By the end of this course, from a knowledge and understanding perspective, students will be able to: ● Describe the graph-theoretic representations of networked system such as adjacency and Laplacian matrices. ● Explain the properties of positive matrices, e.g., the Perron–Frobenius and Gershgorin's theorems, and their role in network dynamics. ● Illustrate the principles of consensus algorithms, including iterative averaging and its convergence conditions. ● Describe the connection between Markov chains, random walks on graphs and consensus dynamics. ● Explain how network topology determines collective behavior in models of opinion dynamics and epidemic spread. ● Formulate the problems of distributed estimation and synchronization of multi-agent systems. Regarding the practical application of the acquired knowledge, students will be able to: ● Construct and manipulate graph models of networked systems in MATLAB®, including random-graph generation and spectral analysis of Laplacian matrices. ● Simulate consensus protocols and synchronization schemes, evaluate their convergence using tools from control theory. ● Model and simulate opinion-dynamics processes (from consensus to disagreement) and epidemic-spread models over networks. ● Design distributed control strategies for multi-agent systems, including formation control and coordinated motion of mobile robots. ● Critically interpret simulation results to assess the impact of network structure on system performance.

For successful completion of this course, the following knowledge and skills are required: ● Linear Algebra: operations on vectors and matrices, complex-number arithmetic, eigenvalues and eigenvectors. ● Calculus: ordinary and partial derivatives, Jacobian matrix, integrals. ● Linear control theory in time and frequency domains: linearization, stability analysis, Laplace transform, transfer functions. Recommended (not mandatory): ♦ Basics of ordinary differential equations. ♦ Basics of probability theory. ♦ Basics of discrete-time systems. Students who have not completed a foundational course in Automatic Control, Controlli Automatici, or equivalent are strongly advised to do so before enrolling in this course.

The course is structured in four modules: ● Module 1 — Foundations (15h lectures + 6h labs) ○ MATLAB fundamentals: solving continuous- and discrete-time systems. ○ Probability basics: random-variable generation and simulation in MATLAB. ○ Algebraic graph theory: adjacency and Laplacian matrices. ○ Advanced matrix analysis: properties of nonnegative matrices, the Perron–Frobenius theorem and Gershgorin's circle theorem. ○ Models of large-scale random graphs. ● Module 2 — Consensus dynamics and Markov chains (25h lectures + 8h labs) ○ Markov chains and random walks on graphs. ○ Consensus via iterative averaging. ○ Applications to control of vehicles and mobile robots. ○ Applications to distributed estimation and computing. ○ Social dynamics: consensus and disagreement. ● Module 3 — Epidemic models (10h lectures + 3h labs) ○ SI, SIS, SIR models, well-posedness and basic properties. ○ Bass model of innovation spread. ○ Deterministic and stochastic epidemics models on networks. ● Module 4 — Synchronization of dynamical systems (10h lectures + 3h labs) ○ Controlled synchronization of identical linear systems. ○ Passivity-based synchronization.

The course is organized into three main activities: Lectures (60h) cover the theoretical foundations — graph-theoretic concepts, state-space models, analysis and control of multi-agent systems, and dynamical processes on networks — augmented by numerical examples and fully worked problems. Laboratory sessions (20h) provide hands-on experience with MATLAB® for the simulation, analysis and design of networked control systems and graph-based processes. Optional group project is typically undertaken by groups of 3–4 students. Each project involves reading a research paper and conducting a numerical simulation of a networked system (e.g., modeling of epidemic spread or opinion formation in a social network). Groups submit a brief report (no more than 10 pages) by a specified deadline and give a 15-minute presentation at the end of the semester.

Reference materials: ● Lecture slides and laboratory handouts, available on the course webpage via the teaching portal. ● F. Bullo, Lectures on Network Systems, freely available at https://fbullo.github.io/lns/ Recommended for further study: ♦ D. Easley and J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, available at https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book.pdf ♦ A.L. Barabási, Network Science, readable online at http://networksciencebook.com/ ♦ F.L. Lewis, H. Zhang, K. Hengster-Movric, A. Das, Cooperative Control of Multi-Agent Systems, Springer, 2014. ♦ W. Mei et al., "On the dynamics of deterministic epidemic propagation over networks," Annual Reviews in Control, vol. 44, pp. 116–128, 2017.

Lecture slides; Exercises; Lab exercises;

Exam: Computer lab-based test; Group project;

The computer-based exam is held in the university's computer lab (LAIB). The exam is designed to verify the acquisition of the knowledge and abilities described in the learning outcomes, including: graph-theoretic modeling of networks, consensus synchronization analysis, distributed control design, epidemic and opinion dynamics models. The format is the same across all examination sessions. ● QUIZ (necessary and sufficient to pass the exam) The computer-based quiz lasts 2 hours and consists of 8 problems, which may be multiple-choice or open-ended. Each problem is worth up to 4 points if fully solved (32 points total). Partial credit is available for open-ended questions. Wrong answers are NOT penalized. Some problems require MATLAB® for their solution. Students must not use their own laptops, tablets or other electronic devices. Use of written or printed materials (books, lecture notes, printed MATLAB® scripts) is strictly prohibited. The quiz can be repeated in any examination session, according to general session rules. ♦ Optional project (group activity — for additional credit) The project is worth a maximum of 6 points, with up to 3 points for the presentation and up to 3 points for the report. The report submission (with all simulation script attached) is mandatory. FINAL MARK: ● Quiz only: the final grade equals the quiz score, capped at 31 points (31 = 30 e lode) ● Quiz + project: the final grade is the sum of the quiz and project scores, capped at 30. The distinction "30 e lode" is awarded when the quiz mark is 28 or above, and the total mark exceeds 30. For example: a student scoring 26 on the quiz and 6 on the project receives a final mark of 30. A student scoring 28 on the Quiz and 3 on the project gets 30 e lode. A student scoring 32 on the quiz without a project receives 30 e lode. (a) If a student prefers the Quiz Only option, their final grade is solely based on the quiz score, capped at 31 points (=30 e lode). (b) The option Quiz + Project is available only to students participating in the optional projects. The final grade is the sum of the quiz and project scores, capped at 31 points (=30 e lode). The Quiz can be repeated multiple times, according to the general session rules.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.