# Otherworldly Graph Theory and its Applications Daniel A. Spielman Dept. of Computer Science Program in Applied Mathema

0
0
1493 days ago, 699 views
PowerPoint PPT Presentation
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..

### Presentation Transcript

Slide 1

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

Slide 2

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

Slide 3

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.

Slide 4

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

Slide 5

3 1 2 4 Example - 1 - 0.618 1

Slide 6

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

Slide 7

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

Slide 8

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

Slide 9

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

Slide 10

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

Slide 11

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.

Slide 12

unimportant 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 14

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

Slide 15

A Graph

Slide 16

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

Slide 18

Spectral attracting of Streets Rome

Slide 19

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

Slide 20

Dodecahedron Best implanted by initial three eigenvectors

Slide 21

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.

Slide 22

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

Slide 23

Fundamental modes: string with settled closures

Slide 24

Fundamental modes: string with free finishes

Slide 25

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

Slide 26

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

Slide 27

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

Slide 28

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)

Slide 29

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.

Slide 30

Isomorphism testing  2 =  3 , eigvecs decided up to turn

Slide 31

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

Slide 32

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.

Slide 33

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.

Slide 34

Random Walks

Slide 35

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

Slide 36

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?

Slide 37

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)

Slide 38

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

Slide 39

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

Slide 40

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

Slide 41

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

Slide 42

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

Slide 43

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

Slide 44

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

Slide 45

Spectral Image Segmentation (Shi-Malik '00)

Slide 46

Spectral Image Segmentation (Shi-Malik '00)

Slide 47

Spectral Image Segmentation (Shi-Malik '00)

Slide 48

Spectral Image Segmentation (Shi-Malik '00)

Slide 49

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

Slide 50

The second eigenvector

Slide 51

Second Eigenvector's sparsest cut

Slide 52

Third Eigenvector

Slide 53

Fourth Eigenvector

Slide 54

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?

Slide 55

Improvement by Miller and Tolliver '06

Slide 56

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

Slide 57

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

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 59

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]

Slide 60

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]

Slide 61

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

Slide 62

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

Slide 63

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

Slide 64

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 

Slide 65

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

Slide 66

edge weight Discretizing Manifold

Slide 67

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

Slide 68

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

SPONSORS

No comments found.

SPONSORS

SPONSORS