1. Structure creates a chance
IK 2013: Wicked Problems, Complexity and Wisdom, Günne, March 2013
Structure creates a chance
D.Sc. (Habil.) D. Volchenkov
(Bielefeld University)
The City, or the game of structure and chance. Mathematical analysis of complex structures and wicked problems

2. “NOW & HERE “ vs. “THEN & THERE”
Searching activity in humans can be characterized by formation of a neighbourhood:
“NOW & HERE “
Environmental structure does not matter;
A characteristic scale (space/time) exists;
“THEN & THERE”
Movements are determined by the environmental structure (strong interactions);
No characteristic time scale (space/time);
20 m
30 sec
“NOW & HERE “
“THEN & THERE”

3. “NOW & HERE “ vs. “THEN & THERE”
Searching activity in humans can be characterized by formation of a neighbourhood:
“NOW & HERE “
Environmental structure does not matter;
A characteristic scale (space/time) exists;
“THEN & THERE”
Movements are determined by the environmental structure (strong interactions);
No characteristic time scale (space/time);
20 m
30 sec
“NOW & HERE “
“THEN & THERE”

4. Structure = adjacency  adjacency  ….  adjacency
“NOW & HERE “
A graph G =(V,E), where V is the set of identical elements called vertices (or nodes) and E VV is a collection of pairs of elements from V called edges.

5. Is it possible to geometrize a structure?
Complexity:
No direct ordering of nodes/ entities;
can contain information about processes evolving at different spatio-temporal scales;
many “semantic levels”.
Lack of global geometric structure!
(binary relations between places, instead of geometry)
“NOW & HERE “
How can we introduce distances and angles of our everyday intuition developed in Euclidean space?

6. Structure = adjacency  adjacency  ….  adjacency
“NOW & HERE “
A walk is a sequence of graph vertices and graph edges such that the graph vertices and graph edges are adjacent.

7. Structure = adjacency  adjacency  ….  adjacency
“NOW & HERE “
Exploration has no characteristic time scale, and therefore…
…a walk is not necessary of the nearest neighbor type: we can move along adjacent edges as long as we like.

8. Representation of Graphs by Matrices
The major advantage of using matrices is that calculations of various graph characteristics can be performed by means of the well known operations of linear algebra.
Aij =1 if i ~ j , and Aij =0 if i ~ j. A=

9. Structure = adjacency  adjacency  ….  adjacency
“NOW & HERE “
Aij =1 if i ~ j , and Aij =0 if i ~ j.
Then the number of walks of length n between i and j is given by Anij

10. Any data interpretation/classification/judgment is always based on introduction of some equivalence relation on the set of walks over the database:
“NOW & HERE “
Rx: walks of the given length n starting at the same node x are equivalent
Ry: walks of the given length n ending at the same node y are equivalent
Rx  Ry : walks of the given length n between the nodes x and y are equivalent

11. Equivalence partition of walks => random walk
Given an equivalence relation on the set of walks and a function such that
we can always normalize it to be a probability function:
all “equivalent” walks are equiprobable.
Partition of walks into equivalence classes
The utility function for each equivalence class
A random walk transition operator between eq. classes
Set of all n-walks

12. We proceed in three steps:
Step 0: Given an equivalence relation between paths, any
transition can be characterized by a probability to belong
to an equivalence class.
Different equivalence relations  Different equivalence classes  Different probabilities
Step 1: “Probabilistic graph theory”
Nodes of a graph, subgraphs (sets of nodes) of the graph, the whole graph are described by probability distributions & characteristic times w.r.t. different Markov chains;
Step 2: “Geometrization of Data Manifolds”
Establish geometric relations between those probability distributions whenever possible;
1. Coarse-graining/reduction of networks & databases → data
analysis; sensitivity to assorted data variations;
2. Transport optimization(Monge-Kontorovich type problems) → distances between distributions;

13. An example of equivalence relation:
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent
Equiprobable walks: the nearest neighbor random walks
Stochastic normalization

14. An example of equivalence relation:
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent
Equiprobable walks: the nearest neighbor random walks
Stochastic normalization
Probability of a n-walk

15. An example of equivalence relation:
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent
Equiprobable walks:
Stochastic normalization
Probability of a n-walk …
“Structure learning”

16. An example of equivalence relation:
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent
Equiprobable walks:
Stochastic normalization
Probability of a n-walk … ≠
“Structure learning”
Stochastic normalization

17. What is a neighbourhood?
Who are my neighbours? …
1.Neighbours are next to me…
2.Neighbours are 2 steps apart from me…
n.Neighbours are n steps apart from me …

18. What is a neighbourhood?
Who are my neighbours? …
1.Neighbours are next to me…
2.Neighbours are 2 steps apart from me…
n.Neighbours are n steps apart from me …
My neighbours are those, which I can visit with equal probability (w.r.t. a chosen equivalence of paths)…

19. An example of equivalence relation:
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent …
Equiprobable walks:
Stochastic matrices:

20. A variety of random walks at different scales
Step 0
A variety of random walks at different scales
An example of equivalence relation:
Rx: walks of the given length n starting at the same node x are equivalent
Left eigenvectors (m=1)
Centrality measures: …
Equiprobable walks:
Stochastic matrices:
The “stationary distribution” of the nearest neighbor RW

21. Random walks of different scales
Time is introduced as powers of transition matrices

22. Random walks of different scales
Time is introduced as powers of transition matrices

23. Random walks of different scales
Time is introduced as powers of transition matrices

24. Random walks of different scales
Time is introduced as powers of transition matrices

25. Random walks of different scales
Time is introduced as powers of transition matrices

26. Random walks of different scales
Time is introduced as powers of transition matrices

27. Random walks of different scales
Time is introduced as powers of transition matrices

28. Random walks of different scales
Time is introduced as powers of transition matrices

29. Random walks of different scales
Time is introduced as powers of transition matrices

30. Random walks of different scales
Time is introduced as powers of transition matrices

31. Random walks of different scales
Time is introduced as powers of transition matrices

32. Random walks of different scales
Time is introduced as powers of transition matrices

33. Random walks of different scales
Time is introduced as powers of transition matrices

34. Random walks of different scales
Time is introduced as powers of transition matrices

35. Random walks of different scales
Time is introduced as powers of transition matrices

36. Random walks of different scales
Time is introduced as powers of transition matrices

37. Random walks of different scales
Time is introduced as powers of transition matrices

38. Random walks of different scales
Time is introduced as powers of transition matrices

39. Random walks of different scales
Time is introduced as powers of transition matrices

40. Random walks of different scales
Time is introduced as powers of transition matrices
Still far from stationary distribution!
Stationary distribution is already reached!
Defect insensitive.
Low centrality (defect) repelling.

41. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

42. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

43. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

44. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

45. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

46. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

47. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

48. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

49. Random walks for different equivalence relations
Nearest neighbor RW
Maximal entropy RW
J. K. Ochab, Z. Burda

50. Random walks for different equivalence relations
Nearest neighbor RW
“Maximal entropy” RW
J. K. Ochab, Z. Burda

51. Step 1: “Probabilistic graph theory”
As soon as we define an equivalence relation …
Graph Subgraph (a subset of nodes) Node
Time scale
Tr T
The probability that the RW stays at the initial node in 1 step.
“Wave functions” (Slater determinants) of transients (traversing nodes and subgraphs within the characteristic scales) return the probability amplitudes whose modulus squared represent the probability density over the subgraphs.
Probabilistic graph invariants = the t-steps recurrence probabilities quantifying the chance to return in t steps. …
| det T |
The probability that the RW revisits the initial node in N steps.
Return times to the subgraphs within transients = 1/Pr{ … }
Centrality measures
(stationary distributions)
Return times to a node
Random target time
Mixing times over subgraphs
(times until the Markov chain is "close" to the steady state distribution)

52. As soon as we get probability distributions…
Step 2: “Geometrization of Data Manifolds”
As soon as we get probability distributions…
Given T, L ≡ 1- T , the linear operators acting on distributions.
The Green function is the natural way to find the relation between two distributions within the diffusion process
Drazin’s generalized inverse:

53. As soon as we get probability distributions…
Step 2: “Geometrization of Data Manifolds”
As soon as we get probability distributions…
Given T, L ≡ 1- T , the linear operators acting on distributions.
The Green function is the natural way to find the relation between two distributions within the diffusion process
Drazin’s generalized inverse:
Given two distributions x,y over the set of nodes, we can define a scalar product,
The (squared) norm of a vector
and the angle
The Euclidean distance:

54. Transport problems of the Monge-Kontorovich type
Step 2: “Geometrization of Data Manifolds”
As soon as we get probability distributions…
Given T, L ≡ 1- T , the linear operators acting on distributions.
The Green function is the natural way to find the relation between two distributions within the diffusion process
Drazin’s generalized inverse:
Given two distributions x,y over the set of nodes, we can define a scalar product,
Transport problems of the Monge-Kontorovich type
The (squared) norm of a vector
and an angle
The Euclidean distance:
“First-passage transportation” from x to y x y
W(x→y)
W(y→x) ≠

55. Transport problems of the Monge-Kontorovich type
Step 2: “Geometrization of Data Manifolds”
As soon as we get probability distributions…
Given T, L ≡ 1- T , the linear operators acting on distributions.
The Green function is the natural way to find the relation between two distributions within the diffusion process
Drazin’s generalized inverse:
Given two distributions x,y over the set of nodes, we can define a scalar product,
Transport problems of the Monge-Kontorovich type
The (squared) norm of a vector
and an angle
The Euclidean distance:
(Mean) first-passage time
Commute time
Electric potential
Effective resistance distance
Tax assessment land price in cities
Musical diatonic scale degree …
Musical tonality scale

56. Example 1: Nearest-neighbor random walks on undirected graphsy1

57. Example 1: Nearest-neighbor random walks on undirected graphsy1
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.
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,

58. Around The City of Big Apple
Federal Hall
Public places
City CORE
Times Square
SoHo
City CORE 10
steps
100
East Village
steps
500
(Mean) first-passage times in the city graph of Manhattan
steps
1,000
steps
Bowery
East Harlem
City Decay
steps
5,000
steps
10,000
SLUM

59. Example 2: Electric Resistance Networks, Resistance distance
An electrical network is considered as an interconnection of resistors:
The currents are described by the Kirchhoff circuit law:

60. Example 2: Electric Resistance Networks, Resistance distance
An electrical network is considered as an interconnection of resistors:
The currents are described by the Kirchhoff circuit law:
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:

61. Impedance networks: The two-point impedance and LC resonances

62. Geodesics paths of language evolution
Levenshtein’s distance (Edit distance): is a measure of the similarity between two strings: the number of deletions, insertions, or substitutions required to transform one string into another.
MILCH K
= MILK
The normalized edit distance between the orthographic realizations of two words can be interpreted as the probability of mismatch between two characters picked from the words at random.

63. The four well-separated monophyletic spines represent the four biggest traditional IE language groups: Romance & Celtic, Germanic, Balto-Slavic, and Indo-Iranian;
The Greek, Romance, Celtic, and Germanic languages form a class characterized by approximately the same azimuth angle (belong to one plane);
The Indo-Iranian, Balto-Slavic, Armenian, and Albanian languages form another class, with respect to the zenith angle.

64. The systematic sound correspondences between the Swadesh’s words across the different languages perfectly coincides with the well-known centum-satem isogloss of the IE family (reflecting the IE numeral ‘100’), related to the evolution in the phonetically unstable palatovelar order.

65. The normal probability plots fitting the distances r of language points from the ‘center of mass’ to univariate normality.
The data points were ranked and then plotted against their expected values under normality, so that departures from linearity signify departures from normality.

66. The univariate normal distribution is closely related to the time evolution of a mass-density function under homogeneous diffusion in one dimension
in which the mean value μ is interpreted as the coordinate of a point where all mass was initially concentrated, and variance σ2 ∝ t grows linearly with time.
The values of variance σ2 give a statistically consistent estimate of age for each language group.
the last Celtic migration (to the Balkans and Asia Minor) (300 BC),
the division of the Roman Empire (500 AD),
the migration of German tribes to the Danube River (100 AD),
the establishment of the Avars Khaganate (590 AD) overspreading Slavic people who did the bulk of the fighting across Europe.
Anchor events:

67. From the time–variance ratio we can retrieve the probable dates for:
The break-up of the Proto-Indo-Iranian continuum.
The migration from the early Andronovo archaeological horizon
(Bryant, 2001).
by 2,400 BC
The end of common Balto-Slavic history
before 1,400 BC
The archaeological dating of Trziniec-Komarov culture
The separation of Indo-Arians from Indo-Iranians.
Probably, as a result of Aryan migration across India to Ceylon, as early as in 483BC (Mcleod, 2002)
before 400 BC
The division of Persian polity into a number of Iranian tribes, after the end of Greco-Persian wars (Green, 1996).
before 400 BC

68. Proto-Indo-Europeans?
The Kurgan scenario postulating the IE origin among the people of “Kurgan culture”(early 4th millennium BC) in the Pontic steppe (Gimbutas,1982) .
Einkorn wheat
The Anatolian hypothesis suggests the origin in the Neolithic Anatolia and associates the expansion with the Neolithic agricultural revolution in the 8th and 6th millennia BC (Renfrew,1987).
The graphical test to check three-variate normality of the distribution of the distances of the five proto-languages from a statistically determined central point is presented by extending the notion of the normal probability plot. The χ-square distribution is used to test for goodness of fit of the observed distribution: the departures from three-variant normality are indicated by departures from linearity.
The use of the previously determined time–variance ratio then dates the initial break-up of the Proto-Indo-Europeans back to 7,400 BC pointing at the early Neolithic date.

69. In search of Polynesian origins
The components probe for a sample of 50 AU languages immediately uncovers the both Formosan (F) and Malayo-Polynesian (MP) branches of the entire language family.
Headhunters

71. An interaction sphere had existed encompassing the whole region
By 550 AD
…pretty well before 600 –1200 AD while descendants from Melanesia settled in the distant apices of the Polynesian triangle as evidenced by archaeological records (Kirch, 2000; Anderson and Sinoto,2002; Hurlesetal.,2003).

72. Mystery of the Tower of Babel
Nonliterate languages evolve EXPONENTIALLY FAST without extensive contacts with the remaining population. Isolation does not preserve a nonliterate language!
Languages spoken in the islands of East Polynesia and of the Atayal language groups seem to evolve without extensive contacts with Melanesian populations, perhaps because of a rapid movement of the ancestors of the Polynesians from South-East Asia as suggested by the ‘express train’ model (Diamond, 1988) consistent with the multiple evidences on comparatively reduced genetic variations among human groups in Remote.
Headhunters

73. Traps and landmarks Exploitation Exploration Recurrence time
First-passage time:
Landmarks, “guiding structures”: firstly reached , seldom revisited
Exploitation
Traps, “confusing environments”:
can take long to reach, but often revisited
Exploration

74. Musical Dice Game
(*)
The relations between notes in (*) are rather described in terms of probabilities and expected numbers of random steps than by physical time. Thus the actual length N of a composition is formally put N → ∞, or as long as you keep rolling the dice.

75. F. Liszt Consolation-No1
Bach_Prelude_BWV999
R. Wagner, Das Rheingold (Entrance of the Gods)
V.A. Mozart, Eine-Kleine-Nachtmusik

76. A “guiding structure”: Tonality scales in Western music
Increase of harmonic interval/ first –passage time
Recurrence time
First-passage time
The recurrence time vs. the first passage time over 804 compositions of 29 Western composers.

77. Network geometry at different scales
First-passage time
Scale of RW …
The node belongs to a network “core”, consolidating with other central nodes
Recurrence times
The node belongs to a “cluster”, loosely connected with the rest of the network.

78. Ricci flows and photo resolution

79. Possible analogy with Ricci flows
“Densification” of the network of “positive curvature”
“Contraction” of a “probabilistic manifold”
First-passage time
Scale of RW …
Recurrence times
A “collapse” of the network
of “negative curvature”

80. References
D.V., Ph. Blanchard, “Introduction to Random Walks on Graphs and Databases”, © Springer Series in Synergetics , Vol. 10, Berlin / Heidelberg , ISBN (2011).
D.V., Ph. Blanchard, Mathematical Analysis of Urban Spatial Networks, © Springer Series Understanding Complex Systems, Berlin / Heidelberg. ISBN , 181 pages (2009).