DOCSLIB.ORG

Spatial Networks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spatial Networks Spatial Networks David Aldous August 30, 2013 Course web site: Google “David Aldous”. [show] I deliberately chose a topic where there is not any clean definition-theorem-proof account – but such accounts exist for various related topics. A metaphor for research styles: external DLA and internal DLA. [show 2 figs]. So start by looking outside the math literature. What does Google Scholar find? Journal Networks and spatial economics [show] This is O.R. style – not quite what we will focus on. Note student requirements: give a talk, either on a research paper you read (easy option), or on some project you do (harder option). [show lists] “Big Picture” on slides. When we get to non-trivial math, I’ll do chalk. Consider points in 2-dimensional space. Mathematicians understand regular patterns. [show Lattice-wikipedia]. Mathematicians also understand completely random points, as modeled by the Poisson point process. [show poisson.jpg] [show Complete-wikipedia]. [show list projects] In mathematics, graph has a fairly precise meaning – vertices (nodes) and edges (links) – I will call this an “abstract graph”. [draw on board]. But network does not have a standard meaning – I say “a network is a graph with context-dependent extra structure”. In our topic, spatial networks, vertices have positions in the plane R2, and the edges (usually made up of line segments) are point-sets in R2. [introduce coordinates]. Familiar “regular pattern networks” [show 9networks] and the kind of questions we will study are (usually) elementary for regular patterns. What can we do with arbitrary configurations of points in the plane? Two classical structures are the Voronoi tessellation and the Delaunay triangulation. [show Voronoi and Delaunay] We can think of the Delaunay triangulation as the prototype example of aspatialnetwork. The Delaunay triangulation (DT) is at the intersection of several academic fields. In the background are “graph theory” in Math much of “theory of algorithms” in C.S. involves algorithms on abstract graphs and in the foreground are Euclidean geometry (the DT itself) computational geometry studies algorithms for problems such as finding the DT stochastic geometry ( geometric probability)studies quantitative aspects (expectations≈ etc) of structures such as the DT built over random points. [show book list] One starting point for this course is to think about random analogs of regular networks. There are several ways to make probability models. I will talk first about schemes for defining a network over arbitrary points (e.g. the DT), applied to random (Poisson) points in the plane but other ways start with a regular network (e.g. square grid) and then randomly delete edges (percolation) randomly add long edges (small worlds) assign random lengths to edges (first passage percolation). Side comments. There is extensive work on (i) random processes built over regular networks – percolation, interacting particle systems – see Grimmett Probability on Graphs for a start. (ii) random models for abstract graphs, generalizing the Erd˝os-R´enyi model – see van der Hofstad Random Graphs and Complex Networks. Something potentially confusing is the notion of a planar graph.Thisis defined as an abstract graph which can be drawn in the plane with non-crossing edges. This verbal definition looks ambiguous – are edges required to be straight lines? – but it’s a fact that the two definitions are equivalent. [draw picture on board] This differs from a spatial network in several ways. A spatial network is drawn in the plane – vertices and edges are at specified coordinates. So there are several possibilities for a spatial network: 1 edges are line segments and do not cross (e,g, the DT) 2 edges are line segments; may cross but you cannot switch (e.g. airplane routes) 3 edges can meet at positions (Steiner points or junctions) that are not the given vertices. Most literature concerns networks satisfying (1) or (2), but we (envisaging inter-city road networks) will often allow (3). [show US-road-map.pdf] Side comment. There is literature on “random planar graphs”: see Benjamini (2010) for a brief overview. There the model is the uniform distribution on the finite set of n-vertex planar graphs, but this model does not seem realistic for any data. Another fact about connected planar graphs is Euler’s formula N E + F =2 − N = number of vertices E = number of edges F = number of faces (= cells = components of plane). [true for tree; induct on extra edges] Taking E 3, a face has at least 3 edges and each edge separates at most 2 faces,≥ so E (3/2)F ;substituteintoEuler’sformula: ≥ E 3N 6. ≤ − So the average degree d¯ in a connected planar graph d¯ =2E/N 6 12 < 6. ≤ − N [show 9networks again] Finite vs Infinite networks Real networks are finite, and algorithmic questions typically ask how many steps are required to solve some problem over n points in the worst case. Mathematicians like me often think in terms of infinite numbers of points in the infinite plane. The latter is perhaps an example of mathematicians putting a lot of effort into being lazy; in particular one can ignore boundary effects. For instance On the infinite square grid the average degree is 4 and the average length per unit area is 2. The Voronoi tessellation on a (reasonable) infinite set of points has only finite-area cells. In any triangulation of a (reasonable) infinite set of points, the average degree is 6. [informal argument on board]. The infinite setting is not so relevant for algorithmic or worst-case studies, but is useful for average-case studies. The natural “average-case” model for n points is to take them IID uniform in a square; to avoid rescaling later we take the scaling convention that the square has area = n. The corresponding infinite model is the rate-1 Poisson point process (PPP) on R2. Most explicit constructions of networks on n points can be used as constructions of networks on the infinite PPP. Precise quantitative connections between the finite and infinite networks are one of the technical topics we’ll discuss later in the course, involving subadditivity local weak convergence stationarity as the formalization of “reasonable” for infinite networks. Comments on triangulations (define: = point set triangulation). not unique [draw n-gon] given a planar graph spatial network which is not a triangulation, we can add edges until it becomes a triangulation in a triangulation, [board] 3(F 1) = 2E C − − where C = number of edges (or vertices) in convex hull. Euler’s formula then gives E =3N 3 C − − and so the average degree is d¯ = 2E =6 6 2C N − N − N which does not depend on the triangulation The DT is in general not the minimum-length triangulation; computing the latter is NP-hard. A planar graph has a (planar) dual graph [show Dual] [show 9networks: multigraph] This duality is the relationship between Voronoi tesselation (considered as a 3-regular graph) and Delaunay triangulation (DT). Mention 4-color theorem. Duality relates this to vertex-coloring of the dual graph. 2 things to note. What is the DT for the square-lattice configuration? An edge AB of the DT does not necessarily pass through the edge separating the cells of A and B; the subgraph of edge that do defines the Gabriel graph. [draw fig 2] [show Gabriel graph = Wiki] AB is edge of Gabriel graph iffdisc with diameter AB contains no other points AB is edge of DT iffthere exists some disc, with A and B on boundary, containing no other points. Average degree of the Gabriel graph depends on configuration; for instance it is 6 for the triangular configuation. We shall soon do a calculation to show that for Poisson points, ave degree := d¯ = 4; length-per-unit-area := =2. These of course are the values for the usual grid network. Is there any more conceptual explanation of this coincidence? Some useful calculus formulas ∞ 1 r exp( Ar 2) dr = · − 2A 0 1/2 ∞ π exp( Ar 2) dr = − 2A1/2 0 Higher moment formulas most easily found by differentiating w.r.t. A 1/2 ∞ π r 2 exp( Ar 2) dr = · − 4A3/2 0 Some calculations with PPP [on board] Definitions of spatial PPP and finite-n model. Link via “random position” and “random point”. PPP as seen from “typical point” is just the PPP with an extra point planted at origin. Use in calculations that are exact for PPP model, and are n limits for finite-n model. →∞ Basic example. In PPP, R = distance from typical point ξ to nearest neighbor ξ has P(R > r)=exp( πr 2)andsoER =1/2. − In finite-n model, consider Rn = distance from random position to nearest point Rn∗ = distance from random point to nearest other point. d d Then R R, R∗ R. n → n → More elaborate example. In PPP, take ξ as typical point, ξ as nearest neighbor of ξ,andξ as nearest neighbor of ξ. Then [board] π q := P(ξ = ξ)= . 4π 3 3 + 4 In finite-n model, let Mn = number of points which are the nearest neighbor of their nearest neighbor. Then 1 n− EMn q. → To re-interpret as a “network”, put an edge from each point of a PPP to its nearest neighbor, the mean number of edges per unit area equals q 2 +(1 q). And [exercise] can also calculate mean length-per-unit-area of this network.− Almost all the networks we will consider, build over the PPP, have distributions invariant under translation and rotation [as will be discussed more carefully later]. For such a network, given the PPP has points at z1 and z2,thechancep(r) that there is an edge (z1, z2) depends only on the distance r between z1 and z2.Thenwecancalculate ∞ ave degree := d¯ = p(r) 2πrdr · 0 ∞ length-per-unit-area := = 1 r p(r) 2πrdr.
Load more

Recommended publications
  • Oracle Database 10G: Oracle Spatial Network Data Model
    Oracle Database 10g: Oracle Spatial Network Data Model An Oracle Technical White Paper May 2005 Oracle Spatial Network Data Model Table of Contents Introduction ....................................................................................................... 3 Design Goals and Architecture ....................................................................... 4 Major Steps Using the Network Data Model................................................ 5 Network Modeling........................................................................................ 5 Network Analysis.......................................................................................... 6 Network Modeling ............................................................................................ 6 Metadata and Schema ....................................................................................... 6 Network Java Representation and API.......................................................... 7 Network Creation Using SQL and PL/SQL ................................................ 8 Creating a Logical Network......................................................................... 9 Creating a Spatial Network.......................................................................... 9 Creating an SDO Geometry Network ................................................ 10 Creating an LRS Geometry Network.................................................. 10 Creating a Topology Geometry Network........................................... 11 Network Creation
    [Show full text]
  • A Network-Of-Networks Percolation Analysis of Cascading Failures in Spatially Co-Located Road-Sewer Infrastructure Networks
    Physica A 538 (2020) 122971 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa A network-of-networks percolation analysis of cascading failures in spatially co-located road-sewer infrastructure networks ∗ ∗ Shangjia Dong a, , Haizhong Wang a, , Alireza Mostafizi a, Xuan Song b a School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331, United States of America b Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China article info a b s t r a c t Article history: This paper presents a network-of-networks analysis framework of interdependent crit- Received 26 September 2018 ical infrastructure systems, with a focus on the co-located road-sewer network. The Received in revised form 28 May 2019 constructed interdependency considers two types of node dynamics: co-located and Available online 27 September 2019 multiple-to-one dependency, with different robustness metrics based on their function Keywords: logic. The objectives of this paper are twofold: (1) to characterize the impact of the Co-located road-sewer network interdependency on networks' robustness performance, and (2) to unveil the critical Network-of-networks percolation transition threshold of the interdependent road-sewer network. The results Percolation modeling show that (1) road and sewer networks are mutually interdependent and are vulnerable Infrastructure interdependency to the cascading failures initiated by sewer system disruption; (2) the network robust- Cascading failure ness decreases as the number of initial failure sources increases in the localized failure scenarios, but the rate declines as the number of failures increase; and (3) the sewer network contains two types of links: zero exposure and severe exposure to liquefaction, and therefore, it leads to a two-phase percolation transition subject to the probabilistic liquefaction-induced failures.
    [Show full text]
  • Planarity As a Driver of Spatial Network Structure
    DOI: http://dx.doi.org/10.7551/978-0-262-33027-5-ch075 Planarity as a driver of Spatial Network structure Garvin Haslett and Markus Brede Institute for Complex Systems Simulation, School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, United Kingdom [email protected] Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2015/27/423/1903775/978-0-262-33027-5-ch075.pdf by guest on 23 September 2021 Abstract include city science (Cardillo et al., 2006; Xie and Levin- son, 2007; Jiang, 2007; Barthelemy´ and Flammini, 2008; In this paper we introduce a new model of spatial network growth in which nodes are placed at randomly selected lo- Masucci et al., 2009; Chan et al., 2011; Courtat et al., 2011; cations in space over time, forming new connections to old Strano et al., 2012; Levinson, 2012; Rui et al., 2013; Gud- nodes subject to the constraint that edges do not cross. The mundsson and Mohajeri, 2013), electronic circuits (i Can- resulting network has a power law degree distribution, high cho et al., 2001; Bassett et al., 2010; Miralles et al., 2010; clustering and the small world property. We argue that these Tan et al., 2014), wireless networks (Huson and Sen, 1995; characteristics are a consequence of two features of our mech- anism, growth and planarity conservation. We further pro- Lotker and Peleg, 2010), leaf venation (Corson, 2010; Kat- pose that our model can be understood as a variant of ran- ifori et al., 2010), navigability (Kleinberg, 2000; Lee and dom Apollonian growth.
    [Show full text]
  • Arxiv:2006.02870V1 [Cs.SI] 4 Jun 2020
    The why, how, and when of representations for complex systems Leo Torres Ann S. Blevins [email protected] [email protected] Network Science Institute, Department of Bioengineering, Northeastern University University of Pennsylvania Danielle S. Bassett Tina Eliassi-Rad [email protected] [email protected] Department of Bioengineering, Network Science Institute and University of Pennsylvania Khoury College of Computer Sciences, Northeastern University June 5, 2020 arXiv:2006.02870v1 [cs.SI] 4 Jun 2020 1 Contents 1 Introduction 4 1.1 Definitions . .5 2 Dependencies by the system, for the system 6 2.1 Subset dependencies . .7 2.2 Temporal dependencies . .8 2.3 Spatial dependencies . 10 2.4 External sources of dependencies . 11 3 Formal representations of complex systems 12 3.1 Graphs . 13 3.2 Simplicial Complexes . 13 3.3 Hypergraphs . 15 3.4 Variations . 15 3.5 Encoding system dependencies . 18 4 Mathematical relationships between formalisms 21 5 Methods suitable for each representation 24 5.1 Methods for graphs . 24 5.2 Methods for simplicial complexes . 25 5.3 Methods for hypergraphs . 27 5.4 Methods and dependencies . 28 6 Examples 29 6.1 Coauthorship . 29 6.2 Email communications . 32 7 Applications 35 8 Discussion and Conclusion 36 9 Acknowledgments 38 10 Citation diversity statement 38 2 Abstract Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language.
    [Show full text]
  • Query Processing in Spatial Network Databases
    Query Processing in Spatial Network Databases Dimitris Papadias Jun Zhang Department of Computer Science Department of Computer Science Hong Kong University of Science and Technology Hong Kong University of Science and Technology Clearwater Bay, Hong Kong Clearwater Bay, Hong Kong [email protected] [email protected] Nikos Mamoulis Yufei Tao Department of Computer Science and Information Systems Department of Computer Science University of Hong Kong City University of Hong Kong Pokfulam Road, Hong Kong Tat Chee Avenue, Hong Kong [email protected] [email protected] than their Euclidean distance. Previous work on spatial Abstract network databases (SNDB) is scarce and too restrictive Despite the importance of spatial networks in for emerging applications such as mobile computing and real-life applications, most of the spatial database location-based commerce. This necessitates the literature focuses on Euclidean spaces. In this development of novel and comprehensive query paper we propose an architecture that integrates processing methods for SNDB. network and Euclidean information, capturing Every conventional spatial query type (e.g., nearest pragmatic constraints. Based on this architecture, neighbors, range search, spatial joins and closest pairs) we develop a Euclidean restriction and a has a counterpart in SNDB. Consider, for instance, the network expansion framework that take road network of Figure 1.1, where the rectangles advantage of location and connectivity to correspond to hotels. If a user at location q poses the efficiently prune the search space. These range query "find the hotels within a 15km range", the frameworks are successfully applied to the most result will contain a, b and c (the numbers in the figure popular spatial queries, namely nearest correspond to network distance).
    [Show full text]
  • Urban Street Network Analysis in a Computational Notebook
    Urban Street Network Analysis in a Computational Notebook Geoff Boeing Department of Urban Planning and Spatial Analysis Sol Price School of Public Policy University of Southern California [email protected] Abstract: Computational notebooks offer researchers, practitioners, students, and educators the ability to interactively conduct analytics and disseminate re- producible workflows that weave together code, visuals, and narratives. This article explores the potential of computational notebooks in urban analytics and planning, demonstrating their utility through a case study of OSMnx and its tu- torials repository. OSMnx is a Python package for working with OpenStreetMap data and modeling, analyzing, and visualizing street networks anywhere in the world. Its official demos and tutorials are distributed as open-source Jupyter notebooks on GitHub. This article showcases this resource by documenting the repository and demonstrating OSMnx interactively through a synoptic tutorial adapted from the repository. It illustrates how to download urban data and model street networks for various study sites, compute network indicators, vi- sualize street centrality, calculate routes, and work with other spatial data such as building footprints and points of interest. Computational notebooks help in- troduce methods to new users and help researchers reach broader audiences in- terested in learning from, adapting, and remixing their work. Due to their utility and versatility, the ongoing adoption of computational notebooks in urban plan- ning, analytics, and related geocomputation disciplines should continue into the future.1 1 Introduction A traditional academic and professional divide has long existed between code creators and code users. The former would develop software tools and workflows for professional or research ap- plications, which the latter would then use to conduct analyses or answer scientific questions.
    [Show full text]
  • Shape of Shortest Paths in Random Spatial Networks Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann
    Shape of shortest paths in random spatial networks Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann To cite this version: Alexander Kartun-Giles, Marc Barthelemy, Carl Dettmann. Shape of shortest paths in random spatial networks. Physical Review E , American Physical Society (APS), 2019, 100 (3), pp.2315. 10.1103/PhysRevE.100.032315. cea-02564008 HAL Id: cea-02564008 https://hal-cea.archives-ouvertes.fr/cea-02564008 Submitted on 2 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The shape of shortest paths in random spatial networks Alexander P. Kartun-Giles,1 Marc Barthelemy,2 and Carl P. Dettmann3 1Max Plank Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig, Germany 2Institut de Physique Th´eorique,CEA, CNRS-URA 2306, Gif-sur-Yvette, France 3School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order Lχ, while the transversal fluctuations, known as wandering, grow as Lξ. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ = 3=5 and χ = 1=5, or ξ = 7=10 and χ = 2=5, depending only on coarse details of the specific connectivity laws used.
    [Show full text]
  • Experimenting with Spatial Networks to Save Time When Experimenting, It Can Be Useful to Create a Subset of the Airport Dataset
    Experimenting with Spatial Networks To save time when experimenting, it can be useful to create a subset of the airport dataset, for instance keeping only the airports of larger degree. Note also that when computing deterrence function, it is sometimes useful to work with random samples instead of computing distances between all pairs of nodes. This dataset is small enough however to compute all values in less than 1 minute on an average personal computer. To the best of my knowledge, there is no good library to work with spatial networks. networkx has functions to generate random spatial graphs (called geometric graphs https://networkx.org/ documentation/stable/reference/generators.html), but they are not designed to fit deterrence functions, and are not adapted to work with geographical coordinates. 1. Deterrence Function on the airport dataset When studying a network with spatial information, a first step to check if it can indeed be considered as a spatial network is to compute its deterrence function. (a) We will need to discretize distances using bins(similar to a histogram). Choose b bins, for instance every 500 km from 0km to 10,000km. Write a function which, provided a list of distances, return a list with b values, the number of values in each bin. A convenient way is to use numpy.digitize and collections.Counter . (b) Write a function which compute the distance between two positions on Earth in km. You can use function haversine from the package of the same name( pip install haversine ). To obtain standard latitude and longitude from the data in the airport dataset, you need to divide latitude and longitude values by 3600.
    [Show full text]
  • Osmnx: New Methods for Acquiring, Constructing, Analyzing, and Visualizing Complex Street Networks
    OSMnx: New Methods for Acquiring, Constructing, Analyzing, and Visualizing Complex Street Networks Geoff Boeing Email: [email protected] Department of City and Regional Planning University of California, Berkeley May 2017 Abstract Urban scholars have studied street networks in various ways, but there are data availability and consistency limitations to the current urban planning/street network analysis literature. To address these challenges, this article presents OSMnx, a new tool to make the collection of data and creation and analysis of street networks simple, consistent, automatable and sound from the perspectives of graph theory, transportation, and urban design. OSMnx contributes five significant capabilities for researchers and practitioners: first, the automated downloading of political boundaries and building footprints; second, the tailored and automated downloading and constructing of street network data from OpenStreetMap; third, the algorithmic correction of network topology; fourth, the ability to save street networks to disk as shapefiles, GraphML, or SVG files; and fifth, the ability to analyze street networks, including calculating routes, projecting and visualizing networks, and calculating metric and topological measures. These measures include those common in urban design and transportation studies, as well as advanced measures of the structure and topology of the network. Finally, this article presents a simple case study using OSMnx to construct and analyze street networks in Portland, Oregon. This is a preprint. Download the published article at: http://geoffboeing.com/publications/osmnx-complex- street-networks/ Cite as: Boeing, G. 2017. “OSMnx: New Methods for Acquiring, Constructing, Analyzing, and Visualizing Complex Street Networks.” Computers, Environment and Urban Systems 65, 126-139. doi:10.1016/j.compenvurbsys.2017.05.004 BOEING 1.
    [Show full text]
  • Spatial Science and Network Science: Review and Outcomes of a Complex Relationship César Ducruet, Laurent Beauguitte
    Spatial science and network science: Review and outcomes of a complex relationship César Ducruet, Laurent Beauguitte To cite this version: César Ducruet, Laurent Beauguitte. Spatial science and network science: Review and outcomes of a complex relationship. Networks and Spatial Economics, Springer Verlag, 2014, 14 (3-4), pp.297-316. 10.1007/s11067-013-9222-6. halshs-01093664 HAL Id: halshs-01093664 https://halshs.archives-ouvertes.fr/halshs-01093664 Submitted on 10 Dec 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spatial science and network science: Review and outcomes of a complex relationship DUCRUET César1 French National Centre for Scientific Research (CNRS) UMR 8054 Géographie-Cités, Paris, France Email: [email protected] BEAUGUITTE Laurent UMR 6266 IDEES, Rouen, France Email: [email protected] Pre-final version of the paper published in Networks and Spatial Economics, Vol. 14, Issue 3-4, pp. 297-316. Abstract For decades, the spatial approach to network analysis has principally focused on planar and technical networks from a classic graph theory perspective. Reference to models and methods developed by other disciplines on non-planar networks, such as sociology and physics, is recent, limited, and dispersed.
    [Show full text]
  • Improved Graph Neural Networks for Spatial Networks Using Structure-Aware Sampling
    International Journal of Geo-Information Article Improved Graph Neural Networks for Spatial Networks Using Structure-Aware Sampling Chidubem Iddianozie * and Gavin McArdle School of Computer Science, University College Dublin, Dublin 4, Ireland; [email protected] * Correspondence: [email protected] Received: 30 September 2020; Accepted: 11 November 2020; Published: 13 November 2020 Abstract: Graph Neural Networks (GNNs) have received wide acclaim in recent times due to their performance on inference tasks for unstructured data. Typically, GNNs operate by exploiting local structural information in graphs and disregarding their global structure. This is influenced by assumptions of homophily and unbiased class distributions. As a result, this could impede model performance on noisy real-world graphs such as spatial graphs where these assumptions may not be sufficiently held. In this article, we study the problem of graph learning on spatial graphs. Particularly, we focus on transductive learning methods for the imbalanced case. Given the nature of these graphs, we hypothesize that taking the global structure of the graph into account when aggregating local information would be beneficial especially with respect to generalisability. Thus, we propose a novel approach to training GNNs for these type of graphs. We achieve this through a sampling technique: Structure-Aware Sampling (SAS), which leverages the intra-class and global-geodesic distances between nodes. We model the problem as a node classification one for street networks with high variance between class sizes. We evaluate our approach using large real-world graphs against state-of-the-art methods. In the majority of cases, our approach outperforms traditional methods by up to a mean F1-score of 20%.
    [Show full text]
  • Nonobtuse Triangulations of Pslgs
    NONOBTUSE TRIANGULATIONS OF PSLGS CHRISTOPHER J. BISHOP Abstract. We show that any planar PSLG with n vertices has a conforming triangulation by O(n2.5) nonobtuse triangles, answering the question of whether a polynomial bound exists. The triangles may be chosen to be all acute or all right. A nonobtuse triangulation is Delaunay, so this result improves a previous O(n3) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n2) elements are needed, improving an O(n4) bound of Bern and Eppstein. We also show that for any ǫ> 0, every PSLG has a conforming triangulation with O(n2/ǫ2) elements and with all angles bounded above by 90◦ + ǫ. This improves a 3 ◦ 7 ◦ result of S. Mitchell when ǫ = 8 π = 67.5 and Tan when ǫ = 30 π = 42 . Date: March 1, 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: Key words and phrases. nonobtuse, acute, triangulation, conforming triangulation, PSLG, De- launay, Gabriel, nearest-neighbor learning, quadrilateral mesh, beta-skeleton, thick-thin decompo- sition, polynomial complexity, The author is partially supported by NSF Grant DMS 10-06309. This is a very preliminary version and should not be distributed. Please contact me for the latest version if you wish to share this with someone. 1 NONOBTUSE TRIANGULATIONS OF PSLGS 1 1. Introduction A planar straight line graph (or PSLG from now on) is the union of a finite number (possibly none) of non-intersecting open line segments together with a disjoint finite set that includes all the endpoints of the line segments, but may include other points as well.
    [Show full text]