Raffaella Burioni

Random walks on graphs: ideas, techniques and results

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.

Random walks on deterministic scale-free networks: Exact results

We study the random walk problem on a class of deterministic scale-free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma=log 3/log 2. We find exact results concerning different first-passage phenomena and, in particular, we calculate the probability of first return to the main hub. These results allow to derive the exact analytic expression for the mean time to first reach the main hub, whose leading behavior is given by tau approximately V1-1/gamma, where V denotes the size of the structure, and the mean is over a set of starting points distributed uniformly over all the other sites of the graph. Interestingly, the process turns out to be particularly efficient. We also discuss the thermodynamic limit of the structure and some local topological properties.

Topological thermal instability and length of proteins

We present an analysis of the effects of global topology on the structural stability of folded proteins in thermal equilibrium with a heat bath. For a large class of single domain proteins, we computed the harmonic spectrum within the Gaussian Network Model (GNM) and determined their spectral dimension, a parameter describing the low frequency behavior of the density of modes. We found a surprisingly strong correlation between the spectral dimension and the number of amino acids in the protein. Considering that larger spectral dimension values relate to more topologically compact folded states, our results indicate that, for a given temperature and length of protein, the folded structure corresponds to a less compact folding, one compatible with thermodynamic stability. Proteins 2004.

Efficiency of information spreading in a population of diffusing agents.

We introduce a model for information spreading among a population of N agents diffusing on a square L x L lattice, starting from an informed agent (Source). Information passing from informed to unaware agents occurs whenever the relative distance is < or = 1. Numerical simulations show that the time required for the information to reach all agents scales as N(-alpha)L(beta), where alpha and beta are noninteger. A decay factor z takes into account the degeneration of information as it passes from one agent to another; the final average degree of information of the population tau(av)(z) is thus history dependent. We find that the behavior of tau(av)(z) is nonmonotonic with respect to N and L and displays a set of minima. Part of the results are recovered with analytical approximations.

Effective target arrangement in a deterministic scale-free graph.

We study the random-walk problem on a deterministic scale-free network, in the presence of a set of static, identical targets; due to the strong inhomogeneity of the underlying structure the mean first-passage time (MFPT), meant as a measure of transport efficiency, is expected to depend sensitively on the position of targets. We consider several spatial arrangements for targets and we calculate, mainly rigorously, the related MFPT, where the average is taken over all possible starting points and over all possible paths. For all the cases studied, the MFPT asymptotically scales like ∼Nθ, being N the volume of the substrate and θ ranging from 1-log 2/log 3, for central target(s), to 1, for a single peripheral target.

Bose-Einstein condensation in inhomogeneous Josephson arrays

We show that spatial Bose-Einstein condensation of non-interacting bosons occurs in dimension d < 2 over discrete structures with inhomogeneous topology and with no need of external confining potentials. Josephson junction arrays provide a physical realization of this mechanism. The topological origin of the phenomenon may open the way to the engineering of quantum devices based on Bose-Einstein condensation. The comb array, which embodies all the relevant features of this effect, is studied in detail.

Existence and theoretical aspects of homomeric and heteromeric dopamine receptor complexes and their relevance for neurological diseases

Dopamine (DA) and other receptors physically interact in the plasma membrane of basal ganglia neurons forming receptor mosaics (RMs). Two types of RMs are discussed, homomers formed only by DA-receptor (DA-R) subtypes and heteromers formed by DA-R associated with other receptors, such as A2A, A1, mGluR5, N-methyl-d-aspartate (NMDA), γ-aminobutryic acid (GABA)-A, and α-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid. By being part of horizontal molecular networks, RMs tune multiple effector systems already at membrane level, such as G protein regulated inward rectifying potassium channels and dopamine transporter activity. Also, ligand-gated ion channels such as GABA-A and NMDA receptors are modulated by DA-R, e.g., in the striatal GABA output neurons through the formation of heteromeric complexes with these receptors. Thus, intramembrane DA-R-receptor interactions play an important role in the information handling in the basal ganglia. On this basis, functional implications of DA RM in physiological and pathological conditions are discussed. The effects of temperature on RM are discussed not only because receptor-decoding mechanisms are temperature sensitive, but also in view of the suggestion that possible ordering effects (i.e., changes in the entropy of a receptor complex) induced by a ligand are as a result of alterations in the receptor oligomerization (i.e., are related to rearrangements of the RM). Hence, brain temperature may have profound effects on brain integrative functions not only because its effects on the kinetics of biochemical reactions, but also for its effects on receptor geometry, building up of RM, and alterations in protein expression, as is the case of H-channels following febrile seizures.

Dynamics within metastable states in a mean-field spin glass

In this letter we present a dynamical study of the structure of metastable states (corresponding to TAP solutions) in a mean-field spin-glass model. After reviewing known results of the statical approach, we use dynamics: starting from an initial condition thermalized at a temperature between the statical and the dynamical transition temperatures, we are able to study the relaxational dynamics within metastable states and we show that they are characterized by a true breaking of ergodicity and exponential relaxation.

Lévy walks and scaling in quenched disordered media.

We study Lévy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean-square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites. Our results are compared with numerical simulations, with excellent agreement.

The dissipative Bose-Hubbard model

Open many-body quantum systems have attracted renewed interest in the context of quantum information science and quantum transport with biological clusters and ultracold atomic gases. The physical relevance in many-particle bosonic systems lies in the realization of counter-intuitive transport phenomena and the stochastic preparation of highly stable and entangled many-body states due to engineered dissipation. We review a variety of approaches to describe an open system of interacting ultracold bosons which can be modeled by a tight-binding Hubbard approximation. Going along with the presentation of theoretical and numerical techniques, we present a series of results in diverse setups, based on a master equation description of the dissipative dynamics of ultracold bosons in a one-dimensional lattice. Next to by now standard numerical methods such as the exact unravelling of the master equation by quantum jumps for small systems and beyond mean-field expansions for larger ones, we present a coherent-state path integral formalism based on Feynman-Vernon theory applied to a many-body context.