Microscopic energy flows in disordered Ising spin systems

An efficient microcanonical dynamics has been recently introduced for Ising spin models embedded in a generic connected graph even in the presence of disorder i.e. with the spin couplings chosen from a random distribution. Such a dynamics allows a coherent definition of local temperatures also when open boundaries are coupled to thermostats, imposing an energy flow. Within this framework, here we introduce a consistent definition for local energy currents and we study their dependence on the disorder. In the linear response regime, when the global gradient between thermostats is small, we also define local conductivities following a Fourier dicretized picture. Then, we work out a linearized "mean-field approximation", where local conductivities are supposed to depend on local couplings and temperatures only. We compare the approximated currents with the exact results of the nonlinear system, showing the reliability range of the mean-field approach, which proves very good at high temperatures and not so efficient in the critical region. In the numerical studies we focus on the disordered cylinder but our results could be extended to an arbitrary, disordered spin model on a generic discrete structures.Comment: 12 pages, 6 figure

Instability and network effects in innovative markets

We consider a network of interacting agents and we model the process of choice on the adoption of a given innovative product by means of statistical-mechanics tools. The modelization allows us to focus on the effects of direct interactions among agents in establishing the success or failure of the product itself. Mimicking real systems, the whole population is divided into two sub-communities called, respectively, Innovators and Followers, where the former are assumed to display more influence power. We study in detail and via numerical simulations on a random graph two different scenarios: no-feedback interaction, where innovators are cohesive and not sensitively affected by the remaining population, and feedback interaction, where the influence of followers on innovators is non negligible. The outcomes are markedly different: in the former case, which corresponds to the creation of a niche in the market, Innovators are able to drive and polarize the whole market. In the latter case the behavior of the market cannot be definitely predicted and become unstable. In both cases we highlight the emergence of collective phenomena and we show how the final outcome, in terms of the number of buyers, is affected by the concentration of innovators and by the interaction strengths among agents.Comment: 20 pages, 6 figures. 7th workshop on "Dynamic Models in Economics and Finance" - MDEF2012 (COST Action IS1104), Urbino (2012

Universal features of information spreading efficiency on dd-dimensional lattices

A model for information spreading in a population of $N$ mobile agents is extended to $d$-dimensional regular lattices. This model, already studied on two-dimensional lattices, also takes into account the degeneration of information as it passes from one agent to the other. Here, we find that the structure of the underlying lattice strongly affects the time $\tau$ at which the whole population has been reached by information. By comparing numerical simulations with mean-field calculations, we show that dimension $d=2$ is marginal for this problem and mean-field calculations become exact for $d > 2$. Nevertheless, the striking nonmonotonic behavior exhibited by the final degree of information with respect to $N$ and the lattice size $L$ appears to be geometry independent.Comment: 8 pages, 9 figure

Exact mean first-passage time on the T-graph

We consider a simple random walk on the T-fractal and we calculate the exact mean time $\tau^g$ to first reach the central node $i_0$. The mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph, except $i_0$. By means of analytic techniques based on decimation procedures, we find the explicit expression for $\tau^g$ as a function of the generation $g$ and of the volume $V$ of the underlying fractal. Our results agree with the asymptotic ones already known for diffusion on the T-fractal and, more generally, they are consistent with the standard laws describing diffusion on low-dimensional structures.Comment: 6 page

A Hebbian approach to complex network generation

Through a redefinition of patterns in an Hopfield-like model, we introduce and develop an approach to model discrete systems made up of many, interacting components with inner degrees of freedom. Our approach clarifies the intrinsic connection between the kind of interactions among components and the emergent topology describing the system itself; also, it allows to effectively address the statistical mechanics on the resulting networks. Indeed, a wide class of analytically treatable, weighted random graphs with a tunable level of correlation can be recovered and controlled. We especially focus on the case of imitative couplings among components endowed with similar patterns (i.e. attributes), which, as we show, naturally and without any a-priori assumption, gives rise to small-world effects. We also solve the thermodynamics (at a replica symmetric level) by extending the double stochastic stability technique: free energy, self consistency relations and fluctuation analysis for a picture of criticality are obtained

A Two-populations Ising model on diluted Random Graphs

We consider the Ising model for two interacting groups of spins embedded in an Erd\"{o}s-R\'{e}nyi random graph. The critical properties of the system are investigated by means of extensive Monte Carlo simulations. Our results evidence the existence of a phase transition at a value of the inter-groups interaction coupling $J_{12}^C$ which depends algebraically on the dilution of the graph and on the relative width of the two populations, as explained by means of scaling arguments. We also measure the critical exponents, which are consistent with those of the Curie-Weiss model, hence suggesting a wide robustness of the universality class.Comment: 11 pages, 4 figure

"Credit Cycle" in an OLG Economy with Money and Bequest

In the late '90s Kiyotaki and Moore (KM) put forward a new framework (Kiyotaki and Moore,1997) to explore the Financial Accelerator hypothesis. The original model was framed in an Infinitely Lived Agent context (ILA-KM economy). As in KM we develop a dynamic model in which the durable asset ("land") is not only an input but also collateralizable wealth to secure lenders from the risk of borrowers' default. In this paper, however, we model an OLG-KM economy whose novel feature is the role of money as a store of value and of bequest as a vehicle of resources to be "invested" in landholding. The dynamics generated by the model are complex. Not only cyclical patterns are routinely generated but the periodicity and amplitude are irregular. A route to chaotic dynamics is open.

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 N^{theta}, being N the volume of the substrate and theta ranging from (1 - log 2/log3), for central target(s), to 1, for a single peripheral target.Comment: 8 pages, 5 figure

Ferromagnetic models for cooperative behavior: Revisiting Universality in complex phenomena

Ferromagnetic models are harmonic oscillators in statistical mechanics. Beyond their original scope in tackling phase transition and symmetry breaking in theoretical physics, they are nowadays experiencing a renewal applicative interest as they capture the main features of disparate complex phenomena, whose quantitative investigation in the past were forbidden due to data lacking. After a streamlined introduction to these models, suitably embedded on random graphs, aim of the present paper is to show their importance in a plethora of widespread research fields, so to highlight the unifying framework reached by using statistical mechanics as a tool for their investigation. Specifically we will deal with examples stemmed from sociology, chemistry, cybernetics (electronics) and biology (immunology).Comment: Contributing to the proceedings of the Conference "Mathematical models and methods for Planet Heart", INdAM, Rome 201