# Networks

## Overview

Many of the phenomena occurring around us (from those influencing our social relationships, to those transforming the overall environment where we live) are just the result of the emergent organisation of systems that, on their turn, involve a multitude of basic constituents (or entities) which selectively interact via somehow complicated patterns. One of the major effort of modern nonlinear physics is therefore providing proper representations of such distributed systems, where constituents are taken to be the nodes (or units) of a network, and interactions are modelled by links of that same network.

The last fifteen years have seen the birth of a movement in science, known under the name of complex networks theory, which today involves the interdisciplinary effort of several communities of scientists, and with the ultimate goal of revealing the unifying principles and mechanisms at the basis of structural and functional organisation of networked systems.

Our research focuses on the intimate relationships between network’s structure and function. We aim at answering questions such as how the network’s collective dynamics can be predicted (or controlled) once the main topological properties of the connectivity structure are known, or how can one infer and reconstruct the microscopic interactions between the network’s units from the observation of the dynamics at certain meso- or macro-scales.

## People involved

## Recent publications

**Giovanni Giacomelli**, **Stefano Lepri**, and Cosimo Trono:

Phys. Rev. A **99**, 023841 (2019) [PDF]

The complex active optical network, or lasing network (LANER) is a recently introduced system capable of laser action. The system is experimentally realized with optical fibers linked each other with couplers and with one or more coherently amplifying sections. The LANER displays a standard laser behavior: When the gain provided by the active sections is high enough to overcome the losses, a coherent emission is produced, with a complicated intensity spectrum. A linear theoretical description shows how the LANER can be considered as a generalization of the laser with the physical network acting as a complicated cavity. Among its main aspects, the system can be represented by directed graphs disclosing the analogies with the problem of quantum chaos on graphs. Moreover, when the links' lengths are all integer multiples of the same value, the LANER framework corresponds to a lattice problem, with the equivalence of the Brillouin zone with the cavity free spectral range.

M. Asllani, T. Carletti, F. Di Patti, **D. Fanelli**, and **F. Piazza**:

Phys. Rev. Lett. 120, 158301 (2018)

A nonlinear operator to model diffusion on a complex undirected network under crowded conditions is introduced. The asymptotic distribution of diffusing agents is a nonlinear function of the nodes’ degree and saturates to a constant value for sufficiently large connectivities, at variance with standard diffusion in the absence of excluded-volume effects. Building on this observation, we define and solve an inverse problem, aimed at reconstructing the connectivity distribution of *a priori* unknown network, by gathering the necessary information at a single node.

G. Cencetti, P. Clusella, **D. Fanelli**:

Scientific Reports, 8, 16226 (2018)

Given a multi-species system interacting via a complex network, we propose two different techniques to modify the network topology while preserving its dynamical behaviour. The newly generated network is isodynamic to the former, meaning that it reproduces the dynamical response to a perturbation, as displayed by the original system.

J. Petit, B. Lauwens, **D. Fanelli**, T. Carletti :

Physical Review Letters 119, 148301 (2017)

In this paper we investigate the process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation, that we cast on solid analytical grounds, to derive a closed prediction for the onset of the instability in the time dependent framework.