DESIGN NETWORK ANALYSIS

Transcription

1 spatial DESIGN NETWORK ANALYSIS (sdna) Specifications DEFINITIONS 1. Preliminaries a. Link b. Junction c. Unlink d. Spatial system 2. Metrics a. Metric b. Geodesic c. Euclidean d. Angular e. Custom metric f. Link origin and link destination 3. Centrality analyses a. Centrality b. Closeness centrality c. Betweenness centrality 3.1. Radius a. Discrete b. Continuous 3.2. Weighting a. no weight b. by origin and destination link length c. custom origin and destination link weight field d. custom origin and destination link weight per unit length field 3.3. Notations a. Closeness b. Betweenness 4. Link description a. Length b. Connectivity c. Link angular curvature d. Notations 5. Radius description a. Link number b. Network length c. Total weight d. Total angular distance of link e. Sum of geodesic f. Mean geodesic g. Number of junction h. Sum of junction connectivity i. Notations 6. Network detour Analyses a. Sum of crow flight b. Mean Diversion ratio c. Diversion ratio d. Notations 7. Network shape analyses a. Convex Hull Area b. Convex Hull Perimeter c. Convex Hull Max Radius d. Convex Hull Bearing e. Shape index f. Notations How to reference this document: Chiaradia A, Cooper C, Webster C 2013, sdna a software for spatial design network analysis. Specifications, 1

2 sdna is a spatial design network analysis tool that quantitatively index shape properties of a spatial network. In sdna the spatial network is represented as a graph where links are nodes (vertices) and the junctions between links are arcs (ties, edges). sdna perform spatial link analysis. 1. Preliminaries DEFINITIONS a. Link A polyline between two junctions, or a junction and an end point. Links vary in length, in angularity (directness/sinuosity), in junction connectivity (number of other links connected at junction). Link can be a polyline centred on road, the road centre line. See ITN, OSM, TIGER line etc. Link can be a polyline centred on cyclist path. Link can be a polyline centred on pedestrian path and pedestrian crossing thus creating a pedestrian centre line network. The largest to date, to our knowledge, has been created for the Commnunauté d Agglomération de Saint Quentin en Yvelines, Paris, France: it is km long. b. Junction A point where at least 3 links are connected. In a special case only two links are connected where both ends of a looped link are connected to a second link. c. Unlink Unlink occurs in the case of a bridge over road or path, tunnel under road, path etc. In sdna this is represented by polylines going over each other without junction. d. Spatial Network System A set of links connected at junctions with no isolated unconnected link or unconnected sub-system of links and junctions. 2. Metrics a. Metric A metric is a function that behaves according to a specific set of rules, and provides a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other. There are three ways in which metric can be defined in sdna, the one chosen affects the nature of any analysis which builds on it. b. Geodesic A geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. In sdna the distance metric between any pair of links can be defined as angular, Euclidean or custom; thus the shortest path (geodesic) between them will vary according to the chosen distance metric. c. Euclidean distance metric The geodesic between two points, or objects on the network using meter as distance unit (or whatever length unit is defined in your host software). Note that in Euclidian metric, turn and curvature along a link have no impact: a wavy link has the same Euclidian distance as a straight link of the same length. d. Angular distance metric The geodesic that minimise the sum of link angular distance and the sum of angular change at each junction between OD link pair. e. Custom distance metric The custom distance metric of each link is defined by a custom field. This metric is assumed to be monotone along the link, so partially traversing the link results in a proportional value. Turns at junctions currently have zero metric in custom metric analysis. 2

3 f. Hybrid distance metric The hybrid distance metric of each link is computed by (i) a user-defined formula for link metrics, and (ii) a user-defined formula for junction metrics. These can combine Euclidean, angular, 3d and custom data. g. Link origin and link destination When Euclidian distance metric is used for analysis link origin and destination are the Euclidean middle. When Angular distance metric is used for analysis link origin and destination are the angular middle. When Custom distance metric is used for analysis, link origin and destination are taken from the Euclidean middle, as custom distances are assumed to be distributed along the Euclidean length of the link. 3. Centrality analyses a. Centrality Within the scope of graph theory and network analysis, there are various types of measures of the centrality of a vertex within a graph that determine the relative importance of that vertex within the graph. Many of the centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin. b. Closeness centrality In graphs the distance metric between all pairs of nodes is defined by the length of their shortest paths. In the network literature the farness of a node is defined as the sum of its distances to all other nodes: sdna provides this measure as Sum of Angular Distance (SAD), Sum of Metric Distance (SMD) and Sum of Custom Distance (SCD) respectively. Closeness is defined as the inverse of the farness. Thus, in theory, nodes which are more central should have lower total distance to other nodes. Unfortunately, this approach does not distinguish between central links that are close to many other links, and peripheral links that are far from just a few other links: Both of these will register high farness / low closeness. To get around this limitation, a solution commonly employed is to measure the mean distance to other links in the network. Hence sdna provides this in the form of Mean Angular/Metric/Custom distance (MAD/MMD/MCD respectively). The limitation of mean distance, however, is that it becomes completely agnostic of network quantity in the radius from the origin link: a link could be surrounded by just one, or a hundred other links; if the distance of reaching each is the same then mean distance will be the same. Therefore we recommend measurement of closeness in one of two ways, according to your needs. 1. Use our new measure Network Quantity Penalized by Distance (NQPD). For each network link in the radius, this computes the quantity of network and divides by the distance of reaching that network, then sums this term over the entire radius. 2. For ArcGIS users interested in detailed statistics: conduct your own multivariate analysis, using MAD/MMD/MCD as one variable, and network quantity in the radius, measured either in terms of number of links or network length or other available control variable in the radius, as another variable. The target variable is the quantity you are trying to model. c. Betweenness centrality In graph betweenness centrality counts the number of geodesic paths that pass through a vertex, i.e, the number of times the vertex lies on the shortest paths between other pairs of vertices. Betweenness centrality is usually displayed as a ranking from the highest level of geodesic overlap (red) for a given radius to the lowest geodesic overlap (blue) for the same radius. 3

4 3.1. Radius Analyses are performed from each link within a user defined radius expressed in metres. (Actually, the unit of length is whatever unit you have configured your host software to use but we have yet to see anybody using a unit other than metres). Radius can be understood as variable floating catchment area. Two modes exist for a link to be part of the radius for a given origin link: Discrete and Continuous. a. Discrete, if a part of a link is in the radius and includes its middle the whole link is part of the radius b. Continuous, only the part that is in the radius is taken into account in the analysis 3.2. Weighting sdna allows for analysis weighting in four different ways: a. no weight, all links have a weight of 1 in both Euclidian and angular metric ( link weighting ) b. weighting by origin and destination link length ( length weighting ) c. custom origin and destination link weight field d. custom origin and destination link weight per unit length field These are controlled by the weight by link length checkbox in combination with the custom weight field. (Currently, custom weights are only available in ArcGIS). The choice of weighting should reflect what you consider to be the best measure of network quantity in your analysis: number of links, length, or a custom defined property (such as population plus number of jobs). The recommended choice for urban networks if you don t have actual census data is number of links (i.e. no weighting). This is because link density increases with number of jobs and population: thus measuring network by the number of links goes some way towards capturing these other variables through network geometry. Weight by link length Custom weight field Result No Left blank All origin and destination links have weight of 1 so this is equivalent to no weight Yes (output named Wl) Left blank All origin and destination links have weight equal to their length No Specifies user weight data All origin and destination links have weight equal to the custom weight provided by user defined field Yes (output named Wl) Analyses options Specifies user weight data Euclidian distance metric 1/ with or without weighted link length (Origin and Destination O & D only) 2/ with or without custom weight 3/ continuous (C) or discrete (D) Angular distance metric 1/ with or without weighted link length (O & D only) 2/ with or without custom weight 3/ continuous (C) or discrete (D) Custom distance metric 1/ with or without weighted link length (O & D only) 2/ with or without custom weight 3/ continuous (C) or discrete (D) Hybrid distance metric All origin and destination links have weight equal to the custom weight provided by user defined field, multiplied by their length. In other words, the user has specified weight per unit length. 4

5 1/ with or without weighted link length (O & D only) 2/ with or without custom weight 3/ continuous (C) or discrete (D) 3.3. Handling of one-way systems When computing geodesics, the analysis respects specified one-way and vertical one-way links. However, when computing the contribution of a single link to its own closeness/betweenness/weight in radius etc, it is assumed that all points on the link are directly reachable from one another regardless of one-way status. This is to maintain consistency with origin approximations and choice of link centres, and the handling of other micromodelling situations within sdna. In the absence of black/white holes this will result in a correct links/length/weight in radius for all sufficiently large radii (that is to say, if the radius exceeds the maximum geodesic from any link to itself respecting the one way system). Origin self-closeness/self-betweenness usually makes only a small contribution to the overall analysis and is included mainly for consistency. 5

6 3.4. Notation The set of links in the global spatial system is denoted N. The set of links in the network Euclidean radius XXX from link x is denoted Rx. The proportion of any link x within the radius is denoted P(x). In discrete space analysis, this always equals 0 or 1, i.e.. In continuous space,. Length of a link y is denoted L(y). Custom weight of a link y is denoted W(y). The distance along a geodesic between an origin link x and a destination link y is denoted d E (x i,y i ) for Euclidian metric d θ (x i,y i ) for angular metric, d C (x i,y i ) for custom metric. In continuous space, link y is cut at the edge of the radius, and geodesic distances are only computed to the centre of the cut link. However, L(y) and W(y) refer to the full, uncut link. Formulae for custom weighting per unit length are not shown, but are equivalent to custom weighting with each custom weight multiplied by the link length in pre-processing. The Euclidian distance along an angular geodesic is noted: de θ (x i,y i ) Measure name Short name Definition No weight Weighted by O & D link length Custom Weighted D C Sum Ang/Euc/Cust/Hybrid Dist RXXX SAD, SED, SCD, SHD XXX Sum of angular/euclidean/cus tom/hybrid distance from all geodesics in the radius from link x (Farness) Mean Ang/Euc/Cust/Hybrid distance RXXX MAD/MED/MCD/MHD XXX Mean geodesic in the radius from link x NetQuantPD Ang/Euc/Cust/Hybrid RXXX NQPDA/NQPDE/NQPDC /NQPDH XXX Network Quantity Penalized for Distance: Destination weight / geodesic for all links in the radius Note that on average, the distance of traversing between two arbitrary points within the same link is 1/3 the distance of traversing the entire link. Thus, the network quantity available in the origin link itself is included in the above summations, but its geodesic is scaled down appropriately. Note that P(y) = 1 in discrete space, not in continuous space 6

7 Measure name Short name Betweenness Ang/Euc/Cust/Hybrid RXXX BtA/BtE/BtC/BtH XXX Alternatively if bidirectional betweenness specified, Betweenness Ang/Euc/Cust/Hybrid Fwd/Bwd RXXX BtAF/BtEF/BtCF/BtHF BtAB/BtEB/BtCB/BtHB XXX Definition No weight Weighted by O & D link length Custom Weighted D C Sum of geodesics that pass through a link x Where Note that y and z are the geodesic endpoints, NOT x the point where betweenness is measured { and the geodesic is defined for angular, Euclidean or custom distance as appropriate. Note that in the literature, unweighted betweenness is commonly defined as Note that y and z are the geodesic endpoints, NOT x the point where betweenness is measured TPBetweenness Ang/Euc/Cust/Hybrid RXXX TPBtA/TPBtE/TPBtC/ TPBtH XXX If bidirectional: TPBetweenness Ang/Euc/Cust/Hybrid Fwd/Bwd RXXX TPBtAF/TPBtEF/TPBtCF/ PBtHF/TPBtAB/TPBtEB/ TPBtCB/TPBtHB XXX Two phase betweenness: sum of geodesics that pass through a link x, weighted by the proportion of network quantity accessible from geodesic origin y that is represented by geodesic destination z. where is the number of geodesics from y to z that pass through x, and is the total number of geodesics from y to z. Our definition is equivalent (except for the extra terms described below) in the case where there is only one geodesic between each pair of links. This is almost always the case with real networks, though not true for regular grids. Extra terms The 1/2 contribution to OD(y,z,x) if or handles the case of journey endpoints and is consistent with computation in Depthmap. The 1/3 contribution to OD(y,z,x) if reflects the fact that individual links can generate their own betweenness, which may have a noticeable effect if links are long. Links(y) is the number of links in radius from each y Where total length(y) is the total network length in radius from each y. Note that y and z are the geodesic endpoints NOT the point x where betweenness is measured. Where total weight(y) is the total custom weight in radius from each y. Note that y and z are the geodesic endpoints NOT the point x where betweenness is measured. Note that TPBt has units of and scales with - network quantity, rather than (network quantity) 2 as is the case with standard Betweenness. While this isn t necessarily a transport model, it corresponds to commonly used two-phase generation-distribution transport models. 7

8 Measure name Short name TPDestination RXXX TPD XXX Definition No weight Weighted by O & D link length Custom Weighted D C Two phase destination assignment: same as TPBt but for the destination of each geodesic only. In a normal betweenness analysis this would be the same thing as Weight RXXX, but with the geodesic weight available from each origin fixed as in the TPBt computation, the weight transferred to destinations along geodesics becomes dependent not only on the weight radius of the destination, but also on what that destination is competing with. Therefore this measure is more discriminating of spatial hierarchy than the Links, Length and Weight measures described below. 8

9 4. Link description For each link the following indices are computed: Measure name Short name Link Length LLen Link Connectivity LConn Link Angular Curvature LAC Hybrid metric fwd HMf Hybrid metric bwd HMb Link Sinuosity LSin Link Bearing LBear Definition Euclidean length of link Number of connections to other links (also called degree centrality). Not the same as number of links connected - a neighbouring link can count 2 towards connectivity if connected at both ends. LConn counts full connectivity for one-way streets i.e. it is independent of any one-way data. The cumulative angular curvature along the full length of a link, in degrees Hybrid metric for individual link, if given (traversing forwards) Hybrid metric for individual link, if given (traversing backwards) Link length / (crow flight length between link endpoints). Similar to diversion ratio but for a single link only. Bearing between link endpoints (returns infinity or null for coincident endpoints) 9

10 5. Radius description For each radius Rxxx and each link the following indices are computed: a. Number of links in the radius b. Network length in the radius c. Total weight in the radius d. Total angular distance in the radius Sum of angular distance of each individual link in the radius (excluding junctions). The sum of Junction Angular Distance can be derived as SAD - Angular distance of links in the radius. e. Sum of geodesic length in the radius (Euclidian, Angular, Custom) f. Mean geodesic length in the radius (Euclidian, Angular, Custom) g. Number of junctions in the radius A network connectivity index can be derived = number of junction / number of link h. Sum of junction connectivity in the radius i. Notations 10

11 Measure name Short name Definition No weight Weighted by O & D link length Custom Weighted D C Links RXXX LnkXXX Length RXXX LenXXX Number of links in the radius Network length in the radius. Note that in continuous space analysis, this quantity can be fractional. Weight RXXX WtXXX Ang Dist RXXX AgD XXX SumGeoLen Ang/Euc/Cust/ Hybrid RXXX SGLA/SGLE/SGLC/ SGLH XXX MeanGeoLen Ang/Euc/Cust/ Hybrid RXXX MGLA/MGLE/MGLC /MGLH XXX Sum of weight in the radius, however computed. Sum of angular distance of links in the radius (excluding junctions) Sum of Euclidean/Angular/Custom /Hybrid geodesic Euclidean length in the radius. SGLE is identical to farness (SED); SGLA/SGLC/SGLH are showing the Euclidean distance that must be travelled to minimize angular/custom/hybrid distance respectively. Mean of Euclidean/Angular/Custom /Hybrid geodesic Euclidean length in the radius. (same as Links RXXX) (same as Length RXXX) If you wish to control any output measure for quantity of network, this is the best control to use as it adapts to the analysis type as appropriate. where, is the angular distance of traversing the portion of link y that falls within the radius Where is the Euclidean length of the angular geodesic between x and y, and is the Euclidean length of the custom geodesic between x and y. So while the geodesic type may vary, this measurement is always Euclidean. Junctions RXXX Connectivity RXXX Number of junctions in the radius Sum of junction connectivity in the radius, where is the set of junctions falling within the radius from link x NA NA, where is the number of ends of links connected to junction j Note that one-way streets count as half a link in this case (unlike LConn where they count fully) NA NA 11

12 6. Network Detour Analysis Network detour analysis compares straight line distance to actual network distance, answering the question, By how much does the network deviate from the most direct path? Different network morphology, e.g. orthogonal, hexagonal, equilateral triangle grid and orthogonal with diagonal will converge to different values. For each radius Rxxx and each link the following indices are computed: a. Sum of crow flight Sum of straight line Euclidean distance between OD pair in the radius b. Mean Diversion ratio The mean of the ratios between sum of crow flight and sum of geodesic for each OD pair in the radius c. Diversion ratio (derived measure) The ratio between sum of Crow flight and sum of Euclidean geodesic in the radius d. Notations Measure name Short name Definition No weight Weighted by O & D link length Custom Weighted D C Sum of Crow Flight RXXX SCF XXX Sum of crow fly distance between the link and all the links in the radius in meter Where CFD(x,y) is the crow flight distance between the centers of links x and y Diversion Ratio Ang/Euc/Cust/ Hybrid RXXX DivA/DivE/DivC /DivH XXX Mean of the ratio of geodesic length over crow flight distance for all links in the radius 12

13 7. Network Shapes Analysis Network shapes refers to the form of the overall spatial footprint of the network. In comparing spatial system, the overall shape of the spatial system has to be taken into account otherwise one run the risk to compare Orange and Carambola (a star shaped fruit). This is generalised to each link in the network and for each radius. E.g. an under scaled part of network (many links over a short area as in a housing estate) can be identified by length in the radius (scale) in relationship to footprint (Convex Hull Area), and footprint shape (Shape index). For each radius Rxxx and each link the following indices are computed: a. Convex Hull Area b. Convex Hull Perimeter c. Convex Hull Max Radius d. Convex Hull Bearing e. Network shape index f. Notations Measure name Short name Convex Hull Area RXXX HullAXXX Convex Hull Perimeter RXXX HullPXXX Convex Hull Max Radius RXXX HullRXXX Convex Hull Bearing RXXX HullBXXX Convex Hull Shape Index RXXX HullSIXXX Definition D C Convex hull area of the entire cut-out network RXXX Hull perimeter in meters (or spatial unit of your host software) Maximum radius of the convex hull (with the centre defined as the centre of the origin link) Bearing of the line of maximum radius of the convex hull in degrees Note that a circle has the minimum possible shape index of 1. 13

14 Update C Cooper Added Link Sinuosity and Link Bearing Corrected error in LAC: LAC is always for complete link and does not depend on radius Update C Cooper Added Hybrid metrics Added handling of one-way streets Added bidirectional betweenness Update Added: How to reference this document Layout: footer, first page and page 9 Update Link measure: LAC = Link Angular Curvature Ang D = Ang Dist = AgD Radius measure: Geodesic length should now be expressed as SumGeoLen and MeanGeoLen. Network detour analysis Mean Diversion ratio notation 14