"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).