# Research

## The networked complexity of real-world systems and processes

Many of the challenges in modern society require a deep understanding of the complexity of biological, economic, financial, physical, social and technological networks. Besides the traditional difficulties which are inherent to the study of the components of these systems (e.g. cells, organizations, atoms, individuals and devices), over the last decades an additional (and often dominant) level of complexity has emerged, which derives from the interactions among components. In a world that is increasingly interconnected at the cultural, digital, economic, physical and social level, the body of knowledge developed by each discipline is becoming less and less exhaustive while the need for innovative, interdisciplinary approaches emerges forcefully.

The Theory of Complex Systems and the Science of Networks are modern approaches to the study of systems characterized by a large number of components, interconnected in irregular architectures, i.e. structures that are quite different from the idealized, oversimplified ones traditionally considered within the natural and social sciences. Indeed, while the interactions among atoms in simple materials can be represented as regular and symmetric lattices, real-world networks of interactions connecting the constituents of cells, organisms, ecosystems, economies, societies and infrastructures turn out to be extremely heterogeneous.

Examples of recurrent structures in empirical networks are: the coexistence of elements (vertices) displaying such diverse numbers of connections that the notion of "average number of neighbours per node" becomes meaningless (scale-free property), the tendency of vertices with "neighbours in common" to be also connected between them (clustering or triadic closure), a larger cohesion within certain sets of vertices (community structure), the abundance of specific substructures (motifs). Complex systems also exhibit collective properties that emerge from the interactionsamong their consituent elements and cannot be traced back uniquely to the intrinsic properties of the latter ones.

Beside the need to characterise the complex structure of large-scale, real-world systems, understanding the consequences of structural complexity for the dynamics of the processes that typically take place on those systems has become more and more important. For instance, recent (economic, financial and health) crises have shown how the highly irregular and inhomogeneous structure of real networks (of firms, banks and people) deeply complicates the management, as well as the prediction, of stress and disease propagation in modern economies and societies: indeed, the phenomenology of these processes crucially depends on which vertices are hit first, how many vertices are directly connected to them and so on.

Finally, in many contexts (e.g. in ecology and economics) a strong interplay is observed between the structure of networks and the dynamics of the processes taking place on them: in fact, not only the underlying structure has an impact on the dynamics but also the dynamics has an impact on the underlying structure.

Within the broad landscape of complex systems science and network theory, our unit specializes on research interests that are both theoretical and applied in nature.

## Our research: theory

Mathematical modelling of complex networks via maximum-entropy ensembles of random graphs with prescribed properties;

statistical physics of systems for which the fundamental assumption of ensemble equivalence is broken by the presence of local constraints;

design of renormalisation schemes for the analysis of networks at multiple scales;

construction of null models of complex systems for statistical pattern detection;

introduction of methods for the detection of mesoscopic levels of organization in complex systems from empirical time series or other features;

refinement of traditional information-theoretic bounds on data compression for large data structures with heterogeneous properties.

## Our research: applications

Reconstruction of financial networks from partial information and reliable estimation of systemic risk from privacy-limited data;

detection of early-warning signals of upcoming instabilities in financial and banking systems;

multi-scale analysis and modelling of economic networks with nontrivial topology;

study of functional brain networks in healthy subjects and diseased patients;

analysis of social networks, (mis)information diffusion, polarization, opinion dynamics;

urban growth, metabolism, and sustainability.