morning 
No lectures

afternoon (14.30-17.30)

Higher-order networks: An introduction to simplicial complexes (Bianconi)

Higher-order networks describe the  many-body interactions of a large variety of complex systems ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow to capture the combinatorial properties, the  topology and the geometry of higher-order networks.  Having been used extensively in quantum gravity to describe discrete or discretized space-time,  simplicial complexes  are becoming the representation of choice for capturing the underlying network topology and geometry of complex systems.Here we provide a comprehensive introduction to this very hot topic of Network Theory covering a wide range of subjects and demonstrating that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

evening (18.00-21.30, including a break for a “welcome light dinner”)
Short talks by students

morning (9.30-12.30)

Modeling the spatial transmission of infectious diseases (Colizza)

Understanding how infectious diseases spread across space is fundamental for anticipating epidemic trajectories and evaluating intervention strategies. Spatial and metapopulation models provide a powerful framework for capturing transmission dynamics in systems where populations are geographically distributed yet connected through mobility and interaction patterns. These models make it possible to assess the role of human movement, population structure, and spatial heterogeneities in shaping epidemic spread and persistence. This lecture will introduce the conceptual foundations of spatial disease modeling, from simple diffusion processes to network-based metapopulation frameworks. We will explore how mobility networks—whether local commuting, long-distance travel, or global air transportation—mediate pathogen dispersal, and how spatial coupling influences epidemic synchrony, seeding events, and spatial hierarchies of transmission. Through real-world examples, we will discuss how these modeling approaches support epidemic risk assessment and guide public health decision-making, from forecasting to targeted interventions.

afternoon
No lectures

morning (9.30-12.30) 

Branching processes for spreading on networks (Gleeson)

Network models may be applied to describe many complex systems, and in the era of online social networks the study of dynamics on networks is an important branch of computational social science.  Cascade dynamics can occur when the state of a node is affected by the states of its neighbours in the network, for example when a Twitter user is inspired to retweet a message that she received from a user she follows, with one event (the retweet) potentially causing further events (retweets by followers of followers) in a chain reaction. In this talk I will review some mathematical techniques and models that can help us understand how social contagion (the spread of cultural fads and the viral diffusion of information) depends upon the structure of the social network and on the dynamics of human behaviour. I will focus in particular on branching processes, but no prior knowledge of stochastic processes is required. Although the models are simple enough to allow for mathematical analysis, I will show examples where they can also provide good matches to empirical observations of cascades on social networks.

afternoon (14.30-17.30)

Dynamics of information spreading in complex environments (Moreno)

The spread of information and misinformation through online social systems has become a central societal concern, with direct implications for public opinion, collective behavior, and decision-making. Understanding how information propagates in these settings is therefore a key problem in network science and computational social science. In this lecture, we will review classical models of information spreading on networks, including epidemic-like processes, threshold models, and rumor dynamics, and discuss how they have been used to capture cascades, virality, and attention decay in social platforms. We will then turn to empirical analyses of online communication networks, highlighting both the successes and the limitations of pairwise interaction models when confronted with real data, particularly in environments dominated by group-based communication. This motivates a shift toward more general modeling frameworks that account for collective interactions. Building on this evolution, we will introduce higher-order models of information spreading, using hypergraphs to represent group communication and incorporating mechanisms of collective saturation and disengagement. We will show how these models naturally explain the coexistence of fast and slow decay regimes observed in data, and discuss their implications for understanding and managing information diffusion in modern social systems.

evening (20.00)
Social dinner

morning (9.30-12.30) 

Temporal networks (Holme)

The power of any network approach lies in its ability to simplify a complex system so that one can better understand its function as a whole. However, sometimes it is beneficial to include more information than a simple graph of nodes and links alone. Adding information about interaction times can improve the accuracy of predictions and mechanistic understanding. The drawback, however, is that there are not many methods available, partly because the study of temporal networks is a relatively young field and partly because it is more difficult to develop such methods than for static networks. In this lecture, I will review methods for analyzing and modeling temporal networks and the processes that occur on them. Applications include the spreading of infectious diseases, opinions, and rumors in social networks; different types of search processes, delay propagation in traffic systems, data packet dynamics in computer networks; various types of signaling in biology, and more. We also discuss future directions.

afternoon
No lectures

morning (9.30-12.30)

Financial networks models: from risk to climate-related risk (Battiston)

Financial network models describe several contexts of the financial system, in particular to understand the conditions for the emergence of systemic risk, which can result in large adverse societal impacts. In a financial network, nodes represent financial actors and links represent different types of financial contracts. Financial network models can also be used to understand how climate-related risks, as for instance the risks of flood or drought, propagate trough the financial network and affect different layers of the economic system. Correlations in space and time and tail risk are particularly relevant in this context. 

afternoon (14.30-17.30)

Analyzing the network of international trade (Piccardi)

Economic transactions between countries are naturally amenable to description and analysis using the tools of network science. Models of increasing complexity are employed, including those involving weighted, directed, bipartite, temporal, multilayer and higher-order networks. These representations are made possible by the systematic collection and availability of a wide range of data on international trade in goods and services by organisations such as the OECD, UN and WTO. The availability of this data, coupled with the development of increasingly sophisticated and diverse analytical methods, has enabled us to study and emphasise significant aspects. In this lecture, we will review a number of results concerning the International (or World) Trade Network, ranging from its fundamental properties and mesoscale structures to the definition of complexity indices for countries and products, the connections between the latter and network topology, the structure of global value chains, and the evolution of countries’ global roles over time.