1. Is it possible to geometrize infinite graphs?
(Diffusion metrics and geometrization of finite graphs and relational databases)
D. Volchenkov

2. Motivation
1. Data interpretation m
m* = ? X Y

3. Monge – Kantorovich transport problem
Motivation
1. Data interpretation m
m* = ?
Monge – Kantorovich transport problem
-- the transportation plan L;
-- X , Y probability measures on
a compact (metrizable) space;
K → the transportation metric X Y

4. Monge – Kantorovich transport problem
Motivation
1. Data interpretation m
m* = ?
Monge – Kantorovich transport problem
-- the transportation plan L;
-- X , Y probability measures on
a compact (metrizable) space;
K → the transportation metric X Y
2. Data “coordinates”
“0”

5. Resistor networks
rij is the measure of the opposition that a conductor presents to a current when a voltage is applied.
(an “empirical distance” between i and j )
Ohm's law:
The current through a conductor between two points is directly proportional to the voltage across the two points.

6. Resistor networks 1 2 3 Kirchhoff's current law:
The algebraic sum of currents in a network of conductors meeting at a point is zero.
Ohm's law:
The current through a conductor between two points is directly proportional to the voltage across the two points.
The (effective) resistance between nodes a and b :

7. Resistor networks 1 2 3 Ohm's law: In the matrix form:
(L – the Kirchhoff matrix)
(f all rij =1 W, L is the graph Laplacian operator)

8. Resistor networks 1 2 3
Let Yi and li be the eigenvectors and eigenvalues of L:
The sum of all columns (or rows) of L is identically zero:
L has no inverse!

9. Resistor networks 1 2 3
How can we calculate the Green function?

10. Resistor networks 1 2 3
How can we calculate the Green function?

11. Resistor networks 1 2 3

12. Resistor networks (Fourier harmonics) A complete graph
(an equidistant set of nodes)
(a superposition of standing waves)

13. Optimization problems
In the context of electrical networks, the OP corresponds to allocating conductance to the branches of a circuit so as to achieve the lowest net resistance between nodes.

14. A probabilistic interpretation1 2 3
Let Pαβ be the probability that the walker starting from node α will reach node β before returning to α, which is the probability of first passage.

15. Study of finite connected graphs by the nearest -neighbor random walks

16. Study of finite connected graphs by the nearest -neighbor random walks
The matrix of the access (hitting) time from i to j:
(with no return to i )
(the first step takes us to a neighbor v of i, and then we have to reach j from there)
The commute time from i to j and back
The first-passage time to i from any other node chosen randomly wrt to the stationary probability of rand walks.
All possible (self-avoiding) paths between the nodes contribute into the “diffusion” distance accordingly their weights (the probability to be chosen by a random walker).

17. Study of finite connected graphs by the nearest -neighbor random walks
First-passage time:
Commute time: y1
Commute time
First-passage time

18. Can we see the first-passage times?
(Mean) First passage time
Tax assessment value of land ($)
Manhattan, 2005
Federal Hall
SoHo
East Village
Bowery
East Harlem
, , ,000
(Mean) first-passage times in the city graph of Manhattan

19. Why are mosques located close to railways?
NEUBECKUM:
Social isolation vs. structural isolation

20. Connection to dynamical systems (Ulam): Music as a time series
W.A. Mozart, Eine Kleine Nachtmusik ,G
= ( )
First-passage time ( )T2
Recurrence time ( ) =1/p , 0p = p G 1
“ “ =
C, “do”:
G major is based on the pitches
G, A, B, C, D, E, and F♯.
“Ricci curvature”:
first-passage times
recurrence times
ANTICIPATION NEIGHBORHOOD (60 sec):
Anticipation of the future events is possible in the data geometric setting
within the intelligible data neighborhoods of positive Ricci curvature
with respect to the transportation metric.
The one-step transition matrix for a musical composition
defines the transportation metric
on the simplex of probability measures corresponding to the musical notes.
The first-passage time to the note by the random walk
from a note randomly chosen over the musical score
is the norm of the musical note with respect to the transportation metric.
The recurrence time of the walk to any note is approximately equal to
the size of musical octave.
The notes the first-passage times to which are shorter than
recurrence time comprise
the basic pitches of the tonality scale of the musical composition.
If one mistakes a note while performing a musical piece,
we catch it immediately as the pitch would step out the tonality scale of composition.
Anticipation is possible within the data neighborhood of positive “Ricci curvature”

21. Is it possible to geometrize infinite graphs?
May be, the resolvent of some (self-adjoint) “transfer operators” would be helpful ??

22. Negative result is a result
(Complex electrical impedance)
zab is the measure of the opposition that an element presents to a current when a voltage is applied.
the resistance
the negative reactance:
Capacitance is a measure of the capacity of storing electric charge for a given potential difference.
Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself and in any nearby conductors by mutual inductance.
the positive reactance:

23. An electrical impedance network1 2 3
The effective impedance between nodes p and q :

24. An electrical impedance network1 2 3
Kirchhoff's current law:
The effective impedance between nodes p and q :
resonances
Electrical resonances occur at a particular resonance frequency when the imaginary parts of impedances of circuit elements cancel each other.

25. Electrical resonance
The collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor, and then the discharging capacitor provides an electric current that builds the magnetic field in the inductor. This process is repeated continually.

26. If edges of an infinite graph have complex weights, what might be its resonances?
(Resonance bands?)

27. Morse structure of first-passage manifolds
The first-passage time to a node is calculated as the mean of all first access (hitting) times:
with respect to the stationary distribution of random walks
For any given starting distribution that differs from the stationary one, we can
calculate the analogous quantity,
We call it the first attaining time to the node j by the random walks starting at the distribution defined by ϕ1.

28. Morse structure of first-passage manifolds
ek are the direction cosines
A manifold locally homeomorphic to Euclidean space

29. First attaining times manifold. The Morse eory
Morse structure of first-passage manifolds
At a vicinity of the stationary distribution (ek ≈0),
each node j is a critical point of the manifold of first attaining times, and the first passage times fj are the correspondent critical values:

30. Morse structure of first-passage manifolds
Following the ideas of the Morse theory, we can perform the standard classification of the critical points, introducing the index g j of the critical point j as the number of negative eigenvalues of H at j.
The index of a critical point is the dimension of the largest subspace of the tangent space to the manifold at j on which the Hessian is negative definite).

31. Morse structure of first-passage manifolds
The Euler characteristic c is an intrinsic property of a manifold that describes its topological space’s shape regardless of the way it is bent.
It is known that the Euler characteristic can be calculated as the alternating sum of Cg , the numbers of critical points of index c of the Hessian function,

32. Morse structure of first-passage manifolds
Amsterdam (57 canals)
Venice (96 canals)
The negative Euler characteristics could either come from a pattern of symmetry in the hyperbolic surfaces, or from a manifold homeomorphic multiple tori.
The large positive value of the Euler characteristic can arise due to the well-known product property of Euler characteristics
for any product space M ×N, or, more generally, from a fibration, when one topological space (a fiber) is being ”parameterized” by another topological space (a base).

33. Thank you!