Nonassortative relationships between groups of nodes are typical in complex networks.

Decomposing a graph into groups of nodes that share similar connectivity properties is essential to understand the organization and function of complex networks. Previous works have focused on groups with specific relationships between group members, such as assortative communities or core-periphery structures, developing computational methods to find these mesoscale structures within a network. Here, we go beyond these two traditional cases and introduce a methodology that is able to identify and systematically classify all possible community types in directed multi graphs, based on the pairwise relationship between groups. We apply our approach to 53 different networks and find that assortative communities are the most common structures, but that previously unexplored types appear in almost every network. A particularly prevalent new type of relationship, which we call a source-basin structure, has information flowing from a sparsely connected group of nodes (source) to a densely connected group (basin). We look in detail at two online social networks-a new network of Twitter users and a well-studied network of political blogs-and find that source-basin structures play an important role in both of them. This confirms not only the widespread appearance of nonassortative structures but also the potential of hitherto unidentified relationships to explain the organization of complex networks.

Dark solitons under higher-order dispersion.

We show theoretically that stable dark solitons can exist in the presence of pure quartic dispersion, and also in the presence of both quadratic and quartic dispersive effects, displaying a much greater variety of possible solutions and dynamics than for pure quadratic dispersion. The interplay of the two dispersion orders may lead to oscillatory non-vanishing tails, which enables the possibility of bound, potentially stable, multi-soliton states. Dark soliton-like states that connect to low-amplitude oscillations are also shown to be possible. Dynamical evolution results corroborate the stability picture obtained, and possible avenues for dark soliton generation are explored.

Scaling laws and dynamics of hashtags on Twitter.

In this paper, we quantify the statistical properties and dynamics of the frequency of hashtag use on Twitter. Hashtags are special words used in social media to attract attention and to organize content. Looking at the collection of all hashtags used in a period of time, we identify the scaling laws underpinning the hashtag frequency distribution (Zipf's law), the number of unique hashtags as a function of sample size (Heaps' law), and the fluctuations around expected values (Taylor's law). While these scaling laws appear to be universal, in the sense that similar exponents are observed irrespective of when the sample is gathered, the volume and the nature of the hashtags depend strongly on time, with the appearance of bursts at the minute scale, fat-tailed noise, and long-range correlations. We quantify this dynamics by computing the Jensen-Shannon divergence between hashtag distributions obtained τ times apart and we find that the speed of change decays roughly as 1/τ. Our findings are based on the analysis of 3.5×109 hashtags used between 2015 and 2016.

High-heat-flux rectification due to a localized thermal diode.

A theoretical implementation of a localized thermal diode with a rectification factor greater than 10^{6} is demonstrated. In reverse thermal bias, extremely low thermal conductivity is achieved through phononic Rayleigh scattering from a finite-depth defect. In forward bias, the diode oscillator escapes the defect and thermal conductivity becomes up to four orders of magnitude higher. The setup provides a minimal model of a localized thermal diode between two identical oscillator chains and opens up a pathway for thermal diode implementations.

Self-similar propagation of optical pulses in fibers with positive quartic dispersion.

We study the propagation of ultrashort pulses in optical fiber with gain and positive (or normal) quartic dispersion by self-similarity analysis of the modified nonlinear Schrödinger equation. We find an exact asymptotic solution, corresponding to a triangle-like T4/3 intensity profile, with a T1/3 chirp, which is confirmed by numerical simulations. This solution follows different amplitude and width scaling compared to the conventional case with quadratic dispersion. We also suggest, and numerically investigate, a fiber laser consisting of components with positive quartic dispersion that emits quartic self-similar pulses.

Stationary and dynamical properties of pure-quartic solitons.

We numerically solve a generalized nonlinear Schrödinger equation and find a family of pure-quartic solitons (PQSs), existing through a balance of positive Kerr nonlinearity and negative quartic dispersion. These solitons have oscillatory tails, which can be understood analytically from the properties of linear waves with quartic dispersion. By computing the linear eigenspectrum of the solitons, we show that they are stable, but that they possess a nontrivial internal mode close to the radiation continuum. We also demonstrate evolution into a PQS from Gaussian initial conditions. The energy-width scaling of PQSs differs strongly from that for conventional solitons, opening up possibilities for PQS lasers.

Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays.

We demonstrate experimentally the localization of broad optical beams in periodic arrays of optical waveguides with defocusing nonlinearity. This observation in optics is linked to nonlinear self-trapping of Bose-Einstein-condensed atoms in stationary periodic potentials being associated with the generation of truncated nonlinear Bloch states, existing in the gaps of the linear transmission spectrum. We reveal that unlike gap solitons, these novel localized states can have an arbitrary width defined solely by the size of the input beam while independent of nonlinearity.

Dynamics of matter-wave solitons in a ratchet potential.

We study the dynamics of bright solitons formed in a Bose-Einstein condensate with attractive atomic interactions perturbed by a weak bichromatic optical lattice potential. The lattice depth is a biperiodic function of time with a zero mean, which realizes a flashing ratchet for matter-wave solitons. We find that the average velocity of a soliton and the soliton current induced by the ratchet depend on the number of atoms in the soliton. As a consequence, soliton transport can be induced through scattering of different solitons. In the regime when matter-wave solitons are narrow compared to the lattice period the dynamics is well described by the effective Hamiltonian theory.

Multivortex solitons in triangular photonic lattices.

We introduce a novel class of stable lattice solitons with a complex phase structure composed of many single-charge discrete vortices in a triangular photonic lattice. We demonstrate that such nonlinear self-trapped states are linked to the resonant Bloch modes, which bear a honeycomb pattern of phase dislocations.

Melting of discrete vortices via quantum fluctuations.

We consider nonlinear boson states with a nontrivial phase structure in the three-site Bose-Hubbard ring, quantum discrete vortices (or q vortices), and study their "melting" under the action of quantum fluctuations. We calculate the spatial correlations in the ground states to show the superfluid-insulator crossover and analyze the fidelity between the exact and variational ground states to explore the validity of the classical analysis. We examine the phase coherence and the effect of quantum fluctuations on q vortices and reveal that the breakdown of these coherent structures through quantum fluctuations accompanies the superfluid-insulator crossover.

Self-trapped nonlinear matter waves in periodic potentials.

We demonstrate that the recent observation of nonlinear self-trapping of matter waves in one-dimensional optical lattices [Th. Anker, Phys. Rev. Lett. 94, 020403 (2005)10.1103/PhysRevLett.94.020403] can be associated with a novel type of broad nonlinear state existing in the gaps of the matter-wave band-gap spectrum. We find these self-trapped localized modes in one-, two-, and three-dimensional periodic potentials, and demonstrate that such novel gap states can be generated experimentally in any dimension.

Asymmetric vortex solitons in nonlinear periodic lattices.

We reveal the existence of asymmetric vortex solitons in ideally symmetric periodic lattices and show how such nonlinear localized structures describing elementary circular flows can be analyzed systematically using the energy-balance relations. We present the examples of rhomboid, rectangular, and triangular vortex solitons on a square lattice and also describe novel coherent states where the populations of clockwise and anticlockwise vortex modes change periodically due to a nonlinearity-induced momentum exchange through the lattice. Asymmetric vortex solitons are expected to exist in different nonlinear lattice systems, including optically induced photonic lattices, nonlinear photonic crystals, and Bose-Einstein condensates in optical lattices.

Observation of discrete vortex solitons in optically induced photonic lattices.

We report on the first experimental observation of discrete vortex solitons in two-dimensional optically induced photonic lattices. We demonstrate strong stabilization of an optical vortex by the lattice in a self-focusing nonlinear medium and study the generation of the discrete vortices from a broad class of singular beams.

Ground states and vortices of matter-wave condensates and optical guided waves.

We analyze the shape and stability of localized states in nonlinear cubic media with space-dependent potentials modeling an inhomogeneity. By means of a static variational approach, we describe the ground states and vortexlike stationary solutions, either in dilute atom gases or in optical cavities, with an emphasis on parabolic-type potentials. First, we determine the existence conditions for soliton and vortex structures for both focusing and defocusing nonlinearity. It is shown that, even for a defocusing medium, soliton modes can exist with a confining potential. Second, step potentials and boundedness effects in hollow capillaries are investigated, which both proceed from a similar analysis. Finally, we discuss applications of this procedure to charged vortices in dilute quantum gases and to Bose-Einstein condensates trapped in the presence of a light-induced Gaussian barrier.