Lucas Figueiredo Programa de Pós-Graduação em Arquitetura e Urbanismo Universidade Federal da Paraíba

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1 Lucas Figueiredo Programa de Pós-Graduação em Arquitetura e Urbanismo Universidade Federal da Paraíba lucasfigueiredo@gmail.com Abstract This paper presents a bridge between line and segment maps, a unified graph model. By using simple justified graphs, it shows that topological steps on axial and continuity maps are equivalent to changes of direction on segment maps in a directional distance model. In other words, line maps are 'compressed versions of segment maps. Line maps store part of the computations in the representation itself while segment maps rely mostly on the measures. A brief discussion about sizerelativisation is also provided. The paper ends with a discussion about potential developments for the ideas presented, which may lead to faster models that preserve data integrity and contribute to a renewed theoretical foundation for street network analysis. Keywords Space syntax, street networks, axial lines, continuity lines, angular-segment analysis, directional distance. 1. Introduction Following Hillier and Hanson's axial lines (1984), scholars have developed several types of street network analysis based on linear features: angular-segment analysis (Hillier and Iida, 2005; Turner, 2007), continuity lines (Figueiredo and Amorim, 2005), intersection continuity negotiation (Porta et al., 2006b), reach and directional distances (Peponis et al., 2008), between others. Batty (2004a; 2004b; 2013, p ) rethought some of these views as a single accessibility problem and showed that these network representations could be interchangeable up to a point. Figueiredo (2009, p ) introduced a number of ideas to explore the relation between graphs of lines and street segments through a single view. This paper builds upon these efforts and investigates a unified model for line and segment maps. It begins with a brief discussion about representations of street networks. Then, it focus on the relations between line and segment maps to examine how topological steps on axial and continuity maps are equivalent to changes of direction on segment maps. Following that, it shows that a single set of equations can be applied to both types of graphs. The paper ends with a discussion about the theoretical and practical consequences of the model, as well as its potential applications. 2. Types of graphs This section examines how streets networks are represented (encoded) as graphs. Readers may want to skip it if they feel comfortable with the subject but some key concepts and details are vital to understand the next section. An effort has been made to present the subject to the general reader. Technical details will be preferentially placed in footnotes. 146:1

2 Figure 1 presents the main types of graphs used to encode street networks using a notational grid defined by six streets. Figure 1b shows a classic graph 1, commonly used by transportation models, where nodes represent crossings and edges represent street segments, without (b1) and with (b2) dummy nodes (i.e. a node that is not a crossing). A street segment is a segment between two crossings whereas a line segment is a segment created by any change of direction. Figure 1c shows a street-segment graph (c1) and a line-segment graph (c2). Angular-segment analysis (Hillier and Iida, 2005; Turner, 2007) is based on the latter 2,3. There is a direct relation between (b1) and (c1), and between (b2) and (c2). Each node in (c) is an edge in (b), each node in (b) results in one or more edges 4 in (c). Figure 1d shows axial and continuity maps 5. Each edge in (d1) is a node in (b1), each edge in (d2) is a node in (b2). The graphs in (d) are smaller. Why? Because they are compressed versions of those in (c). Therefore, there is a direct relation between (c) and (d). Axial maps were created as a geometrical decomposition of open spaces. They are (or used to be) hand-drawn' over common street maps. Therefore, an axial line is drawn at once, as a single linear feature. That is why this compression is not directly noticeable. However, if you think of a line as a an aggregation of segments, this relation becomes clear (for a in-depth discussion, see Figueiredo, 2009). Figure 1 also shows this mechanism. Green nodes in (d) aggregates two nodes from (c) (see labels). Edges are also aggregated as a result (see labels). Therefore, all graph types in Figure 1 are related. A graph in (c) is more or less the inverse' of a graph in (b), i.e. nodes become edges and edges become nodes. A graph in (d) aggregates (compresses) several nodes and edges, i.e. each node in (d) represents one or more nodes in (c) and each edge in (d) represents one or more edges (c). This compression is a process of generalisation 6. Finally, edges in (d) are nodes in (b). As a set of geometrical entities representing open spaces, graphs underlying axial maps were first thought to describe topological relations. Edges may be interpreted as changes of direction but they primarily describe a intersection between two lines, i.e. a connection. The cognitive component was introduced later, from a series of theoretical arguments and experiments until a fully developed distance model 7. In segment graphs, nodes are (lines or street) segments and edges represent continuations or changes of direction between them. Edges are weighted (i.e., each edge is associated to a cost) to implement this external distance model. This represents deep theoretical shift. Line maps (axial or continuity) are geometrical decompositions that describe (or internalise) topological hierarchies that are ultimately generated by line length (Carvalho and Penn 2004). On the other hand, segments (line or street) do not differ much from each other 8. Segment maps need an external model to generate hierarchies. Different models (and costs), for instance, metric and angular distances, will generate different hierarchies (see Gil, 2014). In other words, segment maps need the assumption that the distance model is valid, 1 Porta et al. (2006a; 2006b) call the graphs in Figure 1b primal' and those in Figure 1d dual. This terminology may be misleading because all three types of graphs are related. 2 Segment maps created from axial maps have stubs, small line segments that result from extending axial lines over crossings or from intersections of several axial lines ('trivial rings'). This overloads the graph with meaningless fragments and may affect the analysis (see Turner 2007, p. 542; Figueiredo 2009, p ). 3 There is no technical impediment to use street-segments as a basis for angular analysis. All is needed is to weight the nodes in addition to the edges, i.e. compute angles within street-segments without splitting them. This would speed up computations (by drastically reducing the number of nodes). If implemented, this suggestion could be a indirect contribution to depthmapx (Varoudis, 2012). 4 This could be represented by a 'hyperedge', i.e. an edge that can connect any number of nodes. However, street-segment graphs have the advantage of representing each change of direction as an edge. This characteristic could be very useful for traffic engineering. 5 Streets 'c' and d/e' have a sharp and a smooth change of direction respectively. This could result in different maps depending on angular threshold adopted (for details, see Figueiredo and Amorim, 2005). 6 Thompson (2003) was the first to establish a parallel between axial lines and a generalisation of road-centre lines from a cartographic point of view. See also Mora (2009) and Faria (2010) for a discussion from a cognitive point of view. 7 There is a extensive body of research on the subject. See Hillier (1999), Conroy (2001), Hillier and Iida (2005) and Turner (2007). For a review, see Mora (2009). 8 Lines cross several scales in length and connectivity (see Carvalho and Penn 2004; Figueiredo 2009). Segments may cross some scales in length, but not in connectivity. 146:2

3 i.e. individuals navigating in the system choose their routes using elements incorporated by the model. Other lines of enquiry could provide alternative explanations to validate the model, but they are beyond the scope of this paper and will be explored in future research. This paper assumes that angular (Hillier and Iida, 2005; Turner, 2007) or directional (Peponis et al. 2008) distance models are valid. Figure 1: types of graphs for representing street networks. (a) notational street system composed by six streets; (b) classical graphs used by transportation models: (b1) without dummy nodes, (b2) with dummy nodes (in blue); (c) graphs underlying segment maps: (c1) streetsegment graph (directional distances, continuity maps), (c2) line-segment graph (angular-segment analysis); (d) graphs underlying line maps: (d1) continuity map (d2) axial map. Adapted from: (Figueiredo, 2009, p. 68). 146:3

4 Line maps have several advantages. Their computations are faster because their graphs are much smaller 9. They have a small set of standard cross-disciplinary measures, even though sometimes they have different names. Integration also has a well-discussed size-relativisation 10 mechanism (Hillier and Hanson, 1984, p ; Krüger, 1989; Teklenburg et al., 1993; Krüger and Vieira, 2012). Segment maps produce much bigger graphs, which result in slower computations. It can be argued that there is no standard set of measures for them, nor a definitive size-relativisation mechanism. In part, because the model is more flexible and can implement any distance model working with different types of radii. However, segment maps can capture fine-scale details that cannot be revealed using line maps. That is why line maps are increasingly being replaced by segment maps in space syntax research. Rethinking line maps as aggregated segment maps can lead to a number of developments that will be discussed in the next section. 3. A unified graph model For simplicity, directional distances and continuity maps (Peponis et al., 2008; Figueiredo and Amorim, 2005) will be used to develop the ideas in this section. In a directional distance model, edges have weight (cost) one if they represent changes of direction and weight (cost) zero if they are continuations. More precisely, there is an angular threshold a, where any angle below or equal to a is not a change of direction, it is a continuation, and any angle above a is a change of direction. As it has been demonstrated, line graphs can be seen as compressed versions of their segment counterparts. Figure 2 adds edges weights to Figure 1. As the figure shows, edges that have weight zero (continuations) disappear because the nodes they connect are merged. Edges that have weight one (changes of direction) are aggregated accordingly. For instance, if a node had two edges to two different nodes that were merged, it now has a single edge to the new (compressed) node (see also the labels in Figure 1). Figure 3 uses simple justified graphs (j-graphs) to show that topological steps on axial and continuity maps are equivalent to changes of direction on street-segment maps. In a j-graph, edges are horizontally aligned according to their distance to a given starting (root) node. Using a notational grid as a reference (Figure 1), three j-graphs were built starting from the same root' node (f). The segment j-graph (a) has more nodes and more of them in the deepest level (nodes j, c and i). If the j- graph is rebuilt in a way that 'steps' are replaced by distances in proportion to the weight of the edges 11, continuations will be aligned horizontally because they have weight zero. As a result, four nodes (j, i, e and b) had their depth adjusted in the resulting segment j-graph (b). Comparing the adjusted j-graph (b) with the line j-graph (c) provides a even clearer picture of how the compression of segments into lines operates. In large maps, this compression can drastically reduce the number of nodes and edges (see Figure 4). 9 Although parallel and distributed computing is becoming easier to implement, graph size is still important. Several measures such as integration and choice have non-linear computation costs. For instance, a measure with a computation cost of n * log (n) will quickly escalate as the number of nodes n increases. In addition, performance is also important because huge data sets are now widely available (see Gil, 2014). 10 A relativization that enables (up to a certain point) comparison between systems of different sizes. 11 This representation works for the angular-segment analysis too. All is needed is to build a j-graph with varying 'steps' (distances) according to weight of each edge. 146:4

5 Figure 2: segment and line graphs with edge weights. In the figure, edges have weight 1 (one) if they represent a change of direction and 0 (zero) if they represent a continuation. During the node-compression of segments into lines (continuity or axial), when two nodes are merged, the edge that connect them, a continuation, disappear. The remaining edges are aggregated accordingly. See also the labels in Figure 1. Figure 3: justified graphs. (a) street-segment j-graph (topological). (b) street-segment j-graph adjusted by edge weight. Dashed edges are continuations and have weight 0 (zero), black edges are changes of direction and have weight 1 (one). Nodes connected by continuations are aligned horizontally. (c) standard line j-graph for the final compressed version. 146:5

6 SSS10 Proceedings of the 10th International Space Syntax Symposium Figure 4: line maps as compressed segment maps. Section of Philadelphia, PA, USA (top), its street-segment graph (middle) and its line graph (bottom). The segment graph has nodes and edges, the line graph has 9952 nodes and edges (a compression of approximately 75%). 146:6

7 Therefore, there is a direct equivalence between a line map with an angular threshold of a and a distance model based on directional distances defined by the same angular threshold a. From this equivalence, two propositions can be made: (P1) Continuity maps built with a angular threshold a internalise a directional distance model based on the same threshold a. (P2) Axial maps are a particular case of continuity maps, in which this angular threshold a is equal to zero. Although some particular cases 12 such as bifurcations, turnarounds and ring roads were not taken into consideration, this does not invalidate proposition P1. To ensure the equivalence, the implementation of the model only needs to establish consistent rules for both graphs, line and segment. In other words, line maps such as axial or continuity lines are compressed versions' of segment maps if a directional distance model is adopted 13. Line maps store part of the computations in the representation itself whereas segment maps mostly rely on the measures. From these observations, Figueiredo (2009, p ) introduced a number of ideas that this section will explore further. Equation 1 is the mean depth MD (Hillier and Hanson, 1984, p.108) or the average distance of node to all other nodes in the graph (see also Freeman 1979). In the equation, k is usually described as the number of nodes in the graph. The equation states (k - 1) as the distance between the root to itself is not computed. But there is more to it. Looking again to Figure 3c, (k - 1) is also the minimum number of edges needed to create a shortest path tree 14, i.e. a tree that connects all nodes in the j-graph and still keeps the minimum distances from the root. Since the j-graph (c) is also a weighted j-graph according to (P1), (k - 1) is also the sum of all weights W in the same shortest path tree. Figure 5: shortest path trees. (a) a street-segment shortest path tree extracted from Figure 3b. To ensure Proposition P1, a minimal weight shortest path tree was selected, which is W = (k - 1). Dashed lines have weight zero, black lines have weight one. (b) a line shortest path tree extracted from Figure 3c. The sum of weights W in both trees is (k - 1). 12 For details, see Figueiredo and Amorim (2005), Peponis et al. (2008), Figueiredo (2009) and Dhanani et al. (2012). 13 Even if angles were fully computed (angular distance model), topological steps in line maps or a directional distance model tend to produce similar results. In fact, much of the fine-scale detail produced by segment maps result from adopting segments instead of lines, revealing differences within lines. Therefore, a directional distance model is as robust as an angular distance model. 14 To create a shortest path tree from a j-graph, remove edges while (a) keeping the graph connected and (b) keeping the distances from the root the same. A minimal set of edges of size (k - 1) that keeps these two properties forms a shortest path tree. 146:7

8 Equation 1: mean depth Equation 2: relative distance A j-graph contains at least one shortest path tree. Figure 5 shows two shortest path trees extracted from the j-graphs in Figure 3b and 3c. The sum of all weights in Figure 5a is the minimum weight needed to create the weighted shortest path tree 15. In both trees, W = (k - 1). Therefore, the updated Equation 2 can be applied to both trees in Figure 5. In other words, the denominator in Equation 2 is the minimum distance needed to connect the root to all other nodes in the graph. The numerator in Equation 2 is the total distance covered in individual journeys from the root to each other node. Hence, the meaning of Equation 2 is far less abstract than being an average depth or distance. It measures the centrality of a node taking into account the efficiency of the grid 16 (in terms of interconnectedness and permeability). Until now, this paper brought to line maps a number of lessons taken from segment maps and their distance models. What is striking about (P1), however, is that it is also valid in the opposite way. A segment map is a line map that has not been compressed: (P3) Since W = (k - 1), or k = (W + 1), all classic integration measures and their relativisation mechanisms can be applied to segment graphs in a directional distance model 17. However, proposition (P1) paves the way for more. Returning to line maps as topological graphs, it has been revealed that they share a number of properties with other networked systems 18 (Porta et al, 2006b). In such systems, the average path length (Watts and Strogatz,1998), i.e. the average distance between nodes 19, tends to grow with the logarithm of the size of the system 20, i.e. L ~ ln (k). Following a similar discussion, Park (2006, p. 3) introduced a simplified version of Integration (Equation 3). Figueiredo (2009) suggested, as an alternative, replacing ln (k) for the actual average path length L of the system (Equation 4). In other words, dividing average distance of a node to all others (MD) by the average distance between nodes in the system (L). Since L (or a similar measure) can be calculated in weighted graphs, this type relativisation could be, in principle, applied to any distance model 21, The shortest path tree should not be confused with the minimal weight spanning tree, a slightly different problem in graph theory. 16 A similar rationale is used by Porta et al. (2006a) in their 'straightness' measure. 17 If angles are being measured in radians, the proposition (P3) could also be incorporated by angular-segment analysis. 18 See also Rosvall et al. (2005). 19 While MD calculates the average distance from a node to all others, L calculates the average distance between all pairs of nodes. MD is a measure for an individual node node. L is a measure for the entire system. 20 This has been observed in continuity maps (see Figueiredo, 2009, p. 212). Large axial maps, however, do not follow this because long paths are fragmented in several lines. As a result, the average path length grows faster (in large maps). 21 Dividing a value by the mean value is actually a common size-relativisation procedure. 22 Calculating L may be costly, but there are alternatives: (a) by investigating (or sampling) empirical data it may be possible to estimate L; (b) L may be replaced by another reference value, for instance, the weight W of the minimal weight spanning tree; or (c) a scaled mean depth could be adopted, as suggested by Krüger and Vieira (2012, p. 201, Equation 22). 146:8

9 Equation 3: integration (Park, 2005) Equation 4: integration 4. Discussion This paper presented a bridge between line and segment maps, a unified graph model. The word graph stress very well the limits of what has been discussed here. Beyond these limits, the paper opens far more questions than solve. From a graph-theoretical point of view, as compressed segment maps, line maps store part of the computations in the representation itself. Therefore, all measures extracted from line maps are, to a large extent, artefacts of the representation itself. If line maps internalise a distance model, would be the act of tracing axial or continuity lines a cognitive act in itself? In other words, tracing axial or continuity lines generate part of the answers before any analysis? From the opposite side, if segment maps are line maps that have not been compressed, do their distance models really need cognitive assumptions about individual journeys? Complex network research has shown that networked systems are highly hierarchical 23. A small number of heavily connected nodes in the top of this hierarchy tend concentrate or control flow and communication. Therefore, do people build routes with less changes of direction (angular or directional) because they are composed by main streets in the first place? In space syntax research, correlations 24 with movement data have been used to validate line and segment maps 25. To further complicate matters, the models are often calibrated, which means that different measures and radii are tested against movement data. This clearly is not good enough. If correlations are not good, is the model incorrect? Why? The axial map had a robust theoretical and practical foundation before empirical findings or cognitive explanations. The unified graph model presented in this paper highlights that other line and segment maps, their graphs and measures, also need a robust foundation (theoretical and practical) before being confronted against any kind of data. Are line maps valid because they internalise a distance model based on cognitive assumptions? Or, are segment maps valid because they reconstitute through measures the ubiquitous hierarchies captured by line maps? Fortunately, the practical contributions of the unified graph model are more clear. First, it preserves data integrity 26,27. Using the ideas presented here, both angular-segment and continuity analysis can be computed directly from original road-centre line data without merging or splitting streetsegments. These developments, however, would need a relaxed version of the axial map, since it is 23 See Newman et al. (2006). 24 Although establishing causation is not really necessary to validate a model, this constant use of correlations weakens space syntax research in front of disciplines in which alternative approaches are more common. Batty s research (2013), for instance, uses simulation to speculate about causal mechanisms. In complex network research (see Newman et al. 2006), empirical findings are often followed by theoretical mathematical models that generate similar networks. And so on. 25 For instance, angular-segment maps were first validated in this way (Hillier and Iida, 2005; Turner, 2007). 26 Angular-segment analysis and continuity maps do not preserve data integrity (axial maps and road-centre lines). The former splits lines or street segments and the latter merges them. By using a unified graph model, this can be avoided in most cases. 27 Modifying data is not necessarily a bad thing as the original data may be unreliable (see Dhanani et al., 2012). Ensuring data integrity, however, has a clear advantage: the model implementation can be plugged directly to the data source. In other words, if the data provider updates the maps (input), the analysis (output) can be updated without manual intervention. 146:9

10 not possible to create real axial lines from road-centre lines 28. Continuity maps can also be computed directly from axial maps without merging lines. Second, it is flexible. This paper has shown that continuity maps (Figueiredo and Amorim 2005) and directional distance models (Peponis et al., 2008) are interchangeable through weighted graphs. This mechanism could also function for angular-segment analysis. The discussion about size-relativisation reviewed a number of alternatives that could lead to a general set of equations applicable to any distance model. In fact, the multimodal urban network (MMUN) model developed by Gil (2014) may already demonstrate the feasibility of this proposal. From a unified graph model, future research may build a renewed, unified theoretical foundation for line and segment maps, which could, in principle, be extensible to any kind of distance model. Acknowledgments The author would like to thank Michael Batty and John Peponis for their comments on the first version of this research; and Jorge Gil for an insightful discussion about the subject, summer He must thank the anonymous reviewer for his/her comments. References Batty, M. (2004a), A New Theory of Space Syntax, CASA Working Paper 75. London: Centre for Advanced Spatial Analysis, University College London. Batty, M. (2004b), Distance in Space Syntax, CASA Working Paper 80. London: Centre for Advanced Spatial Analysis, University College London. Batty, M. (2013), The New Science of Cities. Cambridge MA: MIT Press. Carvalho, R. and Penn, A. (2004), Scaling and universality in the micro-structure of urban space. In Physica A: Statistical Mechanics and its Applications, 332, Conroy, R. (2001), Spatial navigation in immersive virtual environments, PhD Thesis, London: Department of Architecture, University College London. Dhanani, A., Vaughan, L., Ellul, C., and Griffiths, S. (2012), 'From the Axial Line to the Walked Line: Evaluating the Utility of Commercial and User-Generated Street Network Datasets in Space Syntax Analysis. In: Greene, M., Reyes, J. and Castro, A. (eds.), Proceedings of the Eighth International Space Syntax Symposium, Santiago de Chile: Pontificia Universidad Católica de Chile. Faria, A. P. N. (2010), Análise configuracional da forma urbana e sua estrutura cognitiva, Tese de Doutorado, Porto Alegre: Programa de Pós-Graduação em Planejamento Urbano e Regional, Universidade Federal do Rio Grande do Sul. Figueiredo, L. and Amorim, L. (2005), Continuity lines in the axial system. In: Van Nes, A.(ed.), Proceedings of Fifth International Space Syntax Symposium. Delft: TU Delft, Faculty of Architecture, Section of Urban Renewal and Management. Figueiredo, L. (2009), Continuity Lines - An investigation of urban form through street networks, PhD Thesis, London: University College London. Freeman, L. C. (1979), Centrality in social networks: conceptual clarification, In Social Networks, Vol. 1 (3), p Gil, J. (2014), Analyzing the Configuration of Multimodal Urban Networks. In Geographical Analysis, Vol. 46 (4), p Hillier, B. and Hanson, J. (1984), The social logic of space, Cambridge: Cambridge University Press. Hillier, B. (1999), The hidden geometry of deformed grids: or, why space syntax works, when it looks as though it shouldn t. In Environment and Planning B - Planning and Design, Vol. 26 (2), p Hillier, B. and Iida, S. (2005), Network and psychological effects in urban movement. In: Cohn, A. G. and Mark, D. M. (eds.), Spatial Information Theory, Berlin: Springer-Verlag, p Krüger, M. J. T. (1989), On node and axial grid maps: distance measures and related topics, London: Bartlett School of Architecture and Planning, University College, London. Krüger, M. J. T., and Vieira, A. P. (2012), 'Scaling relative asymmetry in space syntax analysis'. In The Journal of Space Syntax, Vol. 3 (2), p See Peponis et al (2008), Figueiredo (2009) and Dhanani et al. (2012). Road-centre lines meander even when streets are straight and have complications such as bifurcations and roundabouts. 146:10

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