## Sestieri of Venice (2008)

### BibTeX

@MISC{Volchenkov08sestieriof,

author = {D. Volchenkov and Ph. Blanchard},

title = {Sestieri of Venice},

year = {2008}

}

### OpenURL

### Abstract

We have investigated space syntax of Venice by means of random walks. Random walks being defined on an undirected graph establish the Euclidean space in which distances and angles between nodes acquire the clear statistical interpretation. The properties of nodes with respect to random walks allow partitioning the city canal network into disjoint divisions which may be identified with the traditional divisions of the city (sestieri).

### Citations

1564 |
Applied multiple regression/correlation analysis for the behavioral sciences. Mahwah, NJ: Lawrence Erlbaum Associates
- Cohen, Cohen, et al.
- 2003
(Show Context)
Citation Context ... Covjj instead of the covariance matrix Cov. It is important to note that the diagonal elements of (36) equal 1, while the off-diagonal elements are the Pearson’s coefficients of linear correlations, =-=[33]-=-. Figure 5: The coarse-grained connectivity matrix derived from the low-dimensional representation of Venetian canals given by the transition matrix U6 U ⊤ 6 . The correlation matrix (36) calculated w... |

596 |
Algebraic Graph theory
- Biggs
- 1993
(Show Context)
Citation Context ...coordinates along their edges and outlines. There is a long tradition of research articulating urban environment form using graph-theoretic principles originating from the paper of Leonard Euler (see =-=[2]-=-). Graphs have long been regarded as the basic structures for representing forms where topological relations are firmly embedded within Euclidean space. The widespread use of graph theoretic analysis ... |

407 | Reversible Markov Chains and Random Walks on Graphs, monograph in preparation, available at http://stat-www.berkeley.edu/users/aldous/RWG/book.html
- Aldous, Fill
(Show Context)
Citation Context ...adjoint operator. The orthonormal ordered set of real eigenvectors ψi, i = 1 . . .N, of the symmetric operator � T establishes the basis in H. In quantitative theory of random walks defined on graphs =-=[18, 17]-=- and in spectral graph theory [19], the properties of graphs are studied in relationship to the eigenvalues and eigenvectors of self-adjoint operators defined on them. In particular, the symmetric tra... |

245 |
The Social Logic of Space
- Hillier, Hanson
- 1984
(Show Context)
Citation Context ..., and other structures that physically define a community [37]. Spatial organization of a place has an extremely important effect on the way people move through spaces and meet other people by chance =-=[38]-=-. The patterns of social movement and economical development for a thousand years of Venetian history have been imprinted in space syntax of the city. In the present paper, we have investigated the ca... |

182 | Kernels and regularization on graphs
- Smola, Kondor
- 2003
(Show Context)
Citation Context ...nservation considered in theory of random walks on graphs are among them. Laplace operators describing diffusions on graphs meet the mean value property (mass conservation); they give another example =-=[16]-=- studied in spectral graph theory. Markov’s operators on Hilbert space form the natural language of complex networks theory. Being defined on connected undirected graphs, a Markov transition operator ... |

139 | Kernel kmeans, spectral clustering and normalized cuts
- Dhillon, Guan, et al.
- 2004
(Show Context)
Citation Context ...ng a rotation transformation in R n ). The result (29-30) relates the problem of network segmentations to the investigation of n primary eigenvectors of a symmetric matrix defined on the graph nodes, =-=[26, 27]-=-. The eigenvectors u i> 1 have both positive and negative entries, so that in general the matrix 15s[u1, . . .,un ] differs substantially from that one comprising of the discrete cluster indicator vec... |

123 |
Implementing Discrete Mathematics, Combinatorics and Graph Theory with Mathematica
- Skiena
- 1990
(Show Context)
Citation Context ...ent, [11]. These sets can be naturally interpreted as those of the vertical and horizontal edges in the primary graphs (streets and avenues). In bipartite graphs, all closed paths are of even length, =-=[12]-=-. It is the dual graph transformation which allows to separate the effects of order and of structure while analyzing a transport network on the morphological ground. It converts the repeating geometri... |

99 | Learning spectral clustering
- Bach, Jordan
- 2003
(Show Context)
Citation Context ...n be reduced to an optimization problem with n(n − 1)/2 − 1 parameters [28]. Several methods have been proposed to obtain the partitions from the eigenvectors of various similarity matrices (see [29],=-=[30]-=- for a review). In the next section, we use the ideas of Principal Component Analysis (PCA) in order to bypass the orthonormal transformation. 6 Principal Component Analysis of Venetian Canals In stat... |

97 |
Space Is the Machine: a configurational theory of architecture
- Hillier
- 1996
(Show Context)
Citation Context ...intersections and analyzed as a networks of movement choices. The study of spatial configuration is instrumental in predicting human behavior, for instance, pedestrian movements in urban environments =-=[8]-=-. A set of theories and techniques for the analysis of spatial configurations is called space syntax [9]. Space syntax is established on a quite sophisticated speculation that the evolution of built f... |

73 |
Principal Component Analysis (2nd Edition
- Jolliffe
- 2002
(Show Context)
Citation Context ...he reducing size of a data set. It is achieved by the optimal linear transformation retaining the subspace that has largest variance (a lower-order principal component) and ignoring higher-order ones =-=[31, 32]-=-. Given an operator S self-adjoint with respect to the measure m defined on a connected undirected graph G, it is well known that the eigenvectors of the symmetric matrix S form an ordered orthonormal... |

56 | A unified view of kernel k-means, spectral clustering and graph partitioning
- Dhillon, Guan, et al.
- 2005
(Show Context)
Citation Context ...rs can be reduced to an optimization problem with n(n − 1)/2 − 1 parameters [28]. Several methods have been proposed to obtain the partitions from the eigenvectors of various similarity matrices (see =-=[29]-=-,[30] for a review). In the next section, we use the ideas of Principal Component Analysis (PCA) in order to bypass the orthonormal transformation. 6 Principal Component Analysis of Venetian Canals In... |

53 |
Geographic Information Systems for Transportation: Principles and Applications
- Miller, Shaw
- 2001
(Show Context)
Citation Context ...mputational modelling of an urban network is based on a graph view in which the intersections of linear features are regarded as nodes, and connections between pairs of nodes are represented as edges =-=[5]-=-. Similarly, urban forms are usually represented as the patterns of identifiable urban elements such as locations or areas (forming nodes in a graph) whose relationships to one another are often assoc... |

32 |
Structure of transportation networks: relationships between network geometry and regional characteristics
- Kansky
- 1963
(Show Context)
Citation Context ...embedded within Euclidean space. The widespread use of graph theoretic analysis in geographic science had been reviewed in [3] establishing it as central to spatial analysis of urban environments. In =-=[4]-=-, the basic graph theory methods had been applied to the measurements of transportation networks. Network analysis has long been a basic function of geographic information systems (GIS) for a variety ... |

16 |
The Anthropology of Space and Place: Locating Culture
- Low, Lawrence-Zuniga
- 2003
(Show Context)
Citation Context ...particular, it has been suggested that the urban space combining social, economic, ideological and technological factors is responsible for the technological, socioeconomic, and cultural development, =-=[35]-=-. It is worth to mention that the processes relating urbanization to economic development and knowledge production are very general, being shared by all cities belonging to the same urban system and s... |

13 |
Theory of Reproducing Kernels and Its
- SAITOH
- 1988
(Show Context)
Citation Context ... matrix as measuring similarity between nodes, we can show following [22] that the Euclidean distance (20) leads to Euclidean inner-product similarity which can be replaced by a general Mercer kernel =-=[23, 24]-=- uniquely represented by a positive semi-definite matrix Ki,j. If we then relax the discrete structure of X by assuming that X is an arbitrary orthonormal matrix, the minimization of the objective fun... |

11 |
Lecture notes on spectral graph theory
- Chung
- 1997
(Show Context)
Citation Context ...red set of real eigenvectors ψi, i = 1 . . .N, of the symmetric operator � T establishes the basis in H. In quantitative theory of random walks defined on graphs [18, 17] and in spectral graph theory =-=[19]-=-, the properties of graphs are studied in relationship to the eigenvalues and eigenvectors of self-adjoint operators defined on them. In particular, the symmetric transition operator defined on undire... |

8 | Minimal Volume Entropy on Graphs
- Lim
- 2005
(Show Context)
Citation Context ...e volume entropy of the graph G, while the volume of G is defined as the sum Vol(G) = 1 2 � ℓ(e). (2) The degree of a node v ∈ V is the number of its neighbors in G, deg(v) = kv. It has been shown in =-=[15]-=- that among all undirected connected graphs of normalized volume, Vol(G) = 1, which are not cycles and kv �= 1 for all nodes, the minimal possible value of the volume entropy, min(h) = 1 2 e∈E � kv lo... |

7 |
A space syntax approach to spatial cognition in urban environments”, Position paper for NSF-funded research workshop Cognitive Models of Dynamic Phenomena and Their Representations, October 29 - 31
- Jiang
- 1998
(Show Context)
Citation Context ...rumental in predicting human behavior, for instance, pedestrian movements in urban environments [8]. A set of theories and techniques for the analysis of spatial configurations is called space syntax =-=[9]-=-. Space syntax is established on a quite sophisticated speculation that the evolution of built form can be explained in analogy to the way biological forms unravel [7]. It has been developed as a meth... |

6 |
A New Theory of Space Syntax
- Batty
- 2004
(Show Context)
Citation Context ...of identifiable urban elements such as locations or areas (forming nodes in a graph) whose relationships to one another are often associated with linear transport routes such as streets within cities =-=[6]-=-. Such planar graph representations define locations or points in Euclidean plane as nodes or vertices {i}, i = 1, . . ., N, and the edges linking them together as i ∼ j, in which {i, j} = 1, 2, . . .... |

6 |
Loan Matrix Computations (3rd Edition
- Golub, Van
- 1996
(Show Context)
Citation Context ...ng a rotation transformation in R n ). The result (29-30) relates the problem of network segmentations to the investigation of n primary eigenvectors of a symmetric matrix defined on the graph nodes, =-=[26, 27]-=-. The eigenvectors u i> 1 have both positive and negative entries, so that in general the matrix 15s[u1, . . .,un ] differs substantially from that one comprising of the discrete cluster indicator vec... |

5 |
Submitted to Environment and Planning
- Hillier, B, et al.
- 1984
(Show Context)
Citation Context ...gurations is called space syntax [9]. Space syntax is established on a quite sophisticated speculation that the evolution of built form can be explained in analogy to the way biological forms unravel =-=[7]-=-. It has been developed as a method for analyzing space in an urban environment capturing its quality as being comprehendible and easily navigable [8]. Although, in its initial form, space syntax was ... |

5 |
Environment and Planning B: Planning and Design
- Ratti
- 2004
(Show Context)
Citation Context ... focused mainly on patterns of pedestrian movement in cities, later the various space syntax measures of urban configuration had been found to be correlated with the different aspects of social life, =-=[10]-=-. Decomposition of a space map into a complete set of intersecting axial lines, the fewest and longest lines of sight that pass through every open space comprising any system, produces an axial map or... |

4 |
eds.), Socio-Economic Models in Geography
- Haggett, Chorley
- 1967
(Show Context)
Citation Context ...sic structures for representing forms where topological relations are firmly embedded within Euclidean space. The widespread use of graph theoretic analysis in geographic science had been reviewed in =-=[3]-=- establishing it as central to spatial analysis of urban environments. In [4], the basic graph theory methods had been applied to the measurements of transportation networks. Network analysis has long... |

4 |
On node and axial grid maps: distance measures and related topics
- Kruger
- 1989
(Show Context)
Citation Context ...lete bipartite graph, where the set of vertices can be divided into two disjoint subsets such that no edge has both end-points in 5sthe same subset, and every line joining the two subsets is present, =-=[11]-=-. These sets can be naturally interpreted as those of the vertical and horizontal edges in the primary graphs (streets and avenues). In bipartite graphs, all closed paths are of even length, [12]. It ... |

4 |
Bolyai Society Mathematical Studies 2: Combinatorics, Paul Erdös is Eighty
- Lovász
- 1993
(Show Context)
Citation Context ...adjoint operator. The orthonormal ordered set of real eigenvectors ψi, i = 1 . . .N, of the symmetric operator � T establishes the basis in H. In quantitative theory of random walks defined on graphs =-=[18, 17]-=- and in spectral graph theory [19], the properties of graphs are studied in relationship to the eigenvalues and eigenvectors of self-adjoint operators defined on them. In particular, the symmetric tra... |

4 |
The Syntactical Image of The City: A Reciprocal Definition
- Dalton, Bafna
(Show Context)
Citation Context ...lete subgraphs of the original graph. The implementation of spectral approach removes indeterminacies of the empirical clique concatenation techniques used in space syntax analysis of urban textures, =-=[34]-=-. If the covariance matrix clearly exhibits a block structure, and once the relevant coarse-grained connectivity matrix is computed, we can identify dynamical clusters (blocks) by using a linearized c... |

3 |
Introduction to Statistical
- Fukunaga
(Show Context)
Citation Context ...he reducing size of a data set. It is achieved by the optimal linear transformation retaining the subspace that has largest variance (a lower-order principal component) and ignoring higher-order ones =-=[31, 32]-=-. Given an operator S self-adjoint with respect to the measure m defined on a connected undirected graph G, it is well known that the eigenvectors of the symmetric matrix S form an ordered orthonormal... |

2 |
innovation, scaling, and the pace of life in cities”, PNAS published online Apr 16
- Bettencourt, Lobo, et al.
- 2007
(Show Context)
Citation Context ...relating urbanization to economic development and knowledge production are very general, being shared by all cities belonging to the same urban system and sustained across different nations and times =-=[36]-=-. There is a tied connection between physical activity of humans, their mobility and the layout of buildings, roads, and other structures that physically define a community [37]. Spatial organization ... |

1 |
Spline Models for Observational Data 59
- Wahba
- 1990
(Show Context)
Citation Context ... matrix as measuring similarity between nodes, we can show following [22] that the Euclidean distance (20) leads to Euclidean inner-product similarity which can be replaced by a general Mercer kernel =-=[23, 24]-=- uniquely represented by a positive semi-definite matrix Ki,j. If we then relax the discrete structure of X by assuming that X is an arbitrary orthonormal matrix, the minimization of the objective fun... |