Random remarks about random walks D. Volchenkov 4 Sketches for discussion @ MPI for Mathematics in the Sciences Leipzig, 11.03.2014

1. Identifying important scales by localization of random walks

Disorder of steps vs. disorder of paths Disorder of (infinite) paths:

Disorder of steps vs. disorder of paths Scale of RW, as # transitions increases Disorder of steps: As centrality falls next to boundaries & obstacles Disorder of (infinite) paths:

Disorder of steps vs. disorder of paths

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Random walks of different scales Maximal complexity RW Maximal entropy RW Maximal paths disorder Maximal steps disorder Minimal steps disorder Minimal paths disorder

Structure reinforces order A.) B.) C.) Maximal complexity RWs: Localization within small scale structures Disorder of steps: The complexity-entropy diagram shows how information is stored, organized, and transformed across different scales of the structure.

Structure reinforces order A.) B.) C.) Maximal complexity RWs: Localization within small scale structures Disorder of steps: The complexity-entropy diagram shows how information is stored, organized, and transformed across different scales of the structure.

2. Behaviour of anticipatory systems and random walks

acetylcholine (triggered by known unreliability of predictive cues) Expected uncertainty acetylcholine (triggered by known unreliability of predictive cues) Unexpected uncertainty norepinephrine (triggered by strongly unexpected observations)

acetylcholine (triggered by known unreliability of predictive cues) Lévy flights: Extremely long recurrence times; Short first passage times; Expected uncertainty acetylcholine (triggered by known unreliability of predictive cues) Unexpected uncertainty norepinephrine (triggered by strongly unexpected observations) Brownian walks: Short recurrence times; Long first passage times;

acetylcholine (triggered by known unreliability of predictive cues) ballistic relocations Lévy flights: Extremely long recurrence times; Short first passage times; Expected uncertainty acetylcholine (triggered by known unreliability of predictive cues) Unexpected uncertainty norepinephrine (triggered by strongly unexpected observations) Brownian walks: Short recurrence times; Long first passage times; a meticulous search

Visual intelligibility of environments ballistic relocations Lévy flights: Extremely long recurrence times; Short first passage times; Visual intelligibility of environments Expected uncertainty acetylcholine (triggered by known unreliability of predictive cues) Unexpected uncertainty norepinephrine (triggered by strongly unexpected observations) Confidence in the eventual success Brownian walks: Short recurrence times; Long first passage times; a meticulous search

Structural intelligibility Recurrence time First-passage time Tonality: the hierarchy of harmonic intervals The basic pitches for the E minor scale are "E", "F#", "G", "A", "B". The recurrence time vs. the first passage time over 804 compositions of 29 Western composers. Tonality of music

Structural intelligibility Recurrence time First-passage time Tonality: the hierarchy of harmonic intervals The basic pitches for the E minor scale are "E", "F#", "G", "A", "B". The recurrence time vs. the first passage time over 804 compositions of 29 Western composers. Tonality of music

3. The Schläfli formula for deforming polyhedra and network stability

Path integral distance for graphs & databases Path integral is an analytic continuation of RW summation. Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator; Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant); Removal of ambiguities The Laplace operator diverges, the Green function is not unique: The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator: From path integral to the Riemannian geometry Given two distributions x,y, their scalar product: The (squared) norm of a distribution: The Euclidean distance between two distributions : Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization Transition to self-avoiding random walks (“no loops”).

Path integral distance for graphs & databases Path integral is an analytic continuation of RW summation. Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator; Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant); Removal of ambiguities The Laplace operator diverges, the Green function is not unique: The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator: From path integral to the Riemannian geometry Given two distributions x,y, their scalar product: The (squared) norm of a distribution: The Euclidean distance between two distributions : Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization Transition to self-avoiding random walks (“no loops”).

Path integral distance for graphs & databases Path integral is an analytic continuation of RW summation. Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator; Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant); Removal of ambiguities The Laplace operator diverges, the Green function is not unique: The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator: From path integral to the Riemannian geometry Given two distributions x,y, their scalar product: The (squared) norm of a distribution: The Euclidean distance between two distributions : Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization Transition to self-avoiding random walks (“no loops”).

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;

Fi, the face of the co-dimension 1 opposing i Fij, the face of the co-dimension 2 opposing (i,j) Fj, the face of the co-dimension 1 opposing j

The angle Gram matrix of the simplex

The angle Gram matrix of the simplex Hyperbolic simplex Spherical simplex Euclidean simplex

The angle Gram matrix of the simplex Hyperbolic simplex Spherical simplex Euclidean simplex

The angle Gram matrix of the simplex The Schläfli formula for polyhedra: (continuity of volume)

The angle Gram matrix of the simplex The Schläfli formula for polyhedra: (continuity of volume) The integral of motion:

Another way of formalizing that 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;

4. First attaining times manifolds. The Morse theory

3. First attaining times manifold 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.

First attaining times manifold ek are the direction cosines A manifold locally homeomorphic to Euclidean space

First attaining times manifold. The Morse eory 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.

First attaining times manifold 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 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).

First attaining times manifold. The Morse theory 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,

First attaining times manifold. The Morse theory 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 (called a fiber) is being ”parameterized” by another topological space (called a base).