Searchability of Networks

We investigate the searchability of complex systems in terms of their interconnectedness. Associating searchability with the number and size of branch points along the paths between the nodes, we find that scale-free networks are relatively difficult to search, and thus that the abundance of scale-free networks in nature and society may reflect an attempt to protect local areas in a highly interconnected network from nonrelated communication. In fact, starting from a random node, real-world networks with higher order organization like modular or hierarchical structure are even more difficult to navigate than random scale-free networks. The searchability at the node level opens the possibility for a generalized hierarchy measure that captures both the hierarchy in the usual terms of trees as in military structures, and the intrinsic hierarchical nature of topological hierarchies for scale-free networks as in the Internet.Comment: 9 pages, 10 figure

Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis

Chemical Evolution in Hierarchical Models of Cosmic Structure II: The Formation of the Milky Way Stellar Halo and the Distribution of the Oldest Stars

This paper presents theoretical star formation and chemical enrichment histories for the stellar halo of the Milky Way based on new chemodynamical modeling. The goal of this study is to assess the extent to which metal-poor stars in the halo reflect the star formation conditions that occurred in halo progenitor galaxies at high redshift, before and during the epoch of reionization. Simple prescriptions that translate dark-matter halo mass into baryonic gas budgets and star formation histories yield models that resemble the observed Milky Way halo in its total stellar mass, metallicity distribution, and the luminosity function and chemical enrichment of dwarf satellite galaxies. These model halos in turn allow an exploration of how the populations of interest for probing the epoch of reionization are distributed in physical and phase space, and of how they are related to lower-redshift populations of the same metallicity. The fraction of stars dating from before a particular time or redshift depends strongly on radius within the galaxy, reflecting the "inside-out" growth of cold-dark-matter halos, and on metallicity, reflecting the general trend toward higher metallicity at later times. These results suggest that efforts to discover stars from z > 6 - 10 should select for stars with [Fe/H] <~ -3 and favor stars on more tightly bound orbits in the stellar halo, where the majority are from z > 10 and 15 - 40% are from z > 15. The oldest, most metal-poor stars - those most likely to reveal the chemical abundances of the first stars - are most common in the very center of the Galaxy's halo: they are in the bulge, but not of the bulge. These models have several implications for the larger project of constraining the properties of the first stars and galaxies using data from the local Universe.Comment: Submitted to ApJ, 22 pages emulateapj, 15 color figure

Graphs with Plane Outside-Obstacle Representations

An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations are a recent generalization of classical polygon--vertex visibility graphs, for which the characterization and recognition problems are long-standing open questions. In this paper, we study \emph{plane outside-obstacle representations}, where all obstacles lie in the unbounded face of the representation and no two visibility segments cross. We give a combinatorial characterization of the biconnected graphs that admit such a representation. Based on this characterization, we present a simple linear-time recognition algorithm for these graphs. As a side result, we show that the plane vertex--polygon visibility graphs are exactly the maximal outerplanar graphs and that every chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure