Phantom Graph Theory and its Applications Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity
Slide 2Outline Adjacency network and Laplacian Intuition, ghastly diagram drawing Physical instinct Isomorphism testing Random strolls Graph Partitioning and grouping Distributions of eigenvalues and pressure Computation
Slide 3What I'm Skipping Matrix-tree hypothesis. A large portion of arithmetical chart hypothesis. Extraordinary charts (e.g. Cayley diagrams). Associations with codes and outlines. Loads of work by scholars. Expanders.
Slide 43 1 2 4 The Adjacency Matrix is eigenvalue and v is eigenvector if Think of , or far and away superior Symmetric - > n genuine eigenvalues and genuine eigenvectors frame orthonormal premise
Slide 53 1 2 4 Example - 1 - 0.618 1
Slide 63 1 2 4 Example - 1 - 0.618 1 - 0.618 1 0.382
Slide 73 1 2 4 Example: invariant under re-naming - 1 - 0.618 1
Slide 83 2 1 4 Example: invariant under re-naming - 1 - 0.618 1
Slide 9Operators and Quadratic Forms View of An as an administrator: View of An as quadratic shape: if and afterward
Slide 103 1 2 4 Laplacian: regular quadratic shape on charts where D is corner to corner framework of degrees
Slide 11Laplacian: quick realities along these lines, zero is an eigenvalue If k associated segments, Fiedler ('73) called "logarithmic network of a diagram" The further from 0, the more associated.
Slide 12unimportant arrangement: So, require Solution Atkins, Boman, Hendrickson '97: Gives redress installing for charts like Embedding diagram in line (Hall '70) delineate
Slide 13(here ) Courant-Fischer meaning of eigvals/vecs
Slide 14So, require trifling arrangement: decline arrangement: Embedding chart in plane (Hall '70) outline Also require Solution up to turn
Slide 15A Graph
Slide 16Drawing of the diagram utilizing v 2 , v 3 Plot vertex i at
Slide 18Spectral attracting of Streets Rome
Slide 19Spectral drawing of Erdos diagram: edge between co-creators of papers
Slide 20Dodecahedron Best implanted by initial three eigenvectors
Slide 21Spectral diagram drawing: Tutte legitimization Condition for eigenvector Gives for all i little says x(i) close normal of neighbors Tutte '63: If settle outside face, and let each other vertex be normal of neighbors, get planar inserting of planar chart.
Slide 22Tutte '63 implanting of a diagram. Settle outside face. Edges - > springs. Vertex at focal point of mass of nbrs. 3-associated - > get planar installing
Slide 23Fundamental modes: string with settled closures
Slide 24Fundamental modes: string with free finishes
Slide 25Eigenvectors of way diagram 1: 2: 3: 4: 17:
Slide 26Drawing of the chart utilizing v 3 , v 4 Plot vertex i at
Slide 27Spectral diagram shading from high eigenvectors Embedding of dodecahedron by 19 th and 20 th eigvecs. Shading 3-colorable arbitrary charts [Alon-Kahale '97]
Slide 28Spectral diagram drawing: FEM avocation If apply limited component technique to unravel Laplace's condition in the plane with a Delaunay triangulation Would get chart Laplacian, yet with a few weights on edges Fundamental arrangements are x and y facilitates (see Strang's Introduction to Applied Mathematics)
Slide 29Isomorphism testing 1. distinctive eigenvalues - > non-isomorphic 2. On the off chance that every vertex unmistakable in unearthly inserting, simply need to arrange embeddings. Each eigvec decided up to sign.
Slide 30Isomorphism testing 2 = 3 , eigvecs decided up to turn
Slide 31Isomorphism testing Distinguish by standard in installing 2 = 3 , eigvecs decided up to pivot
Slide 32Isomorphism testing: challenges 1. Numerous vertices can guide to same place in otherworldly implanting, if just utilize few eigenvectors. 2. On the off chance that i has a high dimensional space, eigvecs just decided up to premise revolutions Ex.: Strongly consistent diagrams with just 3 eigenvalues, of multiplicities 1, (n-1)/2 and (n-1)/2 3. A few sets have an exponential number of isomorphisms.
Slide 33Isomorphism testing: achievement [Babai-Grigoryev-Mount '82] If every eigenvalue has variety O(1), can test in polynomial time. Thoughts: Partition vertices into classes by standards in embeddings. Refine allotments utilizing different parcels. Utilize vertex classes to part eigenspaces. Utilize computational gathering hypothesis to circuit data, and create portrayal of all isomorphisms.
Slide 34Random Walks
Slide 35Random strolls and PageRank Adjacency network of coordinated chart: Walk move framework: Walk dispersion at time t: PageRank vector p: Eigenvector of Eigenvalue 1
Slide 36Random strolls and PageRank vector p: Linear variable based math issues: W is not symmetric, not like symmetric, does not really have n eigenvalues If no hubs of out-degree 0, Perron-Frobenius Theorem: Guarantees a special, positive eigevec p of eigenvalue 1. Is there a hypothetically intriguing ghastly hypothesis?
Slide 37Kleinberg and the particular vectors Consider eigenvectors of biggest eigenvalue of and Are left and right solitary estimations of A. Continuously exist. Ordinarily, a more valuable hypothesis than eigenvectors, when not symmetric. (see Strang's Intro. to Linear Algebra)
Slide 38Random strolls on Undirected Graphs Trivial PageRank Vector: Not symmetric, however similiar to symmetrized walk lattice W and S have same eigvals,
Slide 39Random walk merges at rate 1/1- n-1 For sluggish irregular walk (stay put with prob ½): Where is the steady appropriation For symmetric S
Slide 40Normalized Laplacian [Chung] If consider 1- n-1 ought to take a gander at Relationship to cuts:
Slide 41Cheeger's Inequality (Jerrum-Sinclair '89) (Alon-Milman '85, Diaconis-Stroock '91)
Slide 42Cheeger's Inequality (Jerrum-Sinclair '89) (Alon-Milman '85, Diaconis-Stroock '91) Can locate the cut by taking a gander at for some t
Slide 43Only need surmised eigenvector (Mihail '89) Can locate the cut by taking a gander at for some t Guarantee Lanczos period.
Slide 44This way, is an unwinding [see Hagen-Kahng '92]. Standardized Cut Alternative meaning of conductance [Lovasz '96 (?)] Equivalent to Normalized Cut [Shi-Malik '00]
Slide 45Spectral Image Segmentation (Shi-Malik '00)
Slide 46Spectral Image Segmentation (Shi-Malik '00)
Slide 47Spectral Image Segmentation (Shi-Malik '00)
Slide 48Spectral Image Segmentation (Shi-Malik '00)
Slide 49Spectral Image Segmentation (Shi-Malik '00) edge weight
Slide 50The second eigenvector
Slide 51Second Eigenvector's sparsest cut
Slide 52Third Eigenvector
Slide 53Fourth Eigenvector
Slide 54Perspective on Spectral Image Segmentation Ignoring a considerable measure we think about pictures. On non-picture information, gives great instinct. Can we intertwine with what we think about pictures? For the most part, would we be able to intertwine with other information? Shouldn't something be said about better cut calculations?
Slide 55Improvement by Miller and Tolliver '06
Slide 56Improvement by Miller and Tolliver '06 Idea: re-weight (i,j) by Actually, re-weight by Prove: as emphasize 2 - > 0, get 2 parts
Slide 57One way to deal with melding: Dirichlet Eigenvalues Fixing limit qualities to zero [Chung-Langlands '96] Fixing limit qualities to non-zero. [Grady '06] Dominant mode by settling direct condition: figuring electrical stream in resistor organize
Slide 58} p q A } q p B } B An Analysis of Spectral Partitioning Finite Element Meshes (eigvals right) [S-Teng '07] Planted allotments (eigvecs right) [McSherry '01] Prob An edge = p B-B edge = p A-B edge = q < p
Slide 59Other planted issues Finding cn 1/2 inner circle in irregular chart [Alon-Krivelevich-Sudakov '98] Color arbitrary inadequate k-colorable diagram [Alon-Kahale '97] Asymmetric square structure (LSI and HITS) [Azar-Fiat-Karlin-McSherry-Saia '01] Partitioning with broadly differing degrees [Dasgupta-Hopcroft-McSherry '04]
Slide 60p q p Planted issue investigation Sampled An as irritation of Small bothers don't change eigenvalues excessively. Eigenvectors stable as well, if all around isolated from others. Comprehend eigenvalues of irregular frameworks [Furedi-Komlos '81, Alon-Krivelevich-Vu '01, Vu '05]
Slide 61Distribution of eigenvalues of Random Graphs Histogram of eigvals of arbitrary 40-by-40 nearness networks
Slide 62Distribution of eigenvalues of Random Graphs Histogram of eigvals of irregular 40-by-40 contiguousness lattices Predicted bend: Wigner's Semi-Circle Law
Slide 63Eigenvalue disseminations Eigenvalues of walk grid of 50-by-50 lattice chart Number more prominent than 1- corresponding to
Slide 64Eigenvalue circulations Eigenvalues of walk lattice of 50-by-50 matrix diagram All these go to zero when take powers Number more noteworthy than 1- relative to
Slide 65Compression of forces of diagrams [ Coifman, Lafon, Lee, Maggioni, Nadler, Warner, Zucker '05] If most eigenvalues of An and W limited from 1. Most eigenvalues of A t little. Can inexact A t by low-rank framework. Manufacture wavelets bases on diagrams. Understand straight conditions and register eigenvectors. Make thorough by taking chart from discretization of complex
Slide 66edge weight Discretizing Manifold
Slide 67Number more noteworthy than 1- relative to Eigenvalue dispersions Eigenvalues of way diagram on 10k hubs
Slide 68Theorem: Eigenvalue disseminations Theorem: If limited degree, number eigenvalues more prominent than 1- is Proof: can pick vertices to crumple with the goal that conductance gets to be in any event (like including an expander those hubs). New diagram h
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