**Scientific activities of the various Research Units**

The scientific activities of the INFN units participating to the LINCOL project mainly focus on the following topics:

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Statistical and dynamical properties of artificial and real networks (T1)

Dynamics of topologically entangled polymer networks (T2)

Contact networks in epigenetics and protein folding dynamics (T3)

Resilience and control of neural and ecological networks (T4)

Interplay between social and information networks (T5)

Statistics and dynamics of quantum networks (T6)

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**INFN Section: Padova [PD]**

(T1) – The unit is investigating, by using tools and approaches of statistical mechanics, Boltzmann machines, deep neural networks, and other tools for artificial intelligence. The first aim is to unveil relevant yet unknown features leading to their good performances in classification and generalization of data thus developing a stat mech approach for AI explainability.

(T2) – Fluctuating filaments and polymers are examples of complex soft networks displaying self and mutual entanglements. We are studying these networks by means of numerical simulations of effective models and by analytical approaches based on the topological theory of knots and links. The goal is to understand how entanglement affect the relaxation patterns of these system when subject to external forcing.

* *(T3) – 3D protein structures depend on the underlying network of contacts (contact map) but, if these structures are also topologically entangled (i.e. knotted proteins) this description can fail. Since some protein domains are self (and mutual) entangled we will investigate how this entanglement is faithfully captured by the amino acid sequences and how its presence can affect the folding process. Chromatin contact maps are the most experimentally acessible information on chromatin 3D organization and yet their relation with epigenetic marks are poorly understood. Our aim is to introduce a multilayer dynamical network where contact and epigenetic information are taken into account simultaneously during chromatin folding.

* *(T4) – In collaboration with neuroscientists of the Padova Neuroscience Center, we are looking at the spontaneous and induced activity of cortical networks in vivo. We do it by performing statistical inference of neural emergent patterns such as neuronal avalanches and brain rhythms. Next, we will investigate if and how the brain is poised near a critical states, using both interacting particle models and phenomenological renormalisation group.

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**INFN Section: Firenze [FI]**

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* *(T1) – We are studying Statistical and dynamical properties of artificial and real neural networks, in particular processes, design and optimization of networks though spectral methods, in conjunction with the The Biophysics Biophotonics Lab – LENS and the CNR group.

(T4) – We are investigating the possibility of generating networks with given spectral properties, control and design. This is done by using spectral method, control via synchronizatin of processes on networks, interplay between walkers and traps on networks.

(T4) – We are studying the resilience to stressors/perturbation in neural model and in agent-based model of societies, learning the optimal resilience response by classical and innovative methods (simulated annealing, deep learning, unsupervised evolutionary agents).

(T5) *–* We are studying the interplay between social and information networks, in particular the role of human behavior and heuristics on social and information processes, how people behave in real and virtual life and how information is affected/affects humans, in particular with applications to epidemic spreading (in real life and in computer networks).

**INFN Section: Bologna [BO]**

(T4,T5) – This unit has recently considered the problem of studying the spectral properties of random adjacency matrices and Laplacian matrices using the methods of statistical mechanics and the Random Matrix Theory. The main goal is to relate the spectral properties with the structure of the associated complex network and the dynamical properties of the stochastic master equation for evolution of the distribution function. The proposed approach takes advantage from the Central Limit Theorems of the Random Matrix Theory, the Perturbation Theory and the renormalization group techniques to study spectral properties of matrices related to complex networks. The considered applications are the resilience and controllability of neural, ecological networks and mobility networks. In collaboration with PD and FI we are studying the nonlinear stochastic dynamical systems on graphs, when the associated connectivity matrix is the realization of a stochastic process in order to study the statistical properties of the distribution function and the existence of critical phenomena (i.e. the appearance of attractive states) or of synchronization phenomena of the evolution of communities of interacting nodes. The activity will be based on the results of stochastic dynamical systems using perturbative approaches. The considered applications will be the evolution of genetic networks in cancer cells, the study of epidemic spread models based on mobility networks at small spatial and time scales using a Big Data approach (thanks to a collaboration with TIM) and the evolution of ecological networks using generalized Lotka-Volterra equations.

**INFN Section: Rende-Cosenza [RC]**

(T1,T6) – Concerning electric networks and material science we are studying the implementation of the KdV equation into cellular neural networks (CNN), and the spontaneous Synchronization in Two Mutually Coupled Memristor-Based Chua’s Circuits. We next plan to work on mathematical models for charge and heat transport in graphene and metal dichalcogenides. Another future aim is that of describing quantum confinement, which is present when graphene is patterned into nanoribbons (GNRs)

Quantum structures can be identified in several systems including non-quantum systems. For example, they have been identified in financial markets, cognitive sciences, economics, where quantum-like models have been developed. A quantum-like model may be defined as the mathematical description of a non-quantum system by means of the mathematical formalism of quantum physics. What is relevant is the common features these various contexts share with quantum physics, i.e., the fundamental statistical character. It is helpful to recall that both the quantum and the classical models are particular cases of the general statistical models (GSMs). GSMs are used in physics in order to characterize the information processing power of quantum theory and those information-theoretic phenomena that are not classical but more general than quantum. In this respect we aim at developing the theoretical and mathematical structure of GSMs and POVMs (positive operator valued measures) as well as their applications in physics (for example, photon localization). Specifically, we are working on the concept of compatibility of observables in a GSM. We already obtained a characterization of compatibility in the Hilbert space framework and we will look for extensions to more general frameworks. In adddition we aim at providing general statistical models for quantum-like systems by focusing on the evaluation/quantification of their non-classical character. The generality of the GSM approach makes it easy to reinforce the interactions with the other research units. In particular with [FI] (social networks) and [PgCa] unit (quantum networks).

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**INFN Section: Catania [Ca]**

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(T5) – Multilayer networks describe well many real interconnected communication and transportation systems, allowing the description of collective phenomena emerging from the interactions of multiple dynamical processes in social or economics contexts. However, pairwise interactions are often not enough to characterize social processes such as opinion formation, social contagion or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Higher-order models, where a social system is represented by a simplicial complex and the information spreading can occur through interactions in groups of different sizes, promise to better capture the emergence of novel phenomena. In this framework, by means of agent-based simulations, network analysis and analytical models, we are exploring the interplay between social and information networks in several context, from transport planning to politics, from policy management to financial markets, with particular attention to the dynamical aspects of information flows (imitation, viral spreading, critical phenomena, emerging role of noise and randomness, etc..). Further applications to epidemic risk assessment, to seismic vulnerability in urban areas and to the evaluation of efficiency and robustness of trophic networks, will be also addressed.

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**INFN Section: ****Perugia-Camerino [PgCa]**

(T6) – Concerning quantum networks, it is important to know which entangled states are equivalent in the sense that they are capable of performing the same tasks almost equally well. Although it seems natural to seek a classification under Stochastic Local Operations and Classical Communication (SLOCC), this fails for four (or more) qu-dits giving infinitely many (actually uncountable) classes.

We are starting to classify entanglement of qu-dit network pure states in terms of finite number of families and subfamilies by employing tools of algebraic geometry that are SLOCC invariants. The entanglement classification will also offer a perspective for evaluating the computational complexity of quantum algorithms, by analysing how the classes change while running them.

Furthermore, we will pursue the construction of new entanglement witnesses to be used for detecting entanglement in multipartite mixed states. This will be helpful to extend the classification to mixed states.

Concerning the dynamics of quantum networks we aim to study the transfer of quantum information among nodes (either spin-1/2 or bosons) by considering different models (including those referring to biological systems, like alpha-helixes). The goal is to single out conditions under which information transfer capabilities can be improved. This unit collaborates with [RC] for what concerns methods and tools for entanglement characterization and with [FI] and [PD] for biologically oriented quantum network models.