## Areas of Research:

## Neural networks

Proof of the equivalence between formal "Hopfield networks" (i.e., the standard model for associative retrieval) and "Boltzmann machines" (i.e., the standard model for learning processes), and development of a comprehensive theory of the fluctuations of the order parameters to control analytically phase transitions in these models. Extension to more sophisticated multi-species Hopfield models toward a rational for deep learning.

In this field one of the most significant results is the introduction of the first attractor neural network able to process in parallel the information. Extension to non-trivial topologies and to high-storage regimes were also investigate.

Selected publications:

to appear in Journal of Statistical Physics (2018)

Europhysics Letters 117, 28003 (2017)

Physical Review Letters 114, 028103 (2015)

Neural Networks 66, 22–35 (2015)

Neural Networks 38, 52-63 (2013)

Physical Review Letters 109 (26), 268101 (2012

In this field one of the most significant results is the introduction of the first attractor neural network able to process in parallel the information. Extension to non-trivial topologies and to high-storage regimes were also investigate.

Selected publications:

- Non-convex multi-species Hopfield models
**

*E Agliari, D Migliozzi, D Tantari*to appear in Journal of Statistical Physics (2018)

*E Agliari, A Annibale, A Barra, ACC Coolen, D Tantari*Europhysics Letters 117, 28003 (2017)

*E Agliari, A Barra, A Galluzzi, F Guerra, D Tantari, F Tavani*Physical Review Letters 114, 028103 (2015)

*E Agliari, A Barra, A Galluzzi, F Guerra, D Tantari, F Tavani*Neural Networks 66, 22–35 (2015)

*E Agliari, A Barra, A De Antoni, A Galluzzi*Neural Networks 38, 52-63 (2013)

*E Agliari, A Barra, A Galluzzi, F Guerra, F Moauro*Physical Review Letters 109 (26), 268101 (2012

## Statistical Mechanics of Complex Systems and their Applications

Extension of techniques of analytical mechanics to solve complex problems (e.g., evaluation of the free energy of spin-glasses at low temperatures) in mean-field approximation (e.g., method of Hamilton-Jacobi, Noether invariants and Legendre transformations via Cole-Hopf transformation, Fourier methods).

Analytical solution of non-mean-field models (e.g. embedded in hierarchical structures) and characterization of emerging metastable states.

Generalization of two-body problems to the case of many bodies (p-spin), in fully-connected structures as well as diluted structures; these problems were solved by extending classical methods of disordered systems (field cavities and stochastic stability).

Selected publications:

Journal of Physics A: Mathematical and Theoretical 48 (1), 015001 (2015)

Journal of Mathematical Physics 53 (6), 063304 (2012) E Agliari, A Barra, F Camboni,

Reports on Mathematical Physics 68, 1 (2011)

Journal of Statistical Mechanics: Theory and Experiment P10003 (2008)

Analytical solution of non-mean-field models (e.g. embedded in hierarchical structures) and characterization of emerging metastable states.

Generalization of two-body problems to the case of many bodies (p-spin), in fully-connected structures as well as diluted structures; these problems were solved by extending classical methods of disordered systems (field cavities and stochastic stability).

Selected publications:

*E Agliari, A Barra, A Galluzzi, F Guerra, D Tantari, F Tavani*Journal of Physics A: Mathematical and Theoretical 48 (1), 015001 (2015)

*E Agliari, A Barra, R Burioni, A Di Biasio*Journal of Mathematical Physics 53 (6), 063304 (2012) E Agliari, A Barra, F Camboni,

Reports on Mathematical Physics 68, 1 (2011)

*E Agliari, A Barra, F Camboni*Journal of Statistical Mechanics: Theory and Experiment P10003 (2008)

## Random Walks in Inhomogeneous Topologies

Rigorous solution of first-passage problems in low-dimension and/or non-exactly-decimable structures, possibly accounting for memory effects and splitting between local and average phenomena.

Extension of the mean-field techniques for the treatment of processes defined in restricted geometries where the mixing hypothesis does not hold. These results were used in two main areas: "search strategies" and "reaction diffusion".

Selected publications:

Chaos 27, 083108 (2017)

Physical Review E 92, 042156 (2015)

Physical Review E 89 (5), 052147 (2014)

Physical Review E 85 (2), 026113 (2012)

Physical Review E 77 (1), 011128 (2008)

Extension of the mean-field techniques for the treatment of processes defined in restricted geometries where the mixing hypothesis does not hold. These results were used in two main areas: "search strategies" and "reaction diffusion".

Selected publications:

*J Peng, E Agliari*Chaos 27, 083108 (2017)

*L Cattivelli, E Agliari, F Sartori, D Cassi*Physical Review E 92, 042156 (2015)

*E Agliari, A Blumen, D Cassi*Physical Review E 89 (5), 052147 (2014)

*B Meyer, E Agliari, O Bénichou, R Voituriez*Physical Review E 85 (2), 026113 (2012)

*E Agliari*Physical Review E 77 (1), 011128 (2008)

## Quantum walks and quantum algorithms for searching problems

Quantum walks were used to model the coherent motion of excitons in discretizable structures, possibly in the presence of reaction centers.

Development of rigorous analytical techniques based on spectral theory and numerical methods based on the diagonalization of the Laplacian associated to the structure underlying the dynamic process. These results allowed us to show that, in the quantum computational theory, d = 4 is not an upper critical dimension.

Selected publications:

Physical Review A 82 (1), 012305 (2010)

International Journal of Bifurcation and Chaos 20 (02), 271-279 (2010)

Journal of Physics A: Mathematical and Theoretical 41 (44), 445301 (2008)

Development of rigorous analytical techniques based on spectral theory and numerical methods based on the diagonalization of the Laplacian associated to the structure underlying the dynamic process. These results allowed us to show that, in the quantum computational theory, d = 4 is not an upper critical dimension.

Selected publications:

*E Agliari, A Blumen, O Muelken*Physical Review A 82 (1), 012305 (2010)

*E Agliari, O Mülken, A Blumen*International Journal of Bifurcation and Chaos 20 (02), 271-279 (2010)

*E Agliari, A Blumen, O Mülken*Journal of Physics A: Mathematical and Theoretical 41 (44), 445301 (2008)

## Graph Theory and applications

Topological characterization of inhomogeneous structures through algebraic and spectral methods to probe local and average properties (e.g., transience, local dimension, connectivity, mixing time) [14].

The role of these properties on the behavior of random processes defined on the same topologies was also deepened in different contexts [22,28,29].

Selected publications:

Scientific Reports 7, 39962 (2017)E Agliari, C Cioli, E Guadagnini

Physical Review E 84, 031120 (2011)

Europhysics Letters 94 (1), 10002 (2011)

Journal of Statistical Mechanics: Theory and Experiment P09002 (2010)

The role of these properties on the behavior of random processes defined on the same topologies was also deepened in different contexts [22,28,29].

Selected publications:

*E Agliari, F Tavani*Scientific Reports 7, 39962 (2017)E Agliari, C Cioli, E Guadagnini

Physical Review E 84, 031120 (2011)

*E Agliari, A Barra*Europhysics Letters 94 (1), 10002 (2011)

*E Agliari, M Casartelli, E Vivo*Journal of Statistical Mechanics: Theory and Experiment P09002 (2010)

## Applications of Statistical Mechanics

From the analytical investigations described above, we developed several applications in many fields of research: theoretical immunology, infectivology, chemical kinetics, quantitative sociology; the validity of our models was checked versus experimental data.

Selected publications:

Scientific Reports 7, 12737 (2017)

Palgrave Communications 2, 16021 (2016)

Scientific Reports 6, 36314 (2016)

Scientific Reports 5, 9415 (2015)

Journal of theoretical biology 375, 21-31 (2015)

Scientific Reports 4, 6639 (2014)

PloS ONE 8 (1), e55017 (2013)

Selected publications:

*E Biselli,**E Agliari,**A Barra, FR Bertani, A Gerardino et al.*Scientific Reports 7, 12737 (2017)

*A Barra, A Galluzzi, D Tantari, E Agliari, F Requena-Silvente*Palgrave Communications 2, 16021 (2016)

*E Agliari, A Barra, L Dello Schiavo, A Moro*Scientific Reports 6, 36314 (2016)

*E Agliari**, M Altavilla, A Barra,**L Dello Schiavo**, E Katz*Scientific Reports 5, 9415 (2015)

*E Agliari, A Barra, G Del Ferraro, F Guerra, D Tantari*Journal of theoretical biology 375, 21-31 (2015)

*E Agliari, E Biselli, A De Ninno, G Schiavoni, L Gabriele, A Gerardin*o*, F Mattei, A Barra, L Businaro*Scientific Reports 4, 6639 (2014)

*E Agliari, L Asti, A Barra, R Scrivo, G Valesini, RS Wallis*PloS ONE 8 (1), e55017 (2013)