Introduction to Random Walks and Diffusions to Network and Databases: from Electric Networks to Urban Spatial Networks Dimitri Volchenkov (Bielefeld University ） - 1 st 45 ′ talk Markov chain methods

A network is any method of sharing information between systems consisting of many individual units V, a measurable pattern of relationships between entities. What is a network/database? We suggest that these relationships can be expressed by large but finite matrices : A: V×V  R + or at least A T A, AA T are positive, symmetric.

Being often embedded into Euclidean space, graphs/databases nevertheless lack of a metric space structure. The main problem: Thus, we cannot acquire a comprehensive image of the whole network – it looks confusing to us.

Symmetry G  A (adjacency matrix of the graph) Symmetry (exact reflection of form on opposite side) is a striking attribute of a shape or a relation.

Symmetry G  A (adjacency matrix of the graph)   : [ ,A]=0, Automorphisms  A permutation matrix Symmetry (exact reflection of form on opposite side) is a striking attribute of a shape or a relation.

Fractional/Stochastic symmetry   : [ ,A]=0,  only trivial automorphisms  G  A (adjacency matrix of the graph)

Fractional/Stochastic symmetry   : [ ,A]=0,  only trivial automorphisms  G  A (adjacency matrix of the graph) A permutation matrix is a particular case of stochastic matrix:

Fractional/Stochastic symmetry   : [ ,A]=0,  only trivial automorphisms  G  A (adjacency matrix of the graph) A permutation matrix is a particular case of stochastic matrix: Let us extend the notion of automorphisms onto the class of stochastic matrices.  T: [T, A]=0, Fractional automorphisms, or stochastic automorphisms

There are infinitely many fractional automorphisms…  T: [T, A]=0, Fractional automorphisms G  A (adjacency matrix of the graph) Each T can be considered as a transition matrix of some Markov chain, a “random walk” defined on the graph/database.

The shortest-path distance, insensitive to the structure of the graph: The length of a walk The distance = “a Feynman path integral” sensitive to the global structure of the graph. The main idea in “two words” Systems of weights are related to each other in a geometric fashion.

A variety of fractional automorphisms The central question: what types of path do we treat as equi-probable? is a transition matrix of a random walk.

A variety of fractional automorphisms The central question: what types of path do we treat as equi-probable? is a transition matrix of a random walk. “Nearest neighbor random walks” ALL paths to nearest neighbors of i are equi-probable i One end is fixed:

A variety of fractional automorphisms The central question: what types of path do we treat as equi-probable? is a transition matrix of a random walk. “Nearest neighbor random walks” Paths to ALL nearest neighbors of i are equi-probable i “ℓ - neighbor random walks” Paths to ALL neighbors of i at the distance ℓ are equi- probable ℓ One end is fixed:

A variety of fractional automorphisms The central question: what types of path do we treat as equi-probable? is a transition matrix of a random walk. “All paths between i and j of the length ℓ are equi-probable” ℓ Both ends are fixed: i j

A variety of fractional automorphisms The central question: what types of path do we treat as equi-probable? is a transition matrix of a random walk. “All paths between i and j are equi-probable” “All paths between i and j of the length ℓ are equi-probable” i j

General transition operator The generalized transition operator must contain all possible transitions that can take place by the moment t: This is not just any path in a connected graph acquires a statistical weight, but also all strategies of choosing a neighborhood (in which all paths are equi-probable) are characterized by certain probabilities.

“Maximal entropy” RWNearest neighbor RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Maximal entropy RWNearest neighbor RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda

Nearest neighbor RW“Maximal entropy” RW Properties of flows defined by different stochastic automorphisms are very different J. K. Ochab, Z. Burda Homogeneous covering Localization in the best connected places

Graph  A   : [ ,A]=0, Automorphisms  T: [T, A]=0 , the Green function  We can define a scalar product: Metric Structure From stochastic symmetry to metric geometry

Graph  A   : [ ,A]=0, Automorphisms  T: [T, A]=0 , the Green function  We can define a scalar product: Metric Structure (a generalized inverse) From stochastic symmetry to metric geometry The problem is that As being a member of a multiplicative group under the ordinary matrix multiplication, the Laplace operator possesses a group inverse (a special case of Drazin inverse) with respect to this group, L ◊, which satisfies the conditions: The Drazin inverse corresponds to the eigenprojection of the matrix L w.r.t. to the eigenvalue λ 1 = 1−μ 1 = 0 where the product in the idempotent matrix Z is taken over all nonzero eigenvalues of L.

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors of the projective space

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors of the projective space The (squared) norm of a vector and an angle The Euclidean distance

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors of the projective space The (squared) norm of a vector and an angle The Euclidean distance Example 1: Nearest neighbor random walks

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors of the projective space The (squared) norm of a vector and an angle The Euclidean distance Example 1: Nearest neighbor random walks The commute time, the expected number of steps required for a random walker starting at i ∈ V to visit j ∈ V and then to return back to i, The spectral representation of the (mean) first passage time, the expected number of steps required to reach the node i for the first time starting from a node randomly chosen among all nodes of the graph accordingly to the stationary distribution π.

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors in the projective space The (squared) norm of a vector and an angle The Euclidean distance Example 2: Electric Resistance Networks, Resistance distance An electrical network is considered as an interconnection of resistors. Kirchhoff circuit law can be described by the Kirchhoff circuit law,

Probabilistic Euclidean metric structure Every stochastic automorphism T: [T,A]=0 induces a Euclidean metric structure with the inner product between any two vectors in the projective space The (squared) norm of a vector and an angle The Euclidean distance Example 2: Electric Resistance Networks, Resistance distance Given an electric current from a to b of amount 1 A, the effective resistance of a network is the potential difference between a and b, The effective resistance allows for the spectral representation:

Cities are the biggest editors of our life: built environments constrain our visual space and determine our ability to move thorough by structuring movement space. Some places in urban environments are easily accessible, others are not; well accessible places are more favorable to public, while isolated places are either abandoned, or misused. In a long time perspective, inequality in accessibility results in disparity of land prices: the more isolated a place is, the less its price would be. In a lapse of time, structural isolation would cause social isolation, as a host society occupies the structural focus of urban environments, while the guest society would typically reside in outskirts, where the land price is relatively cheap. The (mean) first-passage time in cities (Mean) First passage time Tax assessment value of land ($) Manhattan, 2005 Neubeckum, Germany, 2012

Claude-Nicolas Ledoux (March 21, 1736 – November 18, 1806) Plan for the Ideal City of Chaux

Charles Booth Street maps of London, showing poverty and wealth by color coding, transforming existing methods of social survey and poverty mapping towards the end of the nineteenth century- Charles Booth ( ), London, UK

A modernisation programme of Paris commissioned by Napoléon III and led by the Seine prefect, Baron Georges- Eugène Haussmann, between 1852 and A network of large avenues

40 "We shape our buildings, and afterwards our buildings shape us.“ Sir Winston Churchill Sir Winston Churchill (October 28, 1943: while requesting that the House of Commons be rebuilt exactly as before, remaining insufficient to seat all its members.)

43 A city converts a space pattern into a pattern of relationships. Cities generate more interactions with more people producing denser patterns of connection as the grid constrains proximity. Space Syntax Theory Professors Bill Hillier UCL BARTLETT SCHOOL OF GRADUATE STUDIES SPACE RESEARCH GROUP

Federal Hall Times Square SoHo East Village Bowery East Harlem (Mean) first-passage times in the city graph of Manhattan

mean household income The data on the mean household income per year provided by

The data taken from the

The spreading outwards of a city and its suburbs to its outskirts to low-density and auto-dependent development on rural land, high segregation of uses (e.g. stores and residential), and various design features that encourage car dependency.

CITY STRUCTURAL FOCUS

CITY OLD STRUCTURAL FOCUS NEW STRUCTURAL FOCUS

CITY OLD STRUCTURAL FOCUS NEW STRUCTURAL FOCUS Redlining policy

54 Detroit: in the City Hall 2006 : : 1956 Problem of lost cities Detroit: Madison theater 1942: 2006:

We need the smart urban planning!