## Program

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#### Monday, 27 May

#### 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.

#### Tuesday, 28 May

#### morning (9.30-12.30)

*Network geometry ***(Serrano): **Distances in complex networks. Spatial random graphs. Hyperbolic geometry: the S^{1}/ H^{2} 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

#### Wednesday, 29 May

#### 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*

#### Thursday, 30 May

#### morning (9.30-12.30)

**short talks by students**

#### afternoon

No lectures

#### Friday, 31 May

#### 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.