Analysis meets Graphs Matthew Begue´ Norbert Wiener Center Department of Mathematics University of Maryland, College Park Why study graphs? Graph theory has developed into a useful tool in applied mathematics. Vertices correspond to different sensors, observations, or data points. Edges represent connections, similarities, or correlations among those points. Wikipedia graph Graphs in pure mathematics too! Why study Analysis? Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Why study Analysis? Real Analysis Complex Analysis Partial Differential Equations Ordinary Differential Equations HARMONIC ANALYSIS Probability Differential Geometry Functional Analysis Compressive Sensing Stochastic Processes Measure Theory Brownian Motion Operator Theory Spectral Theory Mathematical Modeling Numerical Analysis Analysis toolbox Translation Convolution Modulation Fourier Transform Analysis toolbox Translation Convolution Modulation Fourier Transform Analysis toolbox Translation Convolution Modulation Fourier Transform Analysis toolbox Translation Convolution Modulation Fourier Transform Outline 1 Fourier Transform 2 Modulation 3 Convolution 4 Translation Motivation In the classical setting, the Fourier transform on R is given by Z ^f (ξ) = f (t)e−2πiξt dt = hf ; e2πiξt i: R This is precisely the inner product of f with the eigenfunctions of the Laplace operator. Motivation In the classical setting, the Fourier transform on R is given by Z ^f (ξ) = f (t)e−2πiξt dt = hf ; e2πiξt i: R This is precisely the inner product of f with the eigenfunctions of the Laplace operator. Graph Laplacian Definition The pointwise formulation for the Laplacian acting on a function f : V ! is R X ∆f (x) = f (x) − f (y): y∼x For a finite graph, the Laplacian can be represented as a matrix. Let D denote the N × N degree matrix, D = diag(dx ). Let A denote the N × N adjacency matrix, 1; if x ∼ x A(i; j) = i j 0; otherwise. Then the unweighted graph Laplacian can be written as L = D − A: Spectrum of the Laplacian L is a real symmetric matrix and therefore has nonnegative N−1 eigenvalues fλk gk=0 with associated orthonormal eigenvectors N−1 f'k gk=0 . If G is finite and connected, then we have 0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λN−1: p Easy to show that '0 ≡ 1= N. Data Sets - Minnesota Road Network 50 50 50 49 49 49 48 48 48 47 47 47 46 46 46 45 45 45 44 44 44 43 43 43 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 (a) λ3 (b) λ4 (c) λ5 50 50 50 49 49 49 48 48 48 47 47 47 46 46 46 45 45 45 44 44 44 43 43 43 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 −98 −97 −96 −95 −94 −93 −92 −91 −90 −89 (d) λ6 (e) λ7 (f) λ8 Figure: Eigenfunctions corresponding to the first six nonzero eigenvalues. Minnesota road graph (2642 vertices) Data Sets - Sierpinski gasket graph approximation 1.8 1.8 1.8 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (a) λ3 (b) λ4 (c) λ5 1.8 1.8 1.8 1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (d) λ6 (e) λ7 (f) λ8 Figure: Eigenfunctions corresponding to the first six nonzero eigenvalues. Level-8 graph approximation to Sierpinski gasket (9843 vertices) Data Sets - Sierpinski gasket graph approximation 0.02 0.01 0.015 0.015 0.005 0.01 0.01 0.005 0 0.005 0 0 −0.005 −0.005 −0.005 −0.01 −0.01 −0.01 −0.015 −0.015 −0.015 −0.02 −0.02 −0.02 2 2 2 1.5 1 1.5 1 1.5 1 0.5 0.5 0.5 1 1 1 0 0 0 0.5 0.5 0.5 −0.5 −0.5 −0.5 0 −1 0 −1 0 −1 (a) λ3 (b) λ4 (c) λ5 0.02 0.02 0.02 0.015 0.015 0.015 0.01 0.01 0.01 0.005 0.005 0.005 0 0 0 −0.005 −0.005 −0.005 −0.01 −0.01 −0.01 −0.015 −0.015 −0.015 −0.02 −0.02 −0.02 2 2 2 1.5 1 1.5 1 1.5 1 0.5 0.5 0.5 1 1 1 0 0 0 0.5 0.5 0.5 −0.5 −0.5 −0.5 0 −1 0 −1 0 −1 (d) λ6 (e) λ7 (f) λ8 Figure: Eigenfunctions corresponding to the first six nonzero eigenvalues. Level-8 graph approximation to Sierpinski gasket (9843 vertices) Graph Fourier Transform Definition The graph Fourier transform is defined as N ^ X ∗ f (λl ) = hf ;'l i = f (n)'l (n): n=1 Notice that the graph Fourier transform is only defined on values of σ(L). The inverse Fourier transform is then given by N−1 X ^ f (n) = f (λl )'l (n): l=0 Outline 1 Fourier Transform 2 Modulation 3 Convolution 4 Translation Graph Modulation In Euclidean setting, modulation is multiplication of a Laplacian eigenfunction. Definition For any k = 0; 1; :::; N − 1 the graph modulation operator Mk , is defined as p (Mk f )(n) = Nf (n)'k (n): On R, modulation in the time domain = translation in the frequency domain ^ Mdξf (!) = f (! − ξ): Example - Classical Setting Example - Movie G =SG6 ^ f (λl ) = δ2(l) =) f = '2.