Last Time: Summary. Hypercube Eigenvalues of hypercube. V = 0,1 d (x,y) E when x and y differ in one bit. Graph G = (V ,E), assume regular graph of degree d. { } ∈ d d 1 V = 2 E = d2 − . E(S,V S) ( ) = | − | ( ) = ( ) | | | | Edge Expansion. h S d min S , V S , h G minS h S Anyone see any symmetry? | | | − | M = A/d adjacency matrix, A Coordinate cuts. +1 on one side, -1 on other. Eigenvector: a vector v where Mv = λv (Mv) = (1 2/d)v . i − i Spectral theorem: Eigenvectors form basis: v1,...,vn. Eigenvalue 1 2/d. d Eigenvectors. Why orthogonal? − x = α v + α v + αnvn. Mx = α λ v + α λ v + αnλnvn 1 1 2 2 ··· 1 1 1 2 2 2 ··· Next eigenvectors? Good cuts? “Coordinate cut”: d of them. Highest eigenvalue: λ = 1. Proof: Plug in 1. d 1 1 2 − 1 Edge expansion: d 1 = Delete edges in two dimensions. Second Eigenvalue: λ2 < 1 if connected. Proof: v2 is not v1. d2 − d Ball cut: All nodes within d/2 of node, say 00 0. Four subcubes: bipartite. Color 1 Eigenvalue gap: µ = λ1 λ2. ··· ± µ − Vertex cut size: d bit strings with d/2 1’s. d Cheeger: 2 h(G) = 2µ d/2 Eigenvalue: 1 4/d. 2 eigenvectors. ≤ ≤ d − 2 d Proof of LHI: Plug in “cut”p vector,x, into Rayleigh Quotient. √
≈ d Eigenvalues: 1 2k/d.
k eigenvectors. xt Mx 1 − µ = 1 maxx 1 t . Vertex expansion: . − ⊥ x x ≈ √d
This expression ’counts’ edges in cut ’x’ plus scales by volume. Edge expansion: d/2 edges to next level. 1 Yields h(S). ≈ 2√d Worse by a factor of √d

T Back to Cheeger. Cycle Find x 1 with Rayleigh quotient, x Mx close to 1. ⊥ xT x

Coordinate Cuts: i n/4 if i n/2 Eigenvalue 1 2/d. d Eigenvectors. xi = − ≤ − (3n/4 i if i > n/2 µ 1 λ2 Tight example for Other side of Cheeger? − = − h(G) 2(1 λ2) = 2µ 2 2 ≤ ≤ − µ 1 λ = − 2 h(G) 2(1 λ ) = 2µ Hit with M. For hypercube: h(G) =p 1 λ λ = p2/d. 2 2 ≤ ≤ − 2 d 1 − 2 Left hand side is tight. Cycle on n nodes. p p n/4 + 1/2 if i = 1,n Note: hamming weight vector also in first eigenspace. Will show other side of Cheeger is tight. − (Mx)i = n/4 1 if i = n/2 Edge expansion:Cut in half. − x otherwise Lose “names” in hypercube, find coordinate cut? S = n/2, E(S,S) = 2 i | | | | Find coordinate cut? h(G) = 2 .  → n xT Mx = xT x(1 O( 1 )) 1 O( 1 ) n2 λ2 n2 Eigenvector v maps to line. Show eigenvalue gap µ 1 . → − → ≥ − ≤ n2 1 Cut along line. µ = λ1 λ2 = O( 2 ) T − n Find x 1 with Rayleigh quotient, x Mx close to 1. Eigenvector algorithm gets a linear combination of coordinate cuts. xT x 2 ⊥ h(G) = n = Θ(√µ) µ 1 λ Something like ball cut. = − 2 h(G) 2(1 λ ) = 2µ 2 2 ≤ ≤ − 2 Find coordinate cut? Tight example for upperp bound for Cheeger.p Eigenvalues of cycle? Random Walk.

p - probability distribution. Probability distrubtion after choose a random neighbor. Eigenvalues: cos 2πk . n Mp. 2πki xi = cos n Converge to uniform distribution. Eigenvalues, random walks, volume estimation, counting. 2πk(i+1) 2πk(i 1) 2πk 2πki t (Mx)i = cos n + cos n− = 2cos n cos n Power method: M x goes to highest eigenvector.         t t 2πk M x = a1λ1v1 + a2λ2v2 + Eigenvalue: cos n . ··· λ λ - rate of convergence. Eigenvalues: 1 − 2 vibration modes of system. Ω(n2) steps to get close to uniform. Fourier basis. Start at node 0, probability distribution, [1,0,0, ,0]. ··· Takes Ω(n2) to get n steps away. Recall drunken sailor.

Sampling. Convex Bodies. Convex Body Graph.

S [k]n is set of grid points inside Convex Body. S [k]n is grid points inside Convex Body. ⊂ ⊂ S Ex: Numerically integrate convex function in d dimensions. Sample Space: . Compute ∑i vi Vol(f (x) > vi ) where vi = iδ. Graph on grid points inside P or on Sample Space. Example: P defined by set of linear inequalities. One neighbor in each direction for each dimension Or other “membership oracle” for P (if neighbor is inside P.) Sampling: Random element of subset S 0,1 n or 0,...,k k . Degree: 2d. ⊂ { } { } S is set of grid points inside Convex Body. Related Problem: Approximate S within factor of 1 + ε. | | Grid points that satisfy linear inequalities. How big is graph? Big! Random walk to do both for some interesting sets S. or “other” membership oracle. So big it ..it INSERT JOKE HERE. Choose a uniformly random elt? O(k n) if coordinates in [k]. Easy to choose randomly from [k]n which is big. That’s a big graph! For convex body? How to find a random node? [k]n P Choose random point in and check if in . Start at a grid point, and take a (random) walk. Works. But P could be exponentially small compared to [k]n . When close to uniform distribution...have a sample point. | | Takes a long time to even find a point in P. How long does this take? More later. But remember power method...which finds first eigenvector. Spanning Trees. Spin systems. Sampling structures and the BIG GRAPH Each element of S may have associated weight. Sample element proportional to weight. Problem: How many? Example? Another Problem: find a random one. 2 or 3 dimensional grid of particles. Particle State 1. System State 1,+1 n. Algorithm: ± {− } Start with spanning tree. E = Sampling Algorithms Random walk on BIG GRAPH. Small degree. Energy on local interactions: ∑(i,j) σi σj . ≡ Repeat: − Vertices Neighbors Degree (ish) “Ferromagnetic regime”: same spin is good. Grid points in convex body. Change one dimension 2d Swap a random nontree edge with a random tree edge. Spanning Trees. Change two edges. V 2 neighbors per node Spin States. Change one spin ≤ | |O(n) neighbors. Gibbs distribution ∝ e E/kT . How long? − Physical properties from Gibbs distribution. Sample space graph (BIG GRAPH) of spanning trees. Node for each tree. Metropolis Algorithm: Neighboring trees differ in two edges. At x, generate y with a single random flip. Go to y with probability min(1,w(y)/w(x)) Algorithm is random walk on BIG GRAPH (sample space graph.) Random walk in sample space graph (BIG GRAPH ALERT) (not random walk in 2d grid of particles.) Markov Chain on statespace of system.

Analyzing random walks on graph. Fix-it-up chappie! Rapid mixing, volume, and surface area.. / Start at vertex, go to random neighbor. “Lazy” random walk: With probability 1 2 stay at current vertex. I+M Recall volume of convex body. For d-regular graph: eventually uniform. Evolution Matrix: 2 if not bipartite. Odd /even step! 1+λi Grid graph on grid points inside convex body. Eigenvalues: 2 µ How to analyse? 1 (I + M)v = 1 (v + λ v ) = 1+λi v Recall Cheeger: h(G) 2µ. 2 i 2 i i i 2 i 2 ≤ ≤ Random Walk Matrix: M. Eigenvalues in interval [0,1]. Lower bound expansion lowerp bounds on spectral gap µ → M - normalized adjacency matrix. 1 λ2 µ Upper bound mixing time. Spectral gap: −2 = 2 . → Symmetric, ∑j M[i,j] = 1. h(G) Surface Area 1 1 Volume M[i,j]- probability of going to j from i. Uniform distribution: π = [ N ,..., N ] ≈ Distance to uniform: d1(vt ,π) = ∑i (vt )i πi Isoperimetric inequality. Probability distribution at time t: vt . | − | 1 ( ( ), ( )) = “Rapidly mixing”: d1(vt ,π) ε in poly(logN,log ) time. min Vol S Vol S vt+1 Mvt Each node is average over neighbors. ≤ ε Voln 1(S,S) diam(P) When is chain rapidly mixing? − ≥ Evolution? Random walk starts at 1, distribution e1 = [1,0,...,0]. 2 Edges ∝ surface area, Assume Diam(P) p0(n) t 1 t Another measure: d2(vt ,π) = ∑i ((vt )i πi ) . ≤ M v1 = N v1 + ∑i>1 λi αi vi . − h(G) 1/p0(n) 1 1 Note: d1(vt ,π) √Nd2(vt ,π) → ≥ 2 v1 = [ N ,..., N ] Uniform distribution. ≤ 1 µ > 1/2p0(n) → n – “size” of vertex, µ for poly p(n), t = O(p(n)logN). → 2 ≥ p(n) O(p0(n) logN) convergence for Markov chain on BIG GRAPH. Doh! What if bipartite? 2t → t 2 (1+λ2) 1 2t 1 Rapidly mixing chain: Negative eigenvalues of value -1: (+1, 1) on two sides. d2(vt ,π) = A e1 π (1 ) − | − | ≤ 2 ≤ − 2p(n) ≤ poly(N) → Side question: Why the same size? Assumed regular graph. Rapidly mixing with big ( 1 ) spectral gap. ≥ p(n) Khachiyan’s algorithm for counting partial orders. Summary.

Given partial order on x1,...,xn.

Sample from uniform distribution over total orders. x1 > x2 x1 > x3 Start at an ordering. x1 > x2

Swap random pair and go if consistent with partial order. (0,0) > x2 x3 Eigenvectors for hypercubes. Rapidly mixing chain? 1 Each order takes n! volume. Tight example for LHI of Cheeger. Eigenvectors for cycle. Map into d-dimensional unit cube. Number of orders volume of intersection of partial order relations. Tight example for RHI of Cheeger. x < x corresponds to halfspace (one side of hyperplane) of cube. ≡ i j Diameter: O(√n) Random Walks and Sampling. “dimension i = dimension j” total order is intersection of n halfspaces. Isoperimetry: Eigenvectors, Isoperimetry of Volume, Mixing. 1 E(S,S) S each of volume: . Vol (S,S) = | | Partial Order Application. n! n 1 (n 1)! n!√n since each total order is disjoint − − ≥ Edge Expansion: the degree d is O(n2), and together cover cube. E(S,S) 1 h(S) = | | Mixes in time O(n7 logN) = O(n8 logn). d S ≥ n7/2 Do the polynomial| | dance!!! x1 > x2 x1 > x3 x1 > x2

(0,0) x2 > x3