Community detection and percolation of information in a geometric setting

We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.Comment: 21 page

Stochastic time-dependent current-density functional theory: a functional theory of open quantum systems

The dynamics of a many-body system coupled to an external environment represents a fundamentally important problem. To this class of open quantum systems pertains the study of energy transport and dissipation, dephasing, quantum measurement and quantum information theory, phase transitions driven by dissipative effects, etc. Here, we discuss in detail an extension of time-dependent current-density-functional theory (TDCDFT), we named stochastic TDCDFT [Phys. Rev. Lett. {\bf 98}, 226403 (2007)], that allows the description of such problems from a microscopic point of view. We discuss the assumptions of the theory, its relation to a density matrix formalism, and the limitations of the latter in the present context. In addition, we describe a numerically convenient way to solve the corresponding equations of motion, and apply this theory to the dynamics of a 1D gas of excited bosons confined in a harmonic potential and in contact with an external bath.Comment: 17 pages, 7 figures, RevTex4; few typos corrected, a figure modifie

Spinodal Instabilities in Nuclear Matter in a Stochastic Relativistic Mean-Field Approach

Spinodal instabilities and early growth of baryon density fluctuations in symmetric nuclear matter are investigated in the basis of stochastic extension of relativistic mean-field approach in the semi-classical approximation. Calculations are compared with the results of non-relativistic calculations based on Skyrme-type effective interactions under similar conditions. A qualitative difference appears in the unstable response of the system: the system exhibits most unstable behavior at higher baryon densities around $\rho_{b}=0.4 ~\rho_{0}$ in the relativistic approach while most unstable behavior occurs at lower baryon densities around $\rho_{b}=0.2 ~\rho_{0}$ in the non-relativistic calculationsComment: 18 pages, 7 figure

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina

Convergence in Models with Bounded Expected Relative Hazard Rates

We provide a general framework to study stochastic sequences related to individual learning in economics, learning automata in computer sciences, social learning in marketing, and other applications. More precisely, we study the asymptotic properties of a class of stochastic sequences that take values in $[0,1]$ and satisfy a property called "bounded expected relative hazard rates." Sequences that satisfy this property and feature "small step-size" or "shrinking step-size" converge to 1 with high probability or almost surely, respectively. These convergence results yield conditions for the learning models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and Schlag (1998) to choose expected payoff maximizing actions with probability one in the long run.Comment: After revision. Accepted for publication by Journal of Economic Theor