1. Data Analysis of Multi-level systems
D. Volchenkov

2. Goals & Objectives
To identify meaningful levels in multi-level systems;
To understand the geometric structure associated to the multimodal data (i.e. how these levels are related to each other);
To analyze the data geometrical manifold and its sensitivity to assorted data variations (evolution and control by Ricci-Hamilton flows).
Processing data from other work packages.

3. We proceed in two steps:
Step 1: “Probabilistic graph theory”
Nodes, subgraphs (sets of nodes), graphs are described by probability distributions & characteristic times w.r.t. inhomogeneous Markov chains based on stochastic/fractional automorphisms (i.e., different definitions of a neighborhood);
Step 2: “Geometrization of Data Manifolds”
Establish geometric relations between those probability distributions whenever possible;
1. Coarse-graining/reduction/PCA for networks/databases → data analysis;
2. Transport optimization(Monge-Kontorovich type problems) → distances between distributions;
3. Network evolution & control via Ricci flows → sensitivity to assorted data
variations;

4. Our heroes: Stochastic / fractional automorphisms
Finite connected graphs;
Finite distance matrices;
Finite relational databases:
the graph of attributes;
the graph of tuples;
GA (adjacency matrix of the graph) 
P: [P,A]=0, P  Aut(G)
Symmetry:

5. Our heroes: Stochastic / fractional automorphisms
Finite connected graphs;
Finite distance matrices;
Finite relational databases:
the graph of attributes;
the graph of tuples;
GA (adjacency matrix of the graph) 
P: [P,A]=0, P  Aut(G)
Symmetry:
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

6. A variety of stochastic automorphisms at different scales
Stochastic matrices: … …

7. A variety of stochastic automorphisms at different scales
Neighborhoods & Scales:
Stochastic matrices:
Equiprobable paths … … … …

8. A variety of stochastic automorphisms at different scales
Neighborhoods & Scales:
Left eigenvectors (m=1) = Centrality measures:
Stochastic matrices:
Equiprobable paths
The “stationary distribution” of the nearest neighbor RW … … … … … …

9. Relation to random walks
Time is introduced as powers of transition matrices

10. Relation to random walks
Time is introduced as powers of transition matrices

11. Relation to random walks
Time is introduced as powers of transition matrices

12. Relation to random walks
Time is introduced as powers of transition matrices

13. Relation to random walks
Time is introduced as powers of transition matrices

14. Relation to random walks
Time is introduced as powers of transition matrices

15. Relation to random walks
Time is introduced as powers of transition matrices

16. Relation to random walks
Time is introduced as powers of transition matrices

17. Relation to random walks
Time is introduced as powers of transition matrices

18. Relation to random walks
Time is introduced as powers of transition matrices

19. Relation to random walks
Time is introduced as powers of transition matrices

20. Relation to random walks
Time is introduced as powers of transition matrices

21. Relation to random walks
Time is introduced as powers of transition matrices

22. Relation to random walks
Time is introduced as powers of transition matrices

23. Relation to random walks
Time is introduced as powers of transition matrices

24. Relation to random walks
Time is introduced as powers of transition matrices

25. Relation to random walks
Time is introduced as powers of transition matrices

26. Relation to random walks
Time is introduced as powers of transition matrices

27. Relation to random walks
Time is introduced as powers of transition matrices

28. Relation to random walks
Time is introduced as powers of transition matrices
Still far from stationary distribution!
Stationary distribution is reached!
Defect irrelevant.
Low centrality (defect) repelling.

29. Research questions on stochastic automorphisms:
1. Generalized transition operator =? S i j
2. Properties of the Fourier transform of the transition operators:
3. Intuition about interscale Ricci flows on the data manifold:
(approached faster)

30. Step 1: “Probabilistic graph theory”
As soon as we define a neighborhood…
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)

31. 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:

32. 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 an angle
The Euclidean distance:

33. 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) ≠

34. 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

35. 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,
L◊ as the PCA kernel
Low dimensional approximations;
Graphic representations;
Coarse-graining& reduction of networks;
Data analysis;
Data interpretation
The (squared) norm of a vector
and an angle
The Euclidean distance:

36. Generalized Ricci curvature
How fast the volume grows w.r.t. to geodesic one…
Ricci curvature (Ollivier):
On a metric space, for any two distinct points (x and y), the Ricci –Ollivier curvature along (x,y) is

37. Generalized Ricci curvature
How fast the volume grows w.r.t. to geodesic one…
Ricci curvature (Ollivier):
On a metric space, for any two distinct points (x and y), the Ricci –Ollivier curvature along (x,y) is
Similar concepts (“Ricci like curvatures”) w.r.t. stochastic automorphisms:
Along two points:
Trees are flat!
At a point:
Trees are flat!

38. An example of “curvature”: Tonality scales of Western music
V.A. Mozart,
Eine-Kleine-Nachtmusik
R. Wagner, Das Rheingold (Entrance of Gods)
Increase of harmonic interval/ first –passage time

39. Geometrization of undirected graphs
Each node is characterized by a vector in RN-1
TxM RN-1 x ui uj p
Riemannian metric:
A node/entry dependent differential operator on the probabilistic manifold:
For the group of translations, the shift operator is
… and then define the Hessian matrix at each node
It happens often that
Q(x) forms a basis of orthogonal sections at x, g is a metric tensor.
→ the standard calculus of differential geometry…

40. Evolution of networks: Ricci-Hamilton flows
We consider the metric tensor to be a function 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 (“Mathew principle”)
“Densification” of the network of “positive curvature”
“Contraction” of a “probabilistic manifold”
A “collapse” and decomposition of the network of “negative curvature”

41. Global connectedness: 1.How do we define a neighborhood?
Concluding remarks
Dealing with multi-level systems requires answering two basic questions:
Global connectedness:
2. How do we connect neighborhoods?
Local connectedness:
1.How do we define a neighborhood?

42. Global connectedness: Intra-scale Ricci flow →
Concluding remarks
Dealing with multi-level systems requires answering two basic questions:
Global connectedness:
2. How do we connect neighborhoods?
Intra-scale Ricci flow → t
• Intra-scale Ricci curvature
• Wasserstein type distances  
Local connectedness:
← Inter-scale Ricci flow
V grows differently over different subgraphs , so that
1.How do we define a neighborhood?
Scales ℓ
Vℓ grows differently at different ℓ, so
Inter-scale Ricci curvature

43. t scales Concluding remarks   Real dynamical flows
Dealing with multi-level systems requires answering two basic questions:
Global connectedness:
2. How do we connect neighborhoods?
Intra-scale Ricci flow → t
• Intra-scale Ricci curvature
• Wasserstein type distances  
Local connectedness:
← Inter-scale Ricci flow
1.How do we define a neighborhood?
scales
Real dynamical flows
Inter-scale Ricci curvature

44. Research questions on the Ricci flows :
Given the inter-scale and intra-scale Ricci flows, is there a “scaling relations”,
2. Is it possible to define “universality classes” of different graphs/networks w.r.t a?
3. Scaffolding of real dynamical flows close to fixed points by flows of inhomogeneous random walks.