Distributed Spectral Decomposition in Networks by Complex Diffusion and Quantum Random Walk INFOCOM, 12 April 2016 Jithin K. Sreedharan ∗ joint work with Konstantin Avrachenkov ∗ and Philippe Jacquet † ∗ INRIA , France † Bell Labs , France
Introduction
Question we address here A scalable way to find largest Jithin K. Sreedharan (INRIA, France) . 1 vectors and the eigen- 1 eigenvalues Problem . 1 Corresponding eigenvectors: 2 1 Eigenvalues: matrix (undirected graph) 3 / 30 ▶ Symmetric graph matrices like adjacency matrix, Laplacian
Question we address here matrix (undirected graph) Problem A scalable way to find largest eigenvalues 1 and the eigen- vectors 1 . Jithin K. Sreedharan (INRIA, France) 3 / 30 ▶ Symmetric graph matrices like adjacency matrix, Laplacian ▶ Eigenvalues: λ 1 ≥ λ 2 ≥ . . . ≥ λ n Corresponding eigenvectors: u 1 , . . . , u n .
Question we address here matrix (undirected graph) Problem Jithin K. Sreedharan (INRIA, France) 3 / 30 ▶ Symmetric graph matrices like adjacency matrix, Laplacian ▶ Eigenvalues: λ 1 ≥ λ 2 ≥ . . . ≥ λ n Corresponding eigenvectors: u 1 , . . . , u n . A scalable way to find largest k eigenvalues λ 1 , . . . , λ k and the eigen- vectors u 1 , . . . , u k .
Total number of triangles in a graph: 1 Why is it relevant? Some uses of graph spectrum and graph 1 Jithin K. Sreedharan (INRIA, France) eigenvector-eigenvector scatter plot of adjacency matrix. Finding near-cliques: phenomenon of Eigenspokes in space. ties: Each node is mapped into a point in Dimensionality reduction, link prediction and Weak and strong 3 2 eigenvectors 1 participated in: Number of triangles that a node 3 . 1 6 4 / 30 ▶ Number of triangles:
Why is it relevant? Some uses of graph spectrum and graph 1 Jithin K. Sreedharan (INRIA, France) eigenvector-eigenvector scatter plot of adjacency matrix. Finding near-cliques: phenomenon of Eigenspokes in space. ties: Each node is mapped into a point in Dimensionality reduction, link prediction and Weak and strong 3 2 eigenvectors 1 participated in: Number of triangles that a node 6 4 / 30 ▶ Number of triangles: ∑ n i = 1 | λ i | 3 . ▶ Total number of triangles in a graph: 1
Why is it relevant? Some uses of graph spectrum and graph eigenvectors Jithin K. Sreedharan (INRIA, France) eigenvector-eigenvector scatter plot of adjacency matrix. Finding near-cliques: phenomenon of Eigenspokes in space. ties: Each node is mapped into a point in Dimensionality reduction, link prediction and Weak and strong 4 / 30 2 1 6 ▶ Number of triangles: ∑ n i = 1 | λ i | 3 . ▶ Total number of triangles in a graph: 1 ▶ Number of triangles that a node m participated in: ∑ i = 1 | λ 3 i | u i ( m )
Why is it relevant? Some uses of graph spectrum and graph 1 Jithin K. Sreedharan (INRIA, France) eigenvector-eigenvector scatter plot of adjacency matrix. Finding near-cliques: phenomenon of Eigenspokes in eigenvectors 2 6 4 / 30 ▶ Number of triangles: ∑ n i = 1 | λ i | 3 . ▶ Total number of triangles in a graph: 1 ▶ Number of triangles that a node m participated in: ∑ i = 1 | λ 3 i | u i ( m ) ▶ Dimensionality reduction, link prediction and Weak and strong ties: Each node is mapped into a point in R k space.
Why is it relevant? Some uses of graph spectrum and graph 1 Jithin K. Sreedharan (INRIA, France) eigenvector-eigenvector scatter plot of adjacency matrix. eigenvectors 2 4 / 30 6 ▶ Number of triangles: ∑ n i = 1 | λ i | 3 . ▶ Total number of triangles in a graph: 1 ▶ Number of triangles that a node m participated in: ∑ i = 1 | λ 3 i | u i ( m ) ▶ Dimensionality reduction, link prediction and Weak and strong ties: Each node is mapped into a point in R k space. ▶ Finding near-cliques: phenomenon of Eigenspokes in
Challenges in classical techniques 1 Jithin K. Sreedharan (INRIA, France) Drawback: Inverse calculation, orthonormalization Eigenvector: 1 : Closest eigenvalue to 1 1 Inverse iteration method: Drawback: Only principal components, orthonormalization 1 1 1 1 5 / 30 ▶ Power iteration: b ℓ + 1 = = ⇒ ∥ b ℓ ∥ Ab ℓ
Challenges in classical techniques Drawback: Only principal components, orthonormalization Jithin K. Sreedharan (INRIA, France) Drawback: Inverse calculation, orthonormalization Eigenvector: 1 : Closest eigenvalue to 1 1 1 Inverse iteration method: 5 / 30 1 ▶ Power iteration: b k + 1 b ⊺ k λ 1 = lim ∥ b k ∥ k →∞ b ℓ + 1 = = ⇒ ∥ b ℓ ∥ Ab ℓ b k u 1 = lim ∥ b k ∥ k →∞
Challenges in classical techniques Drawback: Only principal components, orthonormalization Jithin K. Sreedharan (INRIA, France) Drawback: Inverse calculation, orthonormalization Eigenvector: 1 : Closest eigenvalue to 1 5 / 30 1 ▶ Power iteration: b k + 1 b ⊺ k λ 1 = lim ∥ b k ∥ k →∞ b ℓ + 1 = = ⇒ ∥ b ℓ ∥ Ab ℓ b k u 1 = lim ∥ b k ∥ k →∞ ▶ Inverse iteration method: ∥ b ℓ ∥ ( A − µ I ) − 1 b ℓ = ⇒ b ℓ + 1 =
Challenges in classical techniques Drawback: Only principal components, orthonormalization Jithin K. Sreedharan (INRIA, France) Drawback: Inverse calculation, orthonormalization 1 5 / 30 1 ▶ Power iteration: b k + 1 b ⊺ k λ 1 = lim ∥ b k ∥ k →∞ b ℓ + 1 = = ⇒ ∥ b ℓ ∥ Ab ℓ b k u 1 = lim ∥ b k ∥ k →∞ ▶ Inverse iteration method: ∥ b k ∥ Closest eigenvalue to µ : lim k →∞ µ + b k + 1 b ⊺ k ∥ b ℓ ∥ ( A − µ I ) − 1 b ℓ = ⇒ b ℓ + 1 = b k Eigenvector: lim ∥ b k ∥ k →∞
Connection with Quantum random walks Our contribution Distributed way to find the spectrum At each node, eigenvalues correspond to the frequencies of spectral peaks and respective eigenvector components are the amplitudes at those points. Idea of Complex Power Iterations Diffusion algorithms, Monte Carlo techniques and Random Walk implementation Simulation on real-world networks of varying sizes Jithin K. Sreedharan (INRIA, France) 6 / 30
Our contribution spectral peaks and respective eigenvector components are the amplitudes at those points. Idea of Complex Power Iterations Diffusion algorithms, Monte Carlo techniques and Random Walk implementation Connection with Quantum random walks Simulation on real-world networks of varying sizes Jithin K. Sreedharan (INRIA, France) 6 / 30 ▶ Distributed way to find the spectrum ▶ At each node, eigenvalues correspond to the frequencies of
Our contribution spectral peaks and respective eigenvector components are the amplitudes at those points. Diffusion algorithms, Monte Carlo techniques and Random Walk implementation Connection with Quantum random walks Simulation on real-world networks of varying sizes Jithin K. Sreedharan (INRIA, France) 6 / 30 ▶ Distributed way to find the spectrum ▶ At each node, eigenvalues correspond to the frequencies of ▶ Idea of Complex Power Iterations
Our contribution spectral peaks and respective eigenvector components are the amplitudes at those points. implementation Connection with Quantum random walks Simulation on real-world networks of varying sizes Jithin K. Sreedharan (INRIA, France) 6 / 30 ▶ Distributed way to find the spectrum ▶ At each node, eigenvalues correspond to the frequencies of ▶ Idea of Complex Power Iterations ▶ Diffusion algorithms, Monte Carlo techniques and Random Walk
Our contribution spectral peaks and respective eigenvector components are the amplitudes at those points. implementation Simulation on real-world networks of varying sizes Jithin K. Sreedharan (INRIA, France) 6 / 30 ▶ Distributed way to find the spectrum ▶ At each node, eigenvalues correspond to the frequencies of ▶ Idea of Complex Power Iterations ▶ Diffusion algorithms, Monte Carlo techniques and Random Walk ▶ Connection with Quantum random walks
Our contribution spectral peaks and respective eigenvector components are the amplitudes at those points. implementation Jithin K. Sreedharan (INRIA, France) 6 / 30 ▶ Distributed way to find the spectrum ▶ At each node, eigenvalues correspond to the frequencies of ▶ Idea of Complex Power Iterations ▶ Diffusion algorithms, Monte Carlo techniques and Random Walk ▶ Connection with Quantum random walks ▶ Simulation on real-world networks of varying sizes
Complex Power Iterations
Central idea Harmonics of corresponds to eigenvalues. Details: from spectral theorem, 1 2 1 Jithin K. Sreedharan (INRIA, France) 8 / 30 ▶ Approach based on complex numbers. ▶ Let b t = e i A t b 0 , solution of ∂ ∂t b t = i Ab t .
Central idea Details: from spectral theorem, 1 2 1 Jithin K. Sreedharan (INRIA, France) 8 / 30 ▶ Approach based on complex numbers. ▶ Let b t = e i A t b 0 , solution of ∂ ∂t b t = i Ab t . Harmonics of b t corresponds to eigenvalues.
Central idea 1 Jithin K. Sreedharan (INRIA, France) 8 / 30 ▶ Approach based on complex numbers. ▶ Let b t = e i A t b 0 , solution of ∂ ∂t b t = i Ab t . Harmonics of b t corresponds to eigenvalues. ▶ Details: from spectral theorem, ∫ + ∞ n e i A t e − itθ dt ∑ δ λ j ( θ ) u j u ⊺ = j 2 π −∞ j = 1
Smoothing and a sample plot Idea of Gaussian smoothing: Jithin K. Sreedharan (INRIA, France) Sample plot at an arbitrary node 1 9 / 30 1 ∫ + ∞ n exp( − ( λ j − θ ) 2 e i A t b 0 e − t 2 v/ 2 e − itθ dt ∑ ) u j ( u ⊺ √ = j b 0 ) 2 π 2 v 2 πv −∞ j = 1
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