morning (9.30-12.30)

Introduction to complex networks  (Serrano): What is a complex system? Networks: a change of paradigm. Basic network representations: unweighted/weighted, undirected/directed, unipartite/bipartite, singlelayered/multilayered. Mathematical and computational encodings of networks. Basic network metrics: global, local, and mesoscopic properties. Basic network models: Erdos-Renyi, Configuration Model, Watts-Strogatz, Barabasi-Albert. Dynamical processes: the Voter model; epidemic spreading.

afternoon (14.30-17.30)

Implications of power laws on algorithms in complex networks (Litvak): Background: power laws as a mathematical model for network hubs, highest degrees. I. Centrality: how to find hubs quickly,  PageRank in power law networks.  Connectivity: why power law networks are never disassortative, weighted triangles for detecting geometry.

morning (9.30-12.30)

Network geometry  (Serrano): Distances in complex networks. Spatial random graphs. Hyperbolic geometry: the S¹/ H² model; network maps and embedding techniques. I. The problem of scales. Geometric renormalization: coarse-graining and fine-graining; self-similarity of real multsicale networks; self-similar evolution of real networks. Scaled down and scaled up network replicas. II. The problem of dimensions. Dimensionality of real networks, the S^D/ H^D+1 model. Dimensional reduction, multidimensional hyperbolic maps.

afternoon (14.30-17.30)

No lectures

morning (9.30-12.30)

Advanced network models: multilayer and higher-order networks (I)  (Battiston): Multilayer networks: vectorial formalism, basic node, edge and local properties, shortest paths, correlations, reducibility. Impact of multiplexity on dynamics (public goods game and other examples). Higher-order networks: basic ideas, structural analysis of higher-order networks with HGX (basic node and edge properties, motif analysis, community detection, temporal correlations). Impact of non-pairwise interactions on dynamics (social dilemmas and other examples).

afternoon (14.30-17.30)

Advanced network models: multilayer and higher-order networks (II) (Ferraz de Arruda): Multilayer networks: tensorial formalism, spectral properties, assortativity. Higher-order networks: epidemic spreading and social contagion processes, spectral properties, random walks.

evening (20.00)

social dinner

morning (9.30-12.30)

short talks by students

afternoon

No lectures

morning (9.30-12.30)

Evolutionary game theory and human behaviour (Lenaerts): (1) A general introduction to game structures, solution and equilibria concepts: simultaneous, sequential and stochastic games, social dilemmas, Nash and other equilibria, minimax and regret. (2) Evolutionary game theory in well-mixed and networked populations: simulations, replicator dynamics, Moran processes, evolutionary stability, evolutionary robustness, five rules of cooperation and the impact of topology on dynamics.  (3) Case studies related to cognitive capacities and delegation of decision-making, illustrated with some behavioral experiment data.

afternoon (14.30-17.30)

Networks for Economics and Society (del Rio-Chanona): (1) Application of the network clustering algorithms known as Economic Complexity for analyzing national development and workers’ capabilities. (2) Multilayered Systems: Explores the use of multilayer networks, in particular, the interplay between production and worker networks, with a focus on assortativity in the net-zero transition. (3) Agent-Based Modeling: Details the integration of economic theory into network modeling. We discuss an agent-based model for the labor market dynamics during technological automation and displacement of workers and another for analyzing the health-economy trade-off during the COVID-19 pandemic.