"Hidden geometry of urban areas and interpretation of highly inhomogeneous, incomplete databases" Dmitry Volchenkov

Project FP7 – ICT-318723 MATHEMACS

The city is a spatial network, providing alternatives to our movements …

..and thus converting a space pattern into a pattern of relationships

Street maps of London, showing poverty and wealth by color coding, Charles Booth (1840-1916), London, UK

Royal Saltworks of Chaux Arc-et-Senans, France

Claude-Nicolas Ledoux (1736 –1806) Plan for the Ideal City of Chaux

City of Karlsruhe:

A network of large avenues A modernization program of Paris commissioned by Napoléon III and led by the Seine prefect, Baron GeorgesEugène Haussmann, between 1852 and 1870.

The more isolated is a place, the worse is the situation in that

How to spot isolation?

We used to live in Euclidean space

Volchenkov, D., Ph. Blanchard, Mathematical Analysis of Urban Spatial Networks, © Springer ISBN 978-3540-87828-5, [3564 downloads since January, 2009]

In order to quantify isolation, we have to use such the structural characteristics that fit the Euclidean space structure!

Euclidean space structure of a graph

First –passage time of RW

Commute time

First – passage time of RW

Random Walks: What is that? Physical model

Mathematical meaning

P1  P2  P3  P4  1

, a permutation matrix

Symmetry of route choice: the equivalent paths are equiprobable

if ,        0, then   Aut 

T,   0,

T

ij

 1, a stochastic matrix

j

RW is a stochastic automorphism expressing structural symmetries: Equivalent walks are equiprobable

A “path integral” graph distance

All possible paths are taken into account, some paths are more preferable (for RW) then others.

Geometry of Data & Graphs • Path integral sums over all RWs to compute a propagator. • Propagator is the Green’s function of the diffusion operator: T,   0,

G

 Tn 

n 

1  " L1" 1 T

• The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator: LGL  L, GLG  G,

G, L  0

• Given two distributions x,y, their scalar product:

x, y T

 x, G y 

• The (squared) norm of a distribution: x

2 T

 x, Gx 

• The Euclidean distance: xy

2 T

 x

2 T

 y

2 T

 2x, y T

Probabilistic geometry of graphs Graph  A  T  D1 A, D  diag deg( 1),deg( N ) ˆ  D1 2 AD 1 2 ,  T ˆ    ,   N , T l l l l

1  s ,i  s , j s  2 1   s  1,i  1, j N

Gij  

  2 ,i     1,i 1   2      N ,i     1,i 1   N 

1  1    N ,  12,i   i 

 2, j        1, j 1   2  ,      N, j      1, j 1   N  

First-passage time:  i

2 T

2 N 1  k ,i    i H ij 2 k  2 1  k  1,i i 1 N

 1

Commute time:   Kij  i  j

2 T

  k, j  k ,i      1, j 1  k k  2   1,i 1  k N

   

2



2,i

2, j

, 3, j 

deg i  2E

         i , j T  ei , Ge j       PR N 1

 

 j

, 3,i 

 i

Can we see the first-passage times? Tax assessment value of land ($)

Manhattan, 2005

(Mean) First passage time

(Mean) first-passage times in the city graph of Manhattan SoHo

Federal Hall

10

East Village

100

1,000

Bowery

East Harlem

5,000

10,000

Can we see the first-passage times? (Mean) first-passage times in the city graph of Manhattan Federal Hall

SoHo

East Village

Bowery

East Harlem

10 100 1,000 5,000 10,000 Log of the mean annual household income (×$1,000, 2003) Federal Hall

SoHo

East Village

Bowery

East Harlem

300 100 60 40 20 Log of the annual prison expenditures ( ×$1,000, 2003) Federal Hall

100

SoHo

250

East Village

1,000

2,500

Bowery

10,000

East Harlem

50,000

Why are mosques located close to railways? NEUBECKUM: IsolationM oschee  10  log

first - passage time (M oschee)  12 dB M in first - passage time first - passage time (Kirche) IsolationKirche   10  log  3 dB M in first - passage time

Social isolation vs. structural isolation

Can we hear first-passage times? F. Liszt Consolation-No1

P. Tchaikovsky, Danse Napolitaine

V.A. Mozart, Eine Kleine Nachtmusik

Bach_Prelude_BWV999

R. Wagner, Das Rheingold (Entrance of the Gods)

Can we hear first-passage times? First-passage time Recurrence time

Tonality: the hierarchy of harmonic intervals

Tonality of music

The basic pitches for the E minor scale are "E", "F#", "G", "A", "B".

The recurrence time vs. the first passage time over 804 compositions of 29 Western composers.

Principal components by random walks Representations of graphs & databases in the probabilistic geometric space are essentially multidimensional! 1000 × 1000 data table (or a connected graph of 1000 nodes) is embedded into 999-dimensional space!

Dimensions are unequal!

~

1 , k  2.... N 1  k

Kernel principal component analysis (KPCA) with the kernel G   T  1 1T  " L " n

n 

1

Nonlinear principal components by random walks MILCH

K

= MILK

Matrix of lexical distances, A

Dmilch, milk   2 5 ; 

Stochastic normalizat ion, T

d l1 , l2  

 G

T

n

1 Dl1 , l2   # List of words List of words n



1  " L1"  Kernel PCA 1 T

In contrast to the covariance matrix which best explains the variance in the data with respect to the mean, the kernel G traces out all higher order dependencies among data entries.

Integration of databases for forecasting future trends • Real-world databases are inhomogeneous & incomplete: • The major statistics come after WWII; • The number of polities is ever growing;

Integration of databases for forecasting future trends

Database A

time

Integration of databases for forecasting future trends Relevant databases

Transitions between states Database C Database A

Database B time A graph of states

How can we save Europe? Crisis for Europe as trust hits record low

Is there a common trend for European countries?

No common trends for EU Maddison historical database: GDP per capita

Kalman filter based on GDP data Hypothesis (fitting parameters)

Present

Forecasting

?

Database

Training sequence time

+ Average over many evolution scenarios

… if we play the previous history

No common trends for EU if we play the previous history SCENARIO #1

SCENARIO #2

High trend

High trend

Low trend

Low trend

Economic recovery after the WWII came at different rates in different parts of Europe.

Maddison’s database retells us the story about recovering after the WWII Industrial countries have an edge on competitors if there is no war (GDP variations are limited to ± $500/year)

Traditional capital shelters thrive for larger variations

Maddison’s database predicts bankruptcy to the countries that remained uninvolved in the global recovery process.

IRAQ

To catch up with new tendencies, we have to add more databases Evolution of political Regimes

Democracy/Autocracy indices

Inequality

Top income shares; the largest historical database available concerning the evolution of income inequality

Polity IV tells us that • Six criteria are enough to fully describe a governing regime; • These criteria describe a political state- no matter whether this state is presently occupied, or not; • The historical data on governing polices are well documented (no interruptions/almost no “noise”); • It is possible to quantify the difference between political regimes

Regulation of chief executive recruitment

Unregulated

Openness of Executive Recruitment

Closed

Competitiven ess of Executive Recruitment

Selection Dual executive election

Regulated

Open

Regulation of Participatio n

Competitiv eness of participatio n

Unlimited Authority

Unregulated

Unregulate d

Intermediate

Multiple identity

Repressed

Slight to moderate limitations

Sectarian

Suppressed

Unregulated

Dual executive designation Transitional

Executive constrains

Intermediate

Factional Restricted

Substantial limitations Dual hereditary/co mpetitive

Transitional

Intermediate Regulated Executive Parity

Competitiv e

+ Interruption (foreign occupation) + Interregnum (anarchy) + Transitional = 7,566 “states”

Polity IV tells us that “Political distance” – the minimal number of political changes (reforms) required to convert the political system of one country into that of another Trends in Governance in 1810

Trends in Governance in 2012

the world is always in transience

Polity IV tells us that • There should be a positive feedback, reinforcing the multiplication of polities;

dN N dt • We witness the very beginning of a chain reaction process (of atomization of the polity landscape)

the number of polities is ever growing

The World Top Income database tells us that If the GDP-gain substantially outmatches/ lags below the mean (red line), it apparently comes at the cost of increasing inequality

Global synchronization of inequality dynamics

Parabolic fit(!)

rapidly rising inequality marks wars/conflicts/ instabilities, and instabilities multiply polities.

232 configurations have been observed since 1800 "Tajikistan", 2013 "Nepal", 1945 "Korea North", 2013

"Libya", 2010

Foreign interruption

"Cuba", 2005

"Thailand", 2013 "Korea South", 2013

"United States", 2013

"Czech Republic", 2013

"Estonia", 2013

New configurations arise from time to time

Random walks on the graph of political regimes Transition matrix between types of governance (17,000 historical transitions)

Each political regime has its own dynamics for GDP and IPLC Process starts from the actual data (GDPPC & IPLC) for 2013

+ Averaging over all collected histories

Most transitions happen within the groups of authoritarian states and presidential republics, while liberal democracies and dictatorships are quite “sticky”.

A common state insists on a common economic and political destiny for its citizens.

However, the actual trends of different economic groups might be statistically inconsistent.

Polities proliferation score Possible splitting of a country is visible as the statistically inconsistent trends.

Greece vs. Russia

Expected number of countries

Main factors resulting in multiplying scores: 1. inequality (stretches bandwidth of boxes); 2. Authoritarian regimes are short-lived, quickly transforming to other modes of authoritarianism, provoking instability

There can be a common European trend • if polities are allowed to split within EU without wars; • the workforce are allowed to migrate freely;

Germany vs. Greece

Back to the City-States?

Strong inequality worsens perspectives, authoritarian governance worsens perspectives USA vs. China

IPLC ~ O(GDPpc2)

“In slowly growing economies, past wealth naturally takes on disproportionate importance, because it takes only a small flow of new savings to increase the stock of wealth steadily and substantially.” (Thomas Piketty, Capital in the Twenty-First Century (2014))

Battle in Asia, concord in Europe China (red) vs. Indonesia (blue)

Germany (dark) vs. Austria (light)

Conclusions The city converts a space pattern into a pattern of relationships

RWs represent stochastic automorphisms of a structure; summing up all RWs → Probabilistic geometry

RWs can be used in order to combine different (incomplete) databases

Kernel Principal Component Analysis handles high-order dependences in data

Some references D.V., Ph. Blanchard, Mathematical Analysis of Urban Spatial Networks, © Springer Series Understanding Complex Systems, Berlin / Heidelberg. ISBN 978-3-540-87828-5, 181 pages (2009).

D.V., Ph. Blanchard, “Introduction to Random Walks on Graphs and Databases”, © Springer Series in Synergetics , Vol. 10, Berlin / Heidelberg , ISBN 978-3-64219591-4 (2011).

Volchenkov, D., “Markov Chain Scaffolding of Real World Data”, Discontinuity, Nonlinearity, and Complexity 2(3) 289–299 (2013)| DOI: 10.5890/DNC.2013.08.005. Volchenkov, D., Jean-René Dawin, “Musical Markov Chains ”, International Journal of Modern Physics: Conference Series, 16 (1) , 116-135 (2012) DOI: 10.1142/S2010194512007829. Volchenkov, D., Ph. Blanchard, J.-R. Dawin, “Markov Chains or the Game of Structure and Chance. From Complex Networks, to Language Evolution, to Musical Compositions”, The European Physical Journal Special Topics 184, 1-82 © Springer Berlin / Heidelberg (2010). Volchenkov, D., “Random Walks and Flights over Connected Graphs and Complex Networks”, Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 21–55 http://dx.doi.org/10.1016/j.cnsns.2010.02.016 (2010).