Transitions in spatial networks

Networks embedded in space can display all sorts of transitions when their structure is modified. The nature of these transitions (and in some cases crossovers) can differ from the usual appearance of a giant component as observed for the Erdos-Renyi graph, and spatial networks display a large variety of behaviors. We will discuss here some (mostly recent) results about topological transitions, `localization' transitions seen in the shortest paths pattern, and also about the effect of congestion and fluctuations on the structure of optimal networks. The importance of spatial networks in real-world applications makes these transitions very relevant and this review is meant as a step towards a deeper understanding of the effect of space on network structures.Comment: Corrected version and updated list of reference

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

Structure and dynamics of transportation networks: Models, methods and applications

http://www.uk.sagepub.com/books/Book234882The first section discusses the static dimension (structure) and reviews how transportation networks have been de fined and analyzed with regard to their topology, geometry, morphology, and spatial structure. It presents a critical overview of main global (network level) and local (node level) measures and examines their usefulness for understanding transportation networks. The second section explores the dynamics of transportation networks, their evolution, and the properties underlying such evolutions. Each section provides a brief background of the relevant literature, concrete applications, and policy implications in various transport modes and industries, with an interdisciplinary focus. A discussion is provided evaluating the legacy of reviewed works and potential for further developments in transport studies in general