Abstract: This work introduces the ‘non-Toblerian geographical spaces’, where anisotropic and action-at-a-distance relations are relevant. We consider that many human-built spaces have a strong tendency toward anisotropy. In today´s networked society, we are increasingly influenced by relations that act at a distance, such as connections to markets. The current generation of GIS does not handle such spaces properly. In this paper, we provide practical and theoretical evidence on the commonness of non-Toblerian spaces. We also argue that some relevant critiques on the limits of GIS made by postmodern thinkers can be addressed by adopting a generalized measure of proximity. The paper also discusses the Generalized Proximity Matrix (GPM), a tool for representing non-Toblerian spaces, focusing on its use for dynamical spatial modelling. The authors argue the GPM allows GIS to provide a flexible support for proximity measures, and thus support modelling in non-Toblerian spaces.

• Synthesis
  1. Spatial Relations are fundamental to be represeted in GIS.
     
  2. Topological spatial relations need to be described in a formal way.
     
  3. The notion of proximity provides a practical application of the Tobler's principle (“the first law of geography: everithing is related to everithing else, but near things are more related than distant things”).
     
  4. Problem: In the common-sense, the notion of proximity is related to pont-set topological predicates and isotropic distances in euclidian space, but many spaces can't be soved using this notion. Many human-built process have a strong tendency towards anisotropy, and we need to represent such spaces if we are to capture relations that consider how today's global economic forces influence local actions.
     
  5. Proposal: GIS should adopt a flexible notion of proximity for spaces where anisotropic and action-at-a-distance relations are relevant. Such spaces are called “non-Toblerian”. In these spaces, the proximity is not measured by topological relations or euclidian distances, but is a singular property of each spatial object from that space. For each spatial object, we need to find out the proximity relations to other objects in the same space. Tobler's “first law o geography” continues to be valid in these spaces.
     
  6. Why many human-built spaces are non-Toblerian?
     
     • In many cases, action-at-a-distance relations define how humans occupy space, e. g., when people move into a region, they create and use new access routes, and new roads attract new occupants to it's surroundings. In this process, the resulting patterns are not isotropic, but follow preferencial paths set by the transport network.
       
     • When changes in geographical space depend on networks and routes, we have a non-Toblerian space as result. But how non-Toblerian geographical spaces emerge? The social network theory can explain.
       
     • The social network theory deals with analysing the interplays between individual and groups, and actors at different levels of analysis, and the regularities and patterns of such interactions shape these networks.
       
     • The analysis of empirical data for issues such as collaboration networks, disease spreading, and innovation diffusion is an important part of this theory.
       
     • Recent works have stressed the need to build a spatial expression of networks.
       
     • But how social networks emerge? Most of them have prefferencial attachment, i. e., is more probable that new human settlements will appear close to a route than in a region where exist nothing.