*(updated February 15, 2018)*

**Monday, 14 May, morning (9.30-12.30)**

**Network Science: From Structure to Control** **(Barabasi)**: Systems as diverse as the world wide web, Internet or the cell are described by highly interconnected networks with amazingly complex topology. Recent studies indicate that these networks are the result of self-organizing processes governed by simple but generic laws, resulting in architectural features that makes them much more similar to each other than one would have expected by chance. I will discuss the order characterizing our interconnected world and its implications to network robustness, and control. Indeed, while control theory offers mathematical tools to steer engineered and natural systems towards a desired state, we lack a framework to control complex self-organized systems. I will discuss a recently developed analytical framework to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes whose time-dependent control can guide the system’s dynamics.

**Monday, 14 May, afternoon (14.30-17.30)**

** Mesoscale Analysis of Networks (Rombach)**: Mesoscale structures are structures that are not apparent either at the local scale of nodes and edges or at the global scale of summary statistics. They are computationally harder to detect than local or global structure, but can provide considerable insight into the topological positioning of nodes. The two most widely studied examples are community structure and core-periphery structure. We will review detection methods for these two structures. We will take a detailed look at the use of random graphs with certain global properties as null models. Based on global properties of our network, what mesoscale structure (and how much of it) do we expect?

**Tuesday, 15 May, morning (9.30-12.30)**

**Taming Complexity: Controlling Networks****(Barabasi)**: The ultimate proof of our understanding of biological or technological systems is reflected in our ability to control them. While control theory offers mathematical tools to steer engineered and natural systems towards a desired state, we lack a framework to control complex self-organized systems. Here we develop analytical tools to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes whose time-dependent control can guide the system’s entire dynamics. We apply these tools to several real networks, finding that the number of driver nodes is determined mainly by the network’s degree distribution. We show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control, but dense and homogeneous networks can be controlled via a few driver nodes. Counter-intuitively, we find that in both model and real systems the driver nodes tend to avoid the hubs.

**Tuesday, 15 May, afternoon **

no lectures

**Wednesday, 16 May, morning (9.30-12.30)**

**Centrality in Networks ****(Brandes)**: Centrality is a central concept in network analysis, and variations are termed status, prominence, importance, etc. Traditionally, it is operationalized via indices that assign numerical scores to the nodes such that higher values indicate higher centrality. We will review and analyze important indices, derive general principles, and thus arrive at a framework with much wider applicability. When does an index represent a centrality? Degree, closeness, betweenness, eigenvector centrality. Neighborhood inclusion, threshold graphs, positional dominance. Indirect relations, network positions, centrality in temporal and multivariate networks. Testing hypotheses about centrality.

**Wednesday, 16 May, afternoon (14.30-17.30)**

* short talks by students* (see page Application)

**Wednesday, 16 May, evening (20.00)**

**social dinner**

**Thursday, 17 May, morning (9.30-12.30)**

**Contagion in Networks/1 – Financial Systems ****(Battiston)**: Understanding the dynamics of contagion in financial networks is crucial to design a financial system that is more stable, that does not increase inequality in our society, and that is more aligned with the financing needs of combating climate change in a low-carbon economy. Ten years after the 2008 global financial crisis it is now widely accepted that the financial system is best described as a complex network. Yet, differently for other domains of complex networks, economic agents make decisions based on their own expectations on the future, including being rescued if they become interconnected enough to be considered systemically important. For these reasons, the endogenous dynamics of systemic risk in the financial system is far from being fully understood from a scientific perspective, and it is currently not adequately addressed by policy. Throughout this lecture, students will learn the main theoretical notions to understand network models of financial contagion. During the practical exercises with provided software tools, the students will also have the opportunity to carry out simple stress-test exercises on real financial networks.

**Thursday, 17 May, afternoon**

no lectures

**Friday, 18 May, morning (9.30-12.30)**

**Contagion in Networks/2 – Network Epidemiology****(Vespignani)**: At the core of epidemic modeling approaches is the structure of human interactions, mobility and contacts patterns that finds its best representation in the form of networks. Recent years have witnessed the development of data driven models of infectious diseases rooted in the combination of large–scale data mining techniques, computational approaches and mathematical modeling. These models are defined by the network describing the coupling among individuals and/or populations, along with the intensity of the coupling. In this lecture I will introduce the basic theoretical concepts and tools needed for the analysis of contagion processes taking place on networks. I will review the effect of contact patterns in the spreading of infectious diseases, and show how they manifest in simulated data from realistic models and real-world epidemics. I will also illustrate how networks at different scales are at the core of predictive modeling approaches and forecast to epidemics. Finally I will discuss basic modeling difference emerging from the description of biological and social contagion phenomena.

**Friday, 18 May, afternoon (14.30-17.30)**

**Contagion in Networks/3 – Epidemic Threshold in Structured Host Populations****(Colizza)**: Our understanding of infectious diseases prevention and control is rooted in the theory of host population transmission dynamics. Contacts between hosts (along which transmission can occur) and contacts between populations of hosts (along which spatial diffusion can take place) drive the epidemiology of infectious diseases, determining if and how quickly they spread, and who gets infected. Mathematical epidemiology has made great progress in this area in the last decades, moving from approximations where every host is in contact with anyone else with equal probability (i.e. the homogeneous mixing assumptions) to frameworks where patterns of contacts between hosts or population of hosts are explicitly accounted for through networks, made of nodes representing hosts/populations of hosts and connections representing potential transmission/diffusion links. This lecture will focus on (i) spatial epidemic spread, through the introduction of metapopulation network models with mobility/migration coupling between host populations, and (ii) disease spread when host-to-host contacts evolve in time, through the introduction of temporal networks. Using different techniques and approximations, we will explore how the structure of the host population affects its vulnerability to infectious disease epidemics, by means of the epidemic threshold, a central parameter in epidemiology indicating the critical conditions for the epidemic to occur.