Geometrize everything with Monge-Kantorovich? A topic for discussion D. Volchenkov

Elements -> Distributions The European Union's Seventh Framework Programme, grant no. 318723: Mathematics of Multi-Level Anticipatory Complex Systems (MatheMACS) (2013-2015). Motivation: Extracting generic laws in economics, sociology, neuroscience, by focalizing the description of phenomena to a minimal set of variables and parameters, linked together by evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Elements -> Distributions

The European Union's Seventh Framework Programme, grant no The European Union's Seventh Framework Programme, grant no. 318723: Mathematics of Multi-Level Anticipatory Complex Systems (MatheMACS) (2013-2015). m m* = ?

Big Data m m* = ?

What is a network? How to network? Networks extend across many scales Local, immediate properties Global, long-time, large-scale properties Geometry

From elements to distributions; lifting up the metric

Transportation metrics on networks & databases X Y L Monge – Kantorovich transport problem -- the transportation plan L; -- X , Y probability measures on a compact space; K → the transportation metric inducing the weak topology on the compact space METRIC (60 sec): The Monge – Kantorovich transportation problem searches for the optimal transportation plan L over all Borel measures with marginal measures X and Y on the compact space. The Kantorovich transportation metric induces the weak topology on the simplex of probability measures. However, no explicit formula is known for that in N - dimensions. We have shown that if the compact space has a structure A -- being a graph, a network, or a relational database -- the set of transport geodesics can be defined by a diffusion process through the structure. And the Green function of diffusion process defines the “path-integral” transportation metric on the data manifold, in which all possible paths are taken into account, although some paths are more preferable then others.

Transportation metrics on networks & databases X Y L Monge – Kantorovich transport problem -- the transportation plan L; -- X , Y probability measures on a compact space; K → the transportation metric inducing the weak topology on the compact space -- the (squared) norm of a distribution; -- the Green function (propagator of the self-avoiding walks) -- the stochastic automorphism / a diffusion generator; -- a scalar product of probability measures; METRIC (60 sec): The Monge – Kantorovich transportation problem searches for the optimal transportation plan L over all Borel measures with marginal measures X and Y on the compact space. The Kantorovich transportation metric induces the weak topology on the simplex of probability measures. However, no explicit formula is known for that in N - dimensions. We have shown that if the compact space has a structure A -- being a graph, a network, or a relational database -- the set of transport geodesics can be defined by a diffusion process through the structure. And the Green function of diffusion process defines the “path-integral” transportation metric on the data manifold, in which all possible paths are taken into account, although some paths are more preferable then others. The Green function of diffusion process defines the transportation metric on the data manifold

Y i L X -- the Green function (propagator of the self-avoiding walks) Discretisation/ equiprobable neighborhoods/ geodesics = self-avoiding walks -- the Green function (propagator of the self-avoiding walks) A “construction set” for lifting up m →m* Quantum Mechanics (propagator); Dynamical chaos theory (the Ulam method); uncertainty of steps (“local” scale) uncertainty of ∞ paths (“global” scale)

Probabilistic geometry of finite connected graphs by the nearest -neighbor random walks First-passage time: Commute time: y1 Commute time First-passage time

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 10 100 1,000 5,000 10,000 (Mean) first-passage times in the city graph of Manhattan

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

Transportation (Ricci) curvature

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 , p = p G 1 “ “ = C, “do”: G major is based on the pitches G, A, B, C, D, E, and F♯. “Ricci curvature”: 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

Interaction statistics in organizations The radio-frequency identification sensors reported on occasions of physical proximity

Duration dependent communication graphs 1 2 3 4 5 6 Communication durations (min)

How is the individual communication propensity (a local property) related to global properties?

How is the individual communication propensity (a local property) related to global properties?

Connectedness exceeds connectivity How is the individual communication propensity (a local property) related to global properties? A local property (connectivity) A global property (connectedness) Connectedness exceeds connectivity a “positive Ricci curvature”

A connection with quantum mechanics The Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements For the N-particle case, we have The multi-particle wave function defined by the Slater determinant is no longer distinguishes between the individual particles but returns the probability amplitude whose modulus squared represents a probability density over the entire multi-particle system.

A connection with quantum mechanics The Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements For the N-particle case, we have The multi-particle wave function defined by the Slater determinant is no longer distinguishes between the individual particles but returns the probability amplitude whose modulus squared represents a probability density over the entire multi-particle system. Recurrence times to subgraphs = 1/Pr{ … }

Solving nonlinear networks by Volterra series h(t1, t2,…tl) is the l-th-order Volterra kernel, or l-th-order nonlinear impulse response.

“Poincare recurrence times” |Ik |-Probabilistic graph invariants = the k-steps recurrence probabilities quantifying the chance to return in t steps. IN=| det T | The probability that the RW revisits the initial node in N steps. I1=Tr T The probability that the RW stays at the initial node in 1 step.

Hypergraphs with n-simplexes; simplicial complexes Geostrophic flows Gram matrix can include: a b c

Relation to homology theory The cycle representations disclose the homologic properties of a graph focused on the intrinsic structural interrelations between the edge weights wij and the cycle weights wc , c ∊{Cycles}.

Relation to homology theory The cycle representations disclose the homologic properties of a graph focused on the intrinsic structural interrelations between the edge weights wij and the cycle weights wc , c ∊{Cycles}. Kalpazidou (1995): For an irreducible T, there exists a special orthonormal family G={g1,g2,…} of algebraic cycles connecting the ruling edge-cycle relations with a Fourier representation for T.

Relation to homology theory Theorem (Kalpazidou, 1995) Let T be an irreducible stochastic matrix whose invariant probability distribution is p=(p1, p2,…pN). If the graph G(T) contains a collection G= {g1,g2,…gB} of Betti circuits, then pT has a Fourier representation with respect to G , where the Fourier coefficients are identical with the probabilisitc –homologic cycle weights wg1,…wgB: In terms of (i,j)-coordinates, Fourier & wavelet transforms on graphs; homology of big networks, hypergraphs with n-simplexes The corresponding adjacency matrix is the N × N circulant matrix A

The Schläfli formula for deforming polyhedra and network stability Structure itself can be characterized by some “elastiticity”/ the ability to resist a distorting influence or stress. A finite connected undirected weighted connected graph is a N- simplex, in which: Squared norms of vectors are first-passage times; Squared distances between vectors are commute times;

The Schläfli formula for deforming polyhedra and network stability Fi, the face of the co-dimension 1 opposing i Fj, the face of the co-dimension 1 opposing j Fij, the face of the co-dimension 2 opposing (i,j) The angle Gram matrix of the simplex

The Schläfli formula for deforming polyhedra and network stability Hyperbolic simplex Spherical simplex Euclidean simplex

The Schläfli formula for deforming polyhedra and network stability for polyhedra: (continuity of volume)

The Schläfli formula for deforming polyhedra and network stability for polyhedra: (continuity of volume) The integral of motion:

The Schläfli formula for deforming polyhedra and network stability Another way of writing the Schläfli formula: “Stress tensor” The eigenvalues are the “principal stresses”; The eigenvectors are the principal directions; The principal invariants are the deviatoric stress invariants;

Morse structure of first-passage manifolds The first-passage time can be calculated as the mean of all first 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 ϕ1.

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

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:

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).

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,

Optimization problems In the context of electrical networks, the OP corresponds to allocating conductance to the branches of a circuit so as to achieve low resistance between the nodes. In a Markov chain context, the OP is the problem of selecting the weights on the edges to minimize the average commute (or hitting) time between nodes.