Euclidean Space of City Space Syntax

We demonstrate that random walks embed connected undirected graphs into Euclidean space, in which distances and angles acquire the clear statistical interpretation. Being defined on the spatial network of Venetian canals, random walks afford a portrait of the city space syntax and measure its intelligibility. In the present Letter, we investigate the spatial network of 96 canals in Venice (that stretches across 122 small islands between which the canals serve the function of roads) by means of random walks converting city space syntax into Euclidean space. Studies of urban networks have a long history. In most of researches devoted to the improving of transport routes, the optimization of power grids, and the pedestrian movement surveys the relationships between certain components of the urban texture are often measured along streets and routes considered as edges of a planar graph, while the traffic end points and street junctions are treated as nodes. Such a primary graph representation of urban networks is grounded on relations between junctions through the segments of streets. The usual city map based on Euclidean geometry can be considered as an example of primary city graphs.