the degree distribution via two different methods: Differential equation for the evolution of the degree. Master equations for the in-degree and the total degree distribution. (2) First notions of epidemic spreading. SI and SIR models under the homogeneous mixing hypothesis. Graphical solutions and identification of the epidemic threshold. Basic reproduction number and its link with the epidemic threshold. (3) Epidemics on networks. Transmission probability and mapping the epidemic threshold to the bond percolation. The SIR model on networks. Calculation of the epidemic threshold for random graphs and power-law degree distributions. Considerations on the zero value that the epidemic threshold can acquire for specific networks. Size of the giant component.

Accompanying slides about Networks [PDF]

05/11/2025  (3 h)

Discussion on the general problem of defining a measure for the information content of a “source” of messages or a stochastic process. Definition of the Shannon entropy for a discrete probability distribution. Case of equiprobable states and the general case. Discussion of the Shannon entropy of a Bernoulli distribution. Shannon construction and derivation from the central limit theorem. Equivalence of the Shannon entropy and the thermodynamic entropy in the case of the microcanonical and canonical ensembles. Maximum entropy principle. Derivation of the probability distribution of states in the microcanonical and canonical ensembles from the maximum entropy principle.

Asymptotic equipartition property (AEP). The typical set.

06/11/2025  (3 h)

Discussion on the general problem of encoding. Consequences of AEP on the minimal average length of the encoding of messages from sources producing sequences of i.i.d. random variables. Joint and conditional entropy. Block entropies, differential entropies, and Shannon entropy per character (or entropy rate) of stationary stochastic processes. Shannon-McMillan-Breiman theorem (without demonstration) and its consequences for sequences generated by stationary stochastic processes. Cross-entropy and relative entropy or Kullback-Leibler divergence. Jensen inequality for discrete probability distributions. Positivity of the relative entropy.

12/11/2025  (3 h)

Mutual information. Demonstration that conditioning lowers the entropy.  Symbol codes. Non-singular, uniquely decodable and instantaneous (prefix-free) codes. Codewords in prefix-free codes are leaves of binary trees. Kraft equality in rooted binary trees. Kraft inequality for prefix-free codes. Entropy bounds for the average expected length of prefix- free codes. The Huffman code.

13/11/2025  (3 h)

Entropy estimation. Estimated N-block entropy is a lower bound for the true N-block entropy. Entropy and gambling. The horse races and the maximal doubling rate. The Shannon Game and the Cover and Thomas experiment with proportional gambling to estimate the entropy of the English language. Algorithmic (Kolmogorov) complexity. Definition and its relation to the Shannon entropy. Demonstration of the Kraft inequality for uniquely decodable codes.

19/11/2025  (2 h)

Compressors. LZ77 and its asymptotical optimality. The entropy of a continuous random variable (differential entropy). The entropy of the uniform distribution and of the Gaussian distribution. The Jensen inequality for continuous random variables and the positivity of the Kullback-Leibler divergence for probability densities. Derivation of the exponential and the power law distributions through the maximum entropy principle. Imposing the constraints in the maximum entropy principle and the maximum likelihood principle.

20/11/2025  (2 h)

Consistency of the maximum likelihood principle. Maximum likelihood and Bayesian estimation of the probability of a Bernoulli variable. Uniform prior and Beta prior. The two colours Polya urn process. Joint probability of a sequence generated by the Polya urn process. Exchangeability property. Mean, variance and two-point correlator in the Polya urn process. Mean and variance of the fraction of one colour balls.

Please refer to the Learning Material for the reference book and the PDF files.

26/11/2025 (3h) (1) Prolusion to the study of social dynamics. Historical perspective and main challenges. Models for the formation of consensus. Main mechanisms (social pressure, imitation, homophily, etc.). (2) Kinetic Ising model at zero temperature. Voter model. Equivalence of the Voter model in 1d with the Kinetic Ising Model at T=0 in 1-d. Fokker-Planck equation for the Voter model in the mean field.  Detailed analysis of the Fokker-Planck equation for the Voter model in the mean field. (3) Derivation of the Exit probability and of the Consensus time. Comparison with the Ising model in the mean field regime and with a local field composed of k randomly chosen neighbours. 

Accompanying slides about Social Dynamics [PDF]

27/11/2025 (3h) (1) Classification of Opinion dynamics models. A few examples: the majority rule model, the Sznajd model, and the Deffuant model. Axelrod's model for cultural assimilation. Discussion on the 1st and 2nd order phase transitions. (2+3) Introduction to language dynamics. Naming Game model: consensus and fragmentation. 

Accompanying slides about Language Dynamics [PDF]

3/12/2025 (3 h) Specialistic seminar on "AI and Augmented Creativity" [PDF]

4/12/2025 (3 h)Specialistic seminar on "Infosphere" [PDF]

10/12/2025 (3 h) Specialistic seminar on "Economic Fitness and Complexity" [PDF]

11/12/2025 (3 h) Specialistic seminar on "Sustainable Cities" [PDF] 

17/12/2025 (3 h)  Innovation Dynamics

18/12/2025 (3 h) Innovation Dynamics