The goal in this track is to view networks from the combined perpective of mathematics and physics. The keynote lecture is given by M. Angeles Serrano (Universidad de Barcelona); other contributers are Ginestra Bianconi (Queen Mary University of London) and Tiziano Squartini (Universita di Roma "La Sapienza").

Ideas from statistical physics have proven to be very efficient in the mathematical analysis of complex networks, while methodologies developed within probability theory turn out to be very fruitful in the analysis of network driven by physics, chemistry and biology. Key examples are spatial networks and community networks. Key targets are dynamics of networks, dynamics on networks and simulation of networks.

**M. Ángeles Serrano** - **Keynote lecture**

Universitat de Barcelona-ICREA

*The hidden geometry of complex networks: foundations and applications*

Complex networks display a hidden metric structure, which determines the likelihood and intensity of interactions. This quality has been exploited to map real networks, producing geometric representations that shed light on how pivotal forces --like preferentiality, localization, and hierarchization-- rule their structure and evolution. The first part of the lecture is focused on introducing the foundations of complex networks embedded in hidden metric spaces, and on presenting applications such as efficient navigation and analysis of networks at the mesoscale. The second part of the lecture is devoted to present our last contribution. We have defined a geometric renormalization group for complex networks embedded in an underlying space that allows for a rigorous investigation of networks as viewed at different distance scales. We find that real scale-free networks show geometric scaling under this renormalization group transformation. This feature enables us to unfold them in a self-similar multilayer shell which reveals the coexisting scales and their interplay. The multiscale unfolding brings about immediate practical applications. Among many possibilities, it yields a natural way of building high-fidelity smaller-scale replicas of large real networks, and sustains the design of a new multiscale navigation protocol in hyperbolic space which boosts the success of single-layer versions.

**Ginestra Bianconi**

Queen Mary University of London

*Multilayer networks: a new framework for complex systems*

Multilayer networks describe complex systems formed by different interacting networks. Multilayer networks are ubiquitous and include social networks, financial markets, multimodal transportation systems, infrastructures, the brain and the cell. Multilayer networks cannot be reduced to a large single network. In this talk I will introduce relevant modelling frameworks for multilayer structures and I will discuss how it is possible to extract more relevant information from multilayer networks than from their single layers taken in isolation.Finally I will show the rich interplay between multilayer structure and the dynamics defined on them touching on fundamental questions related to robustness and control of multilayer networks.

IMT School for Advanced Studies Lucca

*Maximum-entropy models for networks*

Entropy maximization represents the unifying concept that underlies the definition of a number of methods, now part of the discipline known as "network theory". Examples are provided by the definition of null models for pattern detection and of an unbiased procedure for inferring the presence of connections from partial information. In this talk, a general method to define constrained maximum-entropy ensembles of networks will be described and examples of techniques for both detecting patterns and reconstructing networks will be provided. In particular, 1) the way the application of the former has led to the detection of early-warning signals of the 2007-2008 worldwide crisis and 2) a comparison of different reconstruction methods will be illustrated. Moreover, different ways of enforcing constraints (i.e., either exactly or on average) will be also proven not to be equivalent. In physical terms, this can be rephrased by saying that the microcanonical and canonical recipes are no longer equivalent, further implying that the approach used to analyse networks indeed makes a difference.

**Dmitri Krioukov
**Northeastern University, Boston

*Clustering Implies Geometry in Networks*

Two common features of many large real networks are that they are sparse and that they have strong clustering, i.e., large number of

triangles homogeneously distributed across all nodes. In many growing real networks for which historical data is available, the average degree and clustering are roughly independent of the growing network size. Recently, (soft) random geometric graphs, also known as latent-space network models, with hyperbolic and de Sitter latent geometries have been used successfully to model these features of real networks, to predict missing and future links in them, and to study their navigability, with applications ranging from designing optimal routing in the Internet, to identification of the information-transmission skeleton in the human brain. Yet it remains unclear if latent-space models are indeed adequate models of real networks, as random graphs in these models may have structural properties that real networks do not have, or vice versa.

We show that the canonical maximum-entropy ensemble of random graphs in which the expected numbers of edges and triangles at every node are fixed to constants, are approximately soft random geometric graphs on the real line. The approximation is exact in the limit of standard random geometric graphs with a sharp connectivity threshold and strongest clustering. This result implies that a large number of triangles homogeneously distributed across all vertices is not only a necessary but also sufficient condition for the presence of a latent/effective metric space in large sparse networks. Strong clustering, ubiquitously observed in real networks, is thus a reflection of their latent geometry.