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Layout. Contiguousness grid and LaplacianIntuition, phantom chart drawingPhysical intuitionIsomorphism testingRandom walksGraph Partitioning and clusteringDistributions of eigenvalues and compressionComputation. What I\'m Skipping. Framework tree theorem.Most of mathematical diagram theory.Special charts (e.g. Cayley graphs).Connections to codes and designs.Lots of work by theorists.Expanders..

Phantom Graph Theory and its Applications Daniel A. Spielman Dept. of Computer Science Program in Applied Mathematics Yale Unviersity

Outline Adjacency network and Laplacian Intuition, ghastly diagram drawing Physical instinct Isomorphism testing Random strolls Graph Partitioning and grouping Distributions of eigenvalues and pressure Computation

What 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.

3 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

3 1 2 4 Example - 1 - 0.618 1

3 1 2 4 Example - 1 - 0.618 1 - 0.618 1 0.382

3 1 2 4 Example: invariant under re-naming - 1 - 0.618 1

3 2 1 4 Example: invariant under re-naming - 1 - 0.618 1

Operators and Quadratic Forms View of An as an administrator: View of An as quadratic shape: if and afterward

3 1 2 4 Laplacian: regular quadratic shape on charts where D is corner to corner framework of degrees

Laplacian: 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.

unimportant arrangement: So, require Solution Atkins, Boman, Hendrickson '97: Gives redress installing for charts like Embedding diagram in line (Hall '70) delineate

(here ) Courant-Fischer meaning of eigvals/vecs

So, require trifling arrangement: decline arrangement: Embedding chart in plane (Hall '70) outline Also require Solution up to turn

A Graph

Drawing of the diagram utilizing v 2 , v 3 Plot vertex i at

Spectral attracting of Streets Rome

Spectral drawing of Erdos diagram: edge between co-creators of papers

Dodecahedron Best implanted by initial three eigenvectors

Spectral 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.

Tutte '63 implanting of a diagram. Settle outside face. Edges - > springs. Vertex at focal point of mass of nbrs. 3-associated - > get planar installing

Fundamental modes: string with settled closures

Fundamental modes: string with free finishes

Eigenvectors of way diagram 1: 2: 3: 4: 17:

Drawing of the chart utilizing v 3 , v 4 Plot vertex i at

Spectral diagram shading from high eigenvectors Embedding of dodecahedron by 19 th and 20 th eigvecs. Shading 3-colorable arbitrary charts [Alon-Kahale '97]

Spectral 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)

Isomorphism 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.

Isomorphism testing 2 = 3 , eigvecs decided up to turn

Isomorphism testing Distinguish by standard in installing 2 = 3 , eigvecs decided up to pivot

Isomorphism 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.

Isomorphism 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.

Random Walks

Random strolls and PageRank Adjacency network of coordinated chart: Walk move framework: Walk dispersion at time t: PageRank vector p: Eigenvector of Eigenvalue 1

Random 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?

Kleinberg 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)

Random strolls on Undirected Graphs Trivial PageRank Vector: Not symmetric, however similiar to symmetrized walk lattice W and S have same eigvals,

Random 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

Normalized Laplacian [Chung] If consider 1- n-1 ought to take a gander at Relationship to cuts:

Cheeger's Inequality (Jerrum-Sinclair '89) (Alon-Milman '85, Diaconis-Stroock '91)

Cheeger's Inequality (Jerrum-Sinclair '89) (Alon-Milman '85, Diaconis-Stroock '91) Can locate the cut by taking a gander at for some t

Only need surmised eigenvector (Mihail '89) Can locate the cut by taking a gander at for some t Guarantee Lanczos period.

This way, is an unwinding [see Hagen-Kahng '92]. Standardized Cut Alternative meaning of conductance [Lovasz '96 (?)] Equivalent to Normalized Cut [Shi-Malik '00]

Spectral Image Segmentation (Shi-Malik '00)

Spectral Image Segmentation (Shi-Malik '00)

Spectral Image Segmentation (Shi-Malik '00)

Spectral Image Segmentation (Shi-Malik '00)

Spectral Image Segmentation (Shi-Malik '00) edge weight

The second eigenvector

Second Eigenvector's sparsest cut

Third Eigenvector

Fourth Eigenvector

Perspective 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?

Improvement by Miller and Tolliver '06

Improvement by Miller and Tolliver '06 Idea: re-weight (i,j) by Actually, re-weight by Prove: as emphasize 2 - > 0, get 2 parts

One 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

} 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

Other 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]

p 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]

Distribution of eigenvalues of Random Graphs Histogram of eigvals of arbitrary 40-by-40 nearness networks

Distribution of eigenvalues of Random Graphs Histogram of eigvals of irregular 40-by-40 contiguousness lattices Predicted bend: Wigner's Semi-Circle Law

Eigenvalue disseminations Eigenvalues of walk grid of 50-by-50 lattice chart Number more prominent than 1- corresponding to

Eigenvalue 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

Compression 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

edge weight Discretizing Manifold

Number more noteworthy than 1- relative to Eigenvalue dispersions Eigenvalues of way diagram on 10k hubs

Theorem: 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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