Resonant Orbits and the High Velocity Peaks Towards the Bulge

We extract the resonant orbits from an N-body bar that is a good representation of the Milky Way, using the method recently introduced by Molloy et al. (2015). By decomposing the bar into its constituent orbit families, we show that they are intimately connected to the boxy-peanut shape of the density. We highlight the imprint due solely to resonant orbits on the kinematic landscape towards the Galactic centre. The resonant orbits are shown to have distinct kinematic features and may be used to explain the cold velocity peak seen in the APOGEE commissioning data (Nidever at al., 2012). We show that high velocity peaks are a natural consequence of the motions of stars in the 2:1 orbit family and that stars on other higher order resonances can contribute to the peaks. The locations of the peaks vary with bar angle and, with the tacit assumption that the observed peaks are due to the 2:1 family, we find that the locations of the high velocity peaks correspond to bar angles in the range 10 < theta_bar < 25 (deg). However, some important questions about the nature of the peaks remain, such as their apparent absence in other surveys of the Bulge and the deviations from symmetry between equivalent fields in the north and south. We show that the absence of a peak in surveys at higher latitudes is likely due to the combination of a less prominent peak and a lower number density of bar supporting orbits at these latitudes.Comment: 7 Figures, 1 Table, Now includes figures & discussion of higher order resonances, Minor revisions to text throughout, Conclusions unchange

Resonant Clumping and Substructure in Galactic Discs

We describe a method to extract resonant orbits from N-body simulations exploiting the fact that they close in a frame rotating with a constant pattern speed. Our method is applied to the N-body simulation of the Milky Way by Shen et al. (2010). This simulation hosts a massive bar, which drives strong resonances and persistent angular momentum exchange. Resonant orbits are found throughout the disc, both close to the bar itself and out to the very edges of the disc. Using Fourier spectrograms, we demonstrate that the bar is driving kinematic substructure even in the very outer parts of the disc. We identify two major orbit families in the outskirts of the disc that make significant contributions to the kinematic landscape, namely the m:l = 3:-2 and 1:-1 families resonating with the pattern speed of the bar. A mechanism is described that produces bimodal distributions of Galactocentric radial velocities at selected azimuths in the outer disc. It occurs as a result of the temporal coherence of particles on the 3:-2 resonant orbits, which causes them to arrive simultaneously at pericentre or apocentre. This resonant clumping, due to the in-phase motion of the particles through their epicycle, leads to both inward and outward moving groups which belong to the same orbital family and consequently produce bimodal radial velocity distributions. This is a possible explanation of the bimodal velocity distributions observed towards the Galactic anti-Centre by Liu et al. (2012). Another consequence is that transient overdensities appear and dissipate (in a symmetric fashion) on timescales equal to the their epicyclic period resulting in a periodic pulsing of the disc's surface density.Comment: 11 Figures, 1 Table. Accepted for publication in ApJ. Version 2 reflects minor changes to the text. Animation referenced in Figure 7 is available at http://hubble.shao.ac.cn/~shen/resonantclumping/DensMovie.mp

Interfaces and the edge percolation map of random directed networks

The traditional node percolation map of directed networks is reanalyzed in terms of edges. In the percolated phase, edges can mainly organize into five distinct giant connected components, interfaces bridging the communication of nodes in the strongly connected component and those in the in- and out-components. Formal equations for the relative sizes in number of edges of these giant structures are derived for arbitrary joint degree distributions in the presence of local and two-point correlations. The uncorrelated null model is fully solved analytically and compared against simulations, finding an excellent agreement between the theoretical predictions and the edge percolation map of synthetically generated networks with exponential or scale-free in-degree distribution and exponential out-degree distribution. Interfaces, and their internal organization giving place from "hairy ball" percolation landscapes to bottleneck straits, could bring new light to the discussion of how structure is interwoven with functionality, in particular in flow networks.Comment: 20 pages, 4 figure

Generalized percolation in random directed networks

We develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges, that is, edges pointing in both directions simultaneously. These two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena. Equations for the percolation threshold and the sizes of the giant components are derived in the most general case. We also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

Giant strongly connected component of directed networks

We describe how to calculate the sizes of all giant connected components of a directed graph, including the {\em strongly} connected one. Just to the class of directed networks, in particular, belongs the World Wide Web. The results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in,out-degree distribution $P(k_i,k_o)$. We show that if $P(k_i,k_o)$ does not factorize, the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in- and out-components. The calculations of the relative sizes of all the giant components are demonstrated using the simplest examples. We explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one.Comment: 4 pages revtex, 4 figure

Evolution equation for a model of surface relaxation in complex networks

In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution $P(k) \sim k^{-\lambda}$ for $\lambda <3$ [Pastore y Piontti {\it et al.}, Phys. Rev. E {\bf 76}, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks non-linear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti {\it et al.} for $\lambda <3$.Comment: 9 pages, 2 figure

Emergence of Clusters in Growing Networks with Aging

We study numerically a model of nonequilibrium networks where nodes and links are added at each time step with aging of nodes and connectivity- and age-dependent attachment of links. By varying the effects of age in the attachment probability we find, with numerical simulations and scaling arguments, that a giant cluster emerges at a first-order critical point and that the problem is in the universality class of one dimensional percolation. This transition is followed by a change in the giant cluster's topology from tree-like to quasi-linear, as inferred from measurements of the average shortest-path length, which scales logarithmically with system size in one phase and linearly in the other.Comment: 8 pages, 6 figures, accepted for publication in JSTA

Transport on weighted Networks: when correlations are independent of degree

Most real-world networks are weighted graphs with the weight of the edges reflecting the relative importance of the connections. In this work, we study non degree dependent correlations between edge weights, generalizing thus the correlations beyond the degree dependent case. We propose a simple method to introduce weight-weight correlations in topologically uncorrelated graphs. This allows us to test different measures to discriminate between the different correlation types and to quantify their intensity. We also discuss here the effect of weight correlations on the transport properties of the networks, showing that positive correlations dramatically improve transport. Finally, we give two examples of real-world networks (social and transport graphs) in which weight-weight correlations are present.Comment: 8 pages, 8 figure