Research ArticleSOCIAL SCIENCES

Toward cities without slums: Topology and the spatial evolution of neighborhoods

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Science Advances  29 Aug 2018:
Vol. 4, no. 8, eaar4644
DOI: 10.1126/sciadv.aar4644
  • Fig. 1 Topology of places and city block complexity.

    (A) Schematic city block (top) with one internal place (red outline) and its characterization in terms of a hierarchy of weak dual graphs, S1, S2, and S3 (bottom). (B) New York City. (C) Prague. (D) Construction of nested dual graphs for a block in the Epworth neighborhood in Harare (Zimbabwe), with block complexity kmax = 3. In this case, internal parcels are only one layer deep relative to existing accesses. Data sources are described in Materials and Methods.

  • Fig. 2 Neighborhood topology and the access networks of informal settlements.

    (A) An informal settlement in Khayelitsha, a township of Cape Town, South Africa. As is typical of most informal settlements, services provided by the city, including power, water, toilets, and trash collection (yellow, blue, orange, and green symbols, respectively), are located exclusively by existing road accesses along the periphery of the block. In contrast, public spaces created by the community, such as a religious and community center (red), are located near the block’s center. Image Credit: DigitalGlobe, copyright 2018. (B) Parcel layout for (A) showing many internal places to the block, outlined in red (top). The corresponding odd-numbered weak dual graphs Sk (see Fig. 1) are shown in different colors, from black to orange, with the latter corresponding to the lowest value of k for which the Sk is not a tree, entailing block complexity kmax = 9.

  • Fig. 3 Growing efficient street networks in underserviced city blocks.

    (A) The topologically optimal solution for the Epworth block of Fig. 1D, with additional street segments shown in blue. The resulting dual graph shows that the S2 dual graph (blue) is now a tree (middle). The parcel-to-parcel travel cost matrix ℑ (right) shows that all parcels are connected but that some remain distant from each other over the network. Each entry of ℑ, Tij shows the minimum on-network travel distance from i to j, where blue and green entries are shorter distances and orange and red entries are longer distances. The matrix has been reordered using a hierarchical clustering algorithm to reveal parcel clusters with short distances over the network. (B) The topological solution for the Khayelitsha neighborhood of Fig. 2, the resulting weak dual graphs (middle), and the corresponding minimal travel cost matrix, ℑ (right). (C) The result of the geometric optimization for (B), where four new bisecting paths (red, orange, yellow, and green) were added (left), resulting in substantial decreases in Embedded Image (middle) by introducing 81 m of new roads and reducing the average parcel-to-parcel travel distance, Embedded Image, from 214 to 145 m (right). Details of the topological and geometric constrained optimization problems and other examples are given in sections SE and SF.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/8/eaar4644/DC1

    Supplementary Text

    Section SA. Topology of access systems

    Section SB. Topology of city blocks

    Section SC. City block topological theorems

    Section SD. Topological optimization: Minimal neighborhood reblocking

    Section SE. Geometric optimization: Travel costs versus road construction

    Section SF. The topology of places is equivalent to the topology of the access system

    Fig. S1. Phule Nagar (Mumbai, India) path width.

    Fig. S2. Topological constructs for the systematic analysis of urban topology.

    Fig. S3. Las Vegas (NV, USA) access system and access network.

    Fig. S4. Schematic bridge graph retraction.

    Fig. S5. Epworth (Harare, Zimbabwe) minimal reblocking.

    Fig. S6. Epworth (Harare, Zimbabwe) before and after reblocking.

    Fig. S7. Epworth (Harare, Zimbabwe): Geometric optimization and travel cost matrices.

    Fig. S8. Khayelitsha (Cape Town, South Africa): Reblocking and travel cost matrices.

    Fig. S9. Prague cadastral map and block parcel layout.

    Movie S1. Urban topological invariance.

  • Supplementary Materials

    The PDF file includes:

    • Supplementary Text
    • Section SA. Topology of access systems
    • Section SB. Topology of city blocks
    • Section SC. City block topological theorems
    • Section SD. Topological optimization: Minimal neighborhood reblocking
    • Section SE. Geometric optimization: Travel costs versus road construction
    • Section SF. The topology of places is equivalent to the topology of the access system
    • Fig. S1. Phule Nagar (Mumbai, India) path width.
    • Fig. S2. Topological constructs for the systematic analysis of urban topology.
    • Fig. S3. Las Vegas (NV, USA) access system and access network.
    • Fig. S4. Schematic bridge graph retraction.
    • Fig. S5. Epworth (Harare, Zimbabwe) minimal reblocking.
    • Fig. S6. Epworth (Harare, Zimbabwe) before and after reblocking.
    • Fig. S7. Epworth (Harare, Zimbabwe): Geometric optimization and travel cost matrices.
    • Fig. S8. Khayelitsha (Cape Town, South Africa): Reblocking and travel cost matrices.
    • Fig. S9. Prague cadastral map and block parcel layout.
    • Legend for movie S1

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Urban topological invariance.

    Files in this Data Supplement:

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