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.

Understanding the origin, nature and functional significance of complex patterns of neural activity, as recorded by diverse electrophysiological and neuroimaging techniques, is a central challenge in Neuroscience. Such patterns include collective oscillations, emerging out of neural synchronization, as well as highly-heterogeneous outbursts of activity interspersed by periods of quiescence, called "neuronal avalanches". Much debate has been generated about the possible scale-invariance or criticality of such avalanches, and its relevance for brain function. Aimed at shedding light onto this, here we analyze the large-scale collective properties of the cortex by using a mesoscopic approach, following the principle of parsimony of Landau-Ginzburg. Our model is similar to that of Wilson-Cowan for neural dynamics but, crucially, including stochasticity and space; synaptic plasticity and inhibition are considered as possible regulatory mechanisms. Detailed analyses uncover a phase diagram including down-states, synchronous, asynchronous, and up-state phases, and reveal that empirical findings for neuronal avalanches are consistently reproduced by tuning our model to the edge of synchronization. This reveals that the putative criticality of cortical dynamics does not correspond to a quiescent-to-active phase transition, as usually assumed in theoretical approaches, but to a synchronization phase transition, at which incipient oscillations and scalefree avalanches coexist. Furthermore, our model also accounts for up and down states as they occur, e.g. during deep sleep. The present approach constitutes a framework to rationalize the possible collective phases and phase transitions of cortical networks in simple terms, thus helping shed light into basic aspects of brain functioning from a very broad perspective.Cortical dynamics | Neuronal avalanches | Criticality | Synaptic plasticity T he cerebral cortex exhibits spontaneous activity even in the absence of any task or external stimuli (1-3). A salient aspect of this, so-called, resting-state dynamics, as revealed by in vivo and in vitro measurements, is that it exhibits outbursts of electrochemical activity, characterized by brief episodes of coherence -during which many neurons fire within a narrow time window-interspaced by periods of relative quiescence, giving rise to collective oscillatory rhythms (4, 5). Shedding light on the origin, nature, and functional meaning of such an intricate dynamics is a fundamental challenge in Neuroscience (6).Upon experimentally enhancing the spatio-temporal resolution of activity recordings, Beggs and Plenz made the remarkable finding that, actually, synchronized outbursts of neural activity could be decomposed into complex spatio-temporal patterns, thereon named "neuronal avalanches" (7). The sizes and durations of such avalanches were reported to be distributed as power-laws, i.e. to be organized in a scale-free way, limited only by network size (7). Furthermore, they obey finite-size scaling (8), a trademark of scale invariance...

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. Random walks in quenched random environments occur in many fields of statistical and condensed matter physics [1], as they represent the simplest model of diffusion phenomena and non-deterministic motion. Disorder and geometrical confinement are known to strongly influence transport properties. In particular, in highly spatial inhomogeneous media, the diffusion process is often characterized by large distance diffusion events, which play a crucial role in transport phenomena and can strongly enhance them [2]. Molecular diffusion at low pressure in porous media is dominated by collision with pore walls, with ballistic motion inside the large pores [3], diffusion in chemical space over polymer chains can be described by a distribution of step length with power law behavior [4]. In addition, recent experiments on new disordered optical materials paved the way to a tuned engineering of Lévy-like distributed step lengths [5]. These and many other processes can often be successfully analyzed using the Lévy walks formalism [6]: The random walker can perform long steps, whose distribution is characterized by a power law behavior λ(r) ∼ 1/r α+1 , with α > 0, for large distance displacements r.An important feature of these experimental settings is that the random walk is in general correlated, and the correlation is induced by the topology of the quenched medium. Diffusing agents moving in highly inhomogeneous regions, where they just experienced a long distance jump without being scattered, have a high probability of being backscattered at the subsequent step undergoing a jump of similar size, and this leads to a correlation in step lengths. While the effect of annealed disorder on transport properties in Lévy walks is quite well understood [7], the role of correlations in Lévy-like motion is still an open problem.If the motion occurs in low dimensional samples, spatial correlations in jump probabilities can deeply influence the diffusion properties. This was first evidenced in models of Lévy flights, [8] and more recently discussed in one-dimensional models for Lévy walks on quenched and correlated random environments. The recent studies focused, respectively, on the mean square displacement in a Lévy-Lorentz gas [9], and on the conductivity and transmission through a chain of barriers with Lévy-distributed spacings [10]. Both studi...

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. LPTENS preprint 95/50 PACS numbers: 05.20-y, 75.10-Nr, 64.60-Cn Submitted to: Europhysics LettersThe recent developments in the theory of spin glass dynamics [1] have made clearer the similarity of behaviour in spin glasses and in glasses [2,3]. In this context it seems at the moment that a certain category of spin glasses, those which are described by a so called one step replica symmetry breaking (RSB) transition [4], are good candidate models for a mean field description of the glass phase [5,6]. In these systems the presence of metastable states generates a purely dynamical transition (which is supposed to be rounded in finite dimensional systems [5,6]) at a temperature T d higher than the one obtained within a theory of static equilibrium, T s .The spherical p-spin spin glass introduced in [7,8] is an interesting example of this category. It is a simple enough system in which the metastable states can be defined and studied by the TAP method [9]. In this paper we want to provide a better understanding of these metastable states, using a dynamical point of view. We shall show the existence of a true ergodicity breaking such that these metastable states, in spite of being excited states with a finite excitation free energy per spin, are actually dynamically stable even at temperatures above T d .The spherical p-spin spin glass describes N real spins s i , i ∈ {1, ..., N } which interact through the Hamiltonian:together with the spherical constraint on the spins:The couplings are gaussian, with zero mean and variance p!/(2N p−1 ). In the p > 2 case it shows an interesting spin glass behaviour, simple enough to allow for detailed analytical treatment.In the static approach, one describes the properties of the Boltzmann probability distribution of this system. The replica method shows the existence of a static transition with a one step RSB at temperature T s [7]. This transition reflects the fact that, below T s , the Boltzmann measure is dominated by a few number of pure states, a scenario which is well known from the random energy model [10].Staying within a static framework, the TAP approach [11,12] provides some more insight into the physical nature of this system. The TAP equations can be derived through a variational principle on the local magnetizations m i =< s i >, from a free energy f ({m i }) which is best written in terms of radial and angular variables, q andŝ i (with m i = √ qŝ i ), in the form [11]:where the angular energy is:

The thermodynamic properties of non interacting bosons on a complex network can be strongly affected by topological inhomogeneities. The latter give rise to anomalies in the density of states that can induce Bose-Einstein condensation in low dimensional systems also in absence of external confining potentials. The anomalies consist in energy regions composed of an infinite number of states with vanishing weight in the thermodynamic limit. We present a rigorous result providing the general conditions for the occurrence of Bose-Einstein condensation on complex networks in presence of anomalous spectral regions in the density of states. We present results on spectral properties for a wide class of graphs where the theorem applies. We study in detail an explicit geometrical realization, the comb lattice, which embodies all the relevant features of this effect and which can be experimentally implemented as an array of Josephson Junctions.

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.PACS numbers: 03.75. Fi, 85.25.Cp, The recent impressive experimental demonstration of Bose-Einstein Condensation (BEC) [1] has stimulated a new wealth of theoretical work aimed to better understanding its basic mechanisms [2] and, possibly, to exploit its consequences for the engineering of quantum devices.It is well known [3] that for an ideal gas of Bose particles BEC does not occur in dimension d ≤ 2, and an ′′ ad hoc ′′ external confining potential is needed to reach the required density of states. The same is true for free bosons living on regular periodic lattices, while the result cannot be extended to more general discrete structures lacking translational invariance.In the following we shall prove that even for d < 2 [4] non-interacting bosons may lead to Bose-Einstein condensation into a single non-degenerate state, provided one resorts to a suitable discrete non-homogeneous support structure: indeed, when the bosonic kinetic degrees of freedom do not depend on metric features only, the particles may feel a sort of effective interaction due to topology. The proposed mechanism for BEC in lower dimensional systems is then a pure effect of the structure of the ambient space and avoids as well the need of resorting to external random potentials as the ones investigated by Huang in [2]; this is a very desirable feature in view of engineering real quantum devices.In practice, the behavior of free bosons over generic discrete structures is made experimentally accessible through the realization of suitable arrays of Josephson junctions. The latter are devices that can be engineered in such a way as to realize a variety of non-homogeneous patterns. We shall show indeed that classical Josephson junction arrays arranged in a non-homogeneous geometry -not even necessarily planar -provide an example of the proposed mechanism for BEC, leading to a single state spatial condensation.Theoretical studies of Josephson junction arrays are based on the short-range Bose-Hubbard model, since the phase diagram of Josephson junction arrays may be derived [5] from an Hamiltonian describing bosons with repulsive interactions over a lattice. In d = 1 the phase diagram has been studied by analytical [6] and quantum Monte Carlo methods [7]; experimentally, Josephson junction arrays are used to study interacting bosons in one dimension. For a generic array the corresponding Hamiltonian is given bywhere A ij is the adjacency matrix: A ij = 1 if the sites i and j are nearest neighbors and A ij = 0 otherwise; a † i creates a boson at site...

The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell. PACS numbers:

The dynamic of social networks is driven by the interplay between diverse mechanisms that still challenge our theoretical and modelling efforts. Amongst them, two are known to play a central role in shaping the networks evolution, namely the heterogeneous propensity of individuals to i) be socially active and ii) establish a new social relationships with their alters. Here, we empirically characterise these two mechanisms in seven real networks describing temporal human interactions in three different settings: scientific collaborations, Twitter mentions, and mobile phone calls. We find that the individuals’ social activity and their strategy in choosing ties where to allocate their social interactions can be quantitatively described and encoded in a simple stochastic network modelling framework. The Master Equation of the model can be solved in the asymptotic limit. The analytical solutions provide an explicit description of both the system dynamic and the dynamical scaling laws characterising crucial aspects about the evolution of the networks. The analytical predictions match with accuracy the empirical observations, thus validating the theoretical approach. Our results provide a rigorous dynamical system framework that can be extended to include other processes shaping social dynamics and to generate data driven predictions for the asymptotic behaviour of social networks.