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Is it possible to geometrize infinite graphs? (Diffusion metrics and geometrization of finite graphs and relational databases) D. Volchenkov Motivation 1. Data interpretation X m* = ? m Y Motivation 1. Data interpretation X m* = ? Y Monge – Kantorovich transport problem K X , Y = inf cost x, y dL L -- the transportation plan L; m -- X , Y probability measures on a compact (metrizable) space; K → the transportation metric Motivation 1. Data interpretation X m* = ? Y Monge – Kantorovich transport problem K X , Y = inf cost x, y dL L -- the transportation plan L; m 2. Data “coordinates” -- X , Y probability measures on a compact (metrizable) space; K → the transportation metric “0” Resistor networks I i Vj Vi rij j 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: I= V j Vi r ij The current through a conductor between two points is directly proportional to the voltage across the two points. Resistor networks i = 1,, N I1 1 rij = rji V1 r12 V 2 r13 3 I3 2 I2 V3 r 23 The (effective) resistance between nodes and : Kirchhoff's current law: N I i =1 i The algebraic sum of currents in a network of conductors meeting at a point is zero. =0 cij = rij1, conductanc e Ohm's law: c V V = I N ij j =1( j i ) R = i j V V I i =? I i = I i i The current through a conductor between two points is directly proportional to the voltage across the two points. ? rij Rij Resistor networks i = 1,, N I1 1 I3 ij j =1( j i ) V1 r12 V 2 2 I2 V3 c V V = I N rij = rji r13 3 Ohm's law: r 23 i i In the matrix form: I = LV c12 c1 c2 c L = 21 cN 1 cN 2 (L – the Kirchhoff matrix) c1N c2 N cN (f all rij =1 W, L is the graph Laplacian operator) R j 1 the Green function " = " L cij = c ji = 1 rij ci = N c j =1 j i ij Resistor networks i = 1,, N I1 1 rij = rji V1 r12 V 2 r13 3 I3 2 I2 V3 r 23 N I i =1 i =0 c12 c1 c2 c L = 21 cN 1 cN 2 c1N c2 N cN cij = c ji = 1 rij ci = N c ij j =1 j i Let Yi and li be the eigenvectors and eigenvalues of L: LYi = li Yi L = L : Yi* , Yj = is* js = ij s The sum of all columns (or rows) of L is identically zero: l1 = 0, 1 = 1 N , = 1,2,, N . L has no inverse! Resistor networks i = 1,, N I1 1 rij = rji I3 I i =1 V1 i =0 c12 c1 c2 c L = 21 cN 1 cN 2 c1N c2 N cN ci = N c ij j =1 j i 2 I2 N L( ) = L 1, 0, G( ) = L ( ), Vi = Gij I j 1 V3 cij = c ji = 1 rij How can we calculate the Green function? r12 V 2 r13 3 N j =1 r 23 U : U LU = Λ, U L( )U = Λ( ) LYi = li Yi U ij = ji , ii ( ) = (li ) ij UG( )U = Λ 1 ( ), G( ) = UΛ1 ( )U N i i* 1 * 1 U i = G ( ) = Ui g ( ), g ( ) = l N i =1 i =2 li i N Resistor networks i = 1,, N rij = rji I1 1 I3 I i =1 V1 i =0 c12 c1 c2 c L = 21 cN 1 cN 2 c1N c2 N cN ci = N c ij j =1 j i 2 I2 N L( ) = L 1, 0, G( ) = L ( ), Vi = Gij I j 1 V3 cij = c ji = 1 rij How can we calculate the Green function? r12 V 2 r13 3 N j =1 r 23 U : U LU = Λ, U L( )U = Λ( ) LYi = li Yi U ij = ji , ii ( ) = (li ) ij N Vi ( ) = Gij ( ) I j = j =1 1 = N N N I g j =1 j j =1 0 ij ( ) I j UG( )U = Λ 1 ( ), G( ) = UΛ1 ( )U N i i* 1 * 1 U i = G ( ) = Ui g ( ), g ( ) = l N i =1 i =2 li i N Resistor networks N I i = 1,, N I1 1 rij = rji r12 V 2 I3 ci = N c ij j =1 j i 0 R = r 23 V V I Ii = I = i i g (0) g (0) g (0) g (0) N * LYi = li Yi , g ( ) = i i i = 2 li N cij = c ji = 1 rij N 1 N Vi = I j lim gij ( ) I j 0 N j =1 j =1 2 I2 V3 =0 c1N c2 N cN V1 r13 3 i =1 i c12 c1 c2 c L = 21 cN 1 cN 2 R = 1 i = 2 li R i 2 i 2 = ,, N lN l2 = i i li i = 2 li N 2 2 ,, N , = lN l2 Resistor networks N 1 1 1 L= r 1 1 1 N 1 1 1 N 1 l1 = 0, l2 = lN = N r n A complete graph R 1 2i = e N r = N N n 1 N , n, = 1, N (Fourier harmonics) n n = n =1 r = N 2 2r N (an equidistant set of nodes) n 1 2i (a superposition of N , , e standing waves) 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. A probabilistic interpretation i = 1,, N rij = rji 1 Pr i j = r12 r13 2 r 23 3 P = 1 1 c R cij = c ji = 1 rij cij ci ci = cij cj c12 c1 0 0 c21 c1 T= cN 1 cN cN 2 cN N c j =1 j i ij = Pr j i c1N c1 c2 N c2 0 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. Study of finite connected graphs by the nearest neighbor random walks Graph A T = D 1 A, D = diag deg(1), deg( N ) ˆ = D1 2 AD 1 2 , T ˆ = m , N , T l l l l 1 s ,i s , j Gij = s = 2 1 m s 1,i 1, j N i 2 T 2 ,i 1,i 1 m 2 = N ,i 1,i 1 m N 2 1 k ,i = i , G i = 2 k =2 1 mk 1,i Kij = i j N 2 T k, j k ,i = 1, j 1 mk k = 2 1,i 1 mk N 2 1 = m1 m N , 12,i = i = 2, j 1, j 1 m 2 , N, j 1, j 1 m N degi 2E = i, j T PR N 1 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 ) 1 H = 1 H vj ij i j deg( i ) v i H ii = 0 (the first step takes us to a neighbor v of i, and then we have to reach j from there) N 1 H ji H ij = k = 2 1 mk kj2 ki kj 2 1 j 1i 1 j k, j k ,i Kij = H ij H ji = 1, j 1 mk k =2 1,i 1 mk N 2 1 k ,i = i H ij = 2 i =1 k =2 1 mk 1,i N Pi N 2 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). Study of finite connected graphs by the nearest neighbor random walks Graph A T = D 1 A, D = diag deg(1), deg( N ) ˆ = D1 2 AD 1 2 , T ˆ = m , N , T l l l l 1 s ,i s , j Gij = s = 2 1 m s 1,i 1, j N 1 = m1 m N , 12,i = i = 2, j 1, j 1 m 2 , N, j 1, j 1 m N 2 ,i 1,i 1 m 2 = N ,i 1,i 1 m N First-passage time: i 2 T 1 k =2 1 mk N = i , G i = 2 k ,i 2 1,i N = i H ij 1 Commute time: Kij = i j 2 T i =1 k, j k ,i = 1, j 1 mk k = 2 1,i 1 mk N 2 2 ,i 2, j , 3, j degi 2E = i, j T PR N 1 j , 3,i i Tax assessment value of land ($) Can we see the first-passage times? Manhattan, 2005 (Mean) First passage time (Mean) first-passage times in the city graph of Manhattan SoHo Federal Hall 10 East Village 100 1,000 Bowery East Harlem 5,000 10,000 Why are mosques located close to railways? NEUBECKUM: Social isolation vs. structural isolation Connection to dynamical systems (Ulam): Music as a time series G major is based on the pitches G, A, B, C, D, E, and F♯. W.A. Mozart, Eine Kleine Nachtmusik T G= Tn = 1-T =" L1" , 1 n C, “do”: first-passage times “ recurrence times 1 0 “ = First-passage time ( Recurrence time ( ) T2 = ( ) =1/ , ,G ) 0 = G “Ricci curvature”: First - passage time = 1 Recurrence time Anticipation is possible within the data neighborhood of positive “Ricci curvature” Is it possible to geometrize infinite graphs? May be, the resolvent of some (self-adjoint) “transfer operators” would be helpful ?? Negative result is a result (Complex electrical impedance) I V V z = z = r ix the resistance r = r 0 z is the measure of the opposition that an element presents to a current when a voltage is applied. the negative reactance: the positive reactance: x = x 0, capacitanc es x = x 0, inductance s 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. An electrical impedance network , = 1,, N 1 z12 z13 2 z 23 3 The effective impedance between nodes p and q : Z pq = V p Vq I =? I = I p q An electrical impedance network N Kirchhoff's current law: , = 1,, N 1 z12 I = LV 2 z 23 3 y = =1 y12 y1 y2 y L = 21 yN 1 yN 2 y = y = 1 z z13 I = 0 N y =1 y1N y2 N y N UT LU = , = diag 0, l2 ,lN The effective impedance between nodes p and q : Z pq = V p Vq I =? I = I p q N Z pq = =2 1 l u p uq , if l 0, 2 2 = , if : l = 0, 2. resonances Electrical resonances occur at a particular resonance frequency when the imaginary parts of impedances of circuit elements cancel each other. Electrical resonance I = LV R = 0 1 1 , L = y12 1 1 y12 = iC 1 iL = iC 1 L L* L 1 = 0, 2 = 4 y12 l1 = 0, l2 = 2 y12 =1 Z12 = 1 y12 =1 2 0 LC LC 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. If edges of an infinite graph have complex weights, what might be its resonances? (Resonance bands?) Morse structure of first-passage manifolds Tˆij = Aij A is s A ˆ = m , O N , l = 1 m , 1 = m m , lT l l l l l 1 N js s The first-passage time to a node is calculated as the mean of all first access (hitting) times: 2 f j = 1,i H ij i 1 with respect to the stationary distribution of random walks i = 12,i 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. Morse structure of first-passage manifolds k are the direction cosines f j = 12,i H ij = q j , q j PR N 1 i 1 A manifold locally homeomorphic to Euclidean space First Morse attaining structure times ofmanifold. first-passage The manifolds Morse eory At a vicinity of the stationary distribution (k ≈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, Morse structure of first-passage manifolds Amsterdam (57 canals) The negative Euler characteristics could either come from a pattern of symmetry in the hyperbolic surfaces, or from a manifold homeomorphic multiple tori. Venice (96 canals) The large positive value of the Euler characteristic can arise due to the wellknown 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). Thank you!