1. Mathematical Analysis of Complex Networks and Databases
Dima Volchenkov

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

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

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

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

6. Fractional/Stochastic symmetry
GA (adjacency matrix of the graph) 
P: [P,A]=0, P =1, only trivial
automorphisms

7. Fractional/Stochastic symmetry
GA (adjacency matrix of the graph) 
P: [P,A]=0, P =1, only trivial
automorphisms
A permutation matrix is a particular case of
stochastic matrix:

8. Fractional/Stochastic symmetry
GA (adjacency matrix of the graph) 
P: [P,A]=0, P =1, only trivial
automorphisms
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

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

10. Plan of my talk A variety of stochastic automorphisms
Probabilistic geometric manifolds
Euclidean metric structure
Probabilistic differential geometry
(+/- ) Curvature  intelligible/ confusing environments.
Ricci-Hamilton flows deforming a probabilistic manifold
Evolution of networks
Which paths are taken to be equi-probable?
Example:
Music: (the cyclic group Z/12Z over the set of frequencies )
From stochastic symmetry to metric geometry
Examples:
Nearest neighbor random walks,
Electric resistance networks,
Urban networks

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

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

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

14. A variety of fractional automorphisms
is a transition matrix of a random walk.
The central question: what types of path do we treat as equi-probable?
One end is fixed:
“Nearest neighbor random walks” ℓ
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

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

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

17. 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.

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

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

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

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

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

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

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

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

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

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

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

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

30. From stochastic symmetry to metric geometry
Graph A  P: [P,A]=0, Automorphisms 
T: [T, A]=0  , the Green function
(a generalized inverse)
We can define a scalar product:
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:
Metric Structure
The Drazin inverse corresponds to the eigenprojection of the matrix L w.r.t. to the eigenvalue λ1 = 1−μ1 = 0
where the product in the idempotent matrix Z is taken over all nonzero eigenvalues of L.

31. 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

32. 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

33. 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
Example 1: Nearest neighbor random walks
The (squared) norm of a vector
and an angle
The Euclidean distance
p=1/4
p=1/4
p=1/4
p=1/4

34. 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
Example 1: Nearest neighbor random walks
The (squared) norm of a vector
and an angle
The Euclidean distance
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,

35. 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
Example 2: Electric Resistance Networks, Resistance distance
The (squared) norm of a vector
and an angle
The Euclidean distance
An electrical network is considered as an interconnection of resistors.
can be described by the Kirchhoff circuit law,

36. 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
Example 2: Electric Resistance Networks, Resistance distance
The (squared) norm of a vector
and an angle
The Euclidean 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:

37. The (mean) first-passage time in cities
Manhattan, 2005
Neubeckum, Germany, 2012
Tax assessment value of land ($)
(Mean) First passage time
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.

38. 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

39. Probabilistic Riemannian geometry
Small changes to data in a database/weights of nodes would rise small changes to the probabilistic geometric representation of database/graph. We can think of them as of smooth manifolds endowed with a Riemannian metric. x ui uj p
Given a function defined at a node x, we can extend it to a vicinity of the node.

40. Probabilistic Riemannian geometry
Small changes to data in a database/weights of nodes would rise small changes to the probabilistic geometric representation of database/graph. We can think of them as of smooth manifolds endowed with a Riemannian metric. x ui uj p
Given a function defined at a node x, we can extend it to a vicinity of the node.
We can determine a node/entry dependent basis of vector fields on the tangential probabilistic manifold:
TxM RN-1

41. Probabilistic Riemannian geometry
Small changes to data in a database/weights of nodes would rise small changes to the probabilistic geometric representation of database/graph. We can think of them as of smooth manifolds endowed with a Riemannian metric. x ui uj p
Given a function defined at a node x, we can extend it to a vicinity of the node.
We can determine a node/entry dependent basis of vector fields on the tangential probabilistic manifold:
TxM RN-1
For the group of translations, the shift operator is given by the exponential map of the differential operator:

42. Probabilistic Riemannian geometry
The node/entry dependent basis of vector fields on the tangential probabilistic manifold: x ui uj p
TxM RN-1
… and then define the metric tensor at each node/entry (of the database) by
The standard calculus of differential geometry…
The Riemann curvature tensor:
The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
The Ricci curvature tensor & the scalar curvature:

43. “Confusing environments”
Probabilistic manifolds of negative curvature
Traps: (Mean) First Passage Time > Recurrence Time . . . i1 ik i2
(mean) first passage time i3
(mean) first passage time .
recurrence times i4 . .
Mazes and labyrinths . .
It might be difficult to reach a place, but we return to the place quite often
provided we reached that.
“Confusing environments”

44. Probabilistic manifolds of positive curvature
Landmarks: (Mean) First Passage Time < Recurrence Time .
(mean) first passage time
recurrence times i1 i2 i3 i4 ik .
Landmarks establishes a wayguiding structure that facilitates understanding of the environment.
“Intelligible environments”

45. Probabilistic manifolds of positive curvature
An example:
Z/12Z
Music = the cyclic group over the discrete space of notes:
Motivated by the logarithmic pitch perception in humans, music theorists represent pitches using a numerical scale based on the logarithm of fundamental frequency.
The resulting linear pitch space in which octaves have size 12, semitones have size 1, and the number 69 is assigned to the note "A4".

46. A discrete model of music (MIDI) as a simple
Markov chain
In a musical dice game, a piece is generated by patching notes Xt taking values from the set of pitches that sound good together into a temporal sequence.

47. First passage times to notes resolve tonality
In music theory, the hierarchical pitch relationships are introduced based on a tonic key, a pitch which is the lowest degree of a scale and that all other notes in a musical composition gravitate toward. A successful tonal piece of music gives a listener a feeling that a particular (tonic) chord is the most stable and final.
Tonality structure
of music
The basic pitches for the E minor scale are "E", "F", "G", "A", "B", "C", and "D".
The E major scale is based on "E", "F", "G", "A", "B", "C", and "D".
The A major scale consists of "A", "B", "C", "D", "E", "F", and "G".
The recurrence time vs. the first passage time over 804 compositions of 29 Western composers.
Namely, every pitch in a musical piece is characterized with respect to the entire structure of the Markov chain by its level of accessibility estimated by the first passage time to it that is the expected length of the shortest path of a random walk toward the pitch from any other pitch randomly chosen over the musical score. The values of first passage times to notes are strictly ordered in accordance to their role in the tone scale of the musical composition.

48. Evolution of networks: Ricci-Hamilton flows
We consider the metric tensor to be functions of a variable which is usually called "time”, then we obtain the geometric evolution equation (which preserves the volume of the metric):
The Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions

49. Evolution of networks: Ricci-Hamilton flows
We consider the metric tensor to be functions of a variable which is usually called "time”, then we obtain the geometric evolution equation (which preserves the volume of the metric):
The Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions
“Densification” of the network of “positive curvature”
“Contraction” of a “probabilistic manifold”
A “collapse” and decomposition of the network of “negative curvature”

50. The scalar curvature for ℓ - neighbor random walks
Paths to ALL neighbors of i at the distance ℓ are equi-probable ℓ
A collapse and decomposition of the network;
localization of walkers in the best connected places ℓ
“Densification” of the network of “positive curvature”

51. ℓ t t ℓ The scalar curvature for ℓ - neighbor random walks
Paths to ALL neighbors of i at the distance ℓ are equi-probable ℓ
A collapse and decomposition of the network;
localization of walkers in the best connected places t
“time” in the Ricci-Hamilton flow t ℓ
“Densification” of the network of “positive curvature”

52. ℓ t t ℓ The scalar curvature for ℓ - neighbor random walks
Paths to ALL neighbors of i at the distance ℓ are equi-probable ℓ
A collapse and decomposition of the network;
localization of walkers in the best connected places t
“time” in the Ricci-Hamilton flow t ℓ
“Densification” of the network of “positive curvature”
The Ricci-Hamilton flow passes through a variety of configurations of the stochastic automorphisms of the graph.

53. Conclusion Probabilistic geometric manifolds
Stochastic automorphisms of graphs/databases/groups
Probabilistic geometric manifolds
Euclidean metric structure
Probabilistic differential geometry
Evolution by Ricci-Hamilton flow expanding negatively curved regions and contracting positively curved regions
“For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken away even that which he hath.”
Matthew 25:29