We explore novel methods to generate and characterize complex networks by means of their embedding on hyperbolic surfaces. Evolution through local elementary moves allows the exploration of the ensemble of networks which share common embeddings and consequently share similar hierarchical properties. This method provides a new perspective to classify network-complexity both on local and global scale. We demonstrate that there is a strong relation between the network properties and the embedding surface. Small world networks and scale free degree distributions emerge spontaneously form the constrained evolution on special manifolds.

Granular materials are everywhere; they are central in a very wide range of domains from agriculture to pharmaceutical industry. The capability to handling, processing, storing and producing granular materials is of paramount importance. Despite such a central role in most fields of human activity and their ubiquitous presence in scientific research areas, yet a proper understanding of their eclectic behaviours and properties remains elusive. Granular materials can flow like liquids in some circumstances but they can act like solids in others. Their understanding requires the development of new paradigms and tools beyond the traditional domains of solid state physics, engineering and material sciences.

Volume fluctuations in granular assemblies
I study the structural organization and correlations in very large packings of equally sized spheres, reconstructed in three dimensions with x-ray computed tomography. I show that the geometrical structure can be conveniently studied as a packing of irregular tetrahedra with volume distribution that must decay exponentially with parameters controlled by the conditions of mechanical stability, nonoverlapping, and space filling. [T. Aste, “Volume fluctuations and geometrical constraints in granular packs”, Phys. Rev. Lett. 96 (2006) 018002.]

The Geometrical Structure of Disordered Sphere Packings
We study very large samples of disorderly packed monosized spheres with the objectives of searching for signatures of disorder, exploring the local organization and the packing efficiency. Bead packs of up to 150,000 mono-sized spheres with packing densities ranging from 0.58 to 0.64 have been studied by means of X-ray Computed Tomography. This study represents the largest and the most accurate empirical analysis of disordered packings at the grain-scale ever performed. See Granular Matter Page ->>>

An introductory reading about ordered and disordered sphere packings can be found in my book: “The Pursuit of Perfect Packing”.

The structure of disordered cellular systems
During my Post Doc. with Prof. Nicolas Rivier and his research group, at the Laboratorie Dinamique des Fluides Complexes, Strabourg, we have introduced a new method to study disordered cellular structures and packings which emphasizes their shell structure. Important new results were obtained for a class of systems that we named shell-structured-inflatable (SSI) froths. We found that relevant structural properties for these systems can be studied by means of an exact iterative relation which is associated with the logistic map. By using this approach all the topological properties of the 24 Frank and Kasper phases were derived and new possible structures were proposed. One of these structures has been recognized by crystallographers to be relevant in the study of ternary crystals and proposed as possible structure of the Th2Cr3Si4 (see, M. O’Keeffe,“On a space-filling polyhedron of Aste et al.”, Phil.Mag.Lett. 76 (1997) p.423-426). Important relations between the shell map and the local cellular organization have been pointed out for 2D soap froths and computer-generated topological networks.

Space curvature and topological properties of froths
We discovered deep connections between the curvature of the manifold tiled by the froth and the orbits of the dynamical map that generate this tiling by inflation, providing a way to define the curvature from topological considerations only. In two dimensions this connection is equivalent to the one provided by the Gauss-Bonnet theorem, but it is an independent derivation. In three dimensions the Gauss-Bonnet theorem does not succeeds in establishing this connection and therefore, in this case, our method is the only one that provides a link between the space curvature and the topological properties of the embedded cellular partition.

Disordered partitions and packings in high dimensional spaces
We studied the statistical properties of cellular partitions in spaces of arbitrary dimensions and curvature. An important result was the discovery of classes of configurations that are stable under elementary topological transformations (first neighbours exchange, cell division and cell coalescence). These classes represent “fixed points” in a dynamical process of generation of cellular partitions by elementary moves. By using our approach all the average statistical properties of these fixed points can be derived. This is important result, because at the present, very little is known about disordered structures in high dimensions. These studies find applications in information theory, signal processing, analogue-digital converters, neural networks and complex systems dynamics.

Computer simulations and statistical analysis of disordered cellular structures
We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These structures have been studied as organized in concentric shells around a given cell and as topological trees rooted on a central cell. We investigate the parameters that are more efficient in characterizing the different degrees of organization in disordered structures.

Random walks in disordered networks
I studied the effect of disorder on the propagation of a ‘walker’ that starts in a given cell of a two- or tree-dimensional froth and jumps randomly on the neighbouring cells. I found that the coefficients of the evolution equation are strongly affected by the presence of ‘topological defects’ (non-SSI configurations) and by the intrinsic dimension of the network. A comparison between the diffusion in ordered crystalline structures and in disordered structures indicates that in disordered systems the diffusion is faster over a short distance and then asymptotically it becomes slower. These studies are now bringing new insights into the mechanism that leads to non-Gaussian distributions in complex systems.

Dynamical growth of cellular structures and their properties
With the research group in Strasbourg we investigated the statistical properties of disordered cellular systems during their dynamical evolution driven by cell subdivision (mitosis) and cell disappearance. The steady state was studied analytically by using a system of rate equations. The resulting predictions for the cell distribution in biological tissues are in qualitative agreement with available experimental data and with computer simulations. These works indicate that a topological information on a short-range scale is sufficient to explain the evolution and stability of biological tissues.

Slow dynamics in disordered cellular systems and biological tissues
With David Sherrington at the Theoretical Physics Department in Oxford we investigated the dynamical evolution of a disordered system controlled by a stochastic Glauber process determined by the deviations of the local configurations from an ordered arrangement. We discovered that, above a critical temperature, evolution is ‘fast’ and toward a common equilibrium state which is independent on the initial configuration, but beneath this temperature there is a dynamical phase transition with a ‘slow’ evolution characterized by a non-equilibrium glassy freezing. The discovery of a glass dynamics in such a simple model -that has no frustration- is a remarkable result that might help to clarify similar behaviours in more complex systems (as structural glasses). Moreover we observed that the amount of disorder in undifferenziated biological tissues is very similar to the minimal amount of disorder that we obtain in our model before the glassy freezing. So we speculated that the ‘ideal’ biological tissue must fit the compromize between low disorder (homogeneity) and fast dynamics (efficient recovering of perturbations); in our model such a compromize is realized by a structure in equilbrium at a temperature just above the glass transition.

Relation between structural and functional proprieties in disordered cellular systems
These studies were mainly performed during my Ph.D. at the Interdepartmental Centre of Material Science and Engineering, under the supervision of Prof. Dario Beruto. My Ph.D. thesis is a study of the effects of the geometrical and topological constraints on the formation, evolution and functional properties of disordered cellular solids. In the thesis original results concerning random granular systems are applied to the study of the structural and functional properties of granular films for gas sensors. Original results are obtained by analyzing, in terms of maximum packing limited by geometrical constraints, the structural morphogenesys of Sn films reotaxially grown by PVD deposition. In particular, I studied the formation of “breath figures” in films of Sn deposited in high vacuum on a substrate heated at a temperature higher than the tin melting point. Despite the complicated mechanism associated with the dynamical formation of these structures, several relevant properties of these systems have been derived by studying the topological and geometrical constraints resulting from the 2D close packings of circles (or drops) with broad polydispersity. In particular I predicted power law behaviours with exponent equal to - 2 for the drop sizes distribution in “breath figures”. SEM analysis indicate that this distribution is the one observed in such Sn film.
The relation between disorder and functional properties in SnO2 granular sensors were investigated by analytical studies an computer simulation. We found that the sensitivity is improved in devices that work close to the percolation threshold. An original derivation for such a threshold of 2D random networks was proposed.
Finally, an innovative humidity-condensation sensor based on the coalescence mechanism was developed and patented.

Interplay between paramagnetic fluctuations and superconductivity
These studies were performed during my Ms degree thesis and during the following year with Prof. E. Galleani d’Agliano end F. Napoli. The thesis is a theoretical model for high temperature superconductivity induced by the exchange of antiferromagnetic fluctuations in CuO2 layers. The 3-site Hubbard model in presence of holes is used, together with the Path Integral formalism, in order to derive an effective action for the oxygen holes. A mechanism of antiferromagnons exchange between two holes is individuated and an effective attractive potential, which forms the superconducting couples, is derived. We show that this potential is strictly related with the magnetic suscettivity of the Cu-spin system.
After the thesis, and before the Ph.D., I studied the magnetic properties of two dimensional spin systems and the interplay between magnetism and superconductivity in High Tc Superconductors. In particular, we investigated magnetic phases and correlation length in the 3-site Hubbard model by using the Renormalization Group technique applied to the action of the Non-Linear s-Model. A particular effort was devoted to study the effects of doping on the magnetic order and the effect of diamagnetic substitutions of Cu on the critical temperature.