Almost-Linear-Time Algorithms for Markov Chains and New Spectral Primitives for Directed Graphs Richard Peng Georgia Tech Michael Cohen Jon Kelner John Peebles Aaron Sidford Adrian Vladu Anup B. Rao 1

OUtline Undirected vs. Directed Graphs Eulerian graphs Sparsification for iterative methods Full algorithm for Lx = b

The Laplacian Paradigm b c Graph Laplacians occur in Spectral graph theory Optimization Markov chains b 1 2 -1 -1 -1 1 0 -1 0 1 a a b c 1 c Directly related: Elliptic systems Lx=b Few iterations: Eigenvectors, Heat kernels Many iterations / modify algorithm Graph problems, Image processing

Directed vs. Undirected Is it really the undirected Laplacian paradigm? Undirected Directed Linear systems mlog1/2n nω (before this) Approx. max flow mlog41n m10/7 Approx shortest path in parallel (no(1) depth) mlog3n mn1/2 Spectral clustering Cheeger’s inequality Less clear, [Chung `05]

Pieces of the Laplacian Paradigm Graph sparsifiers Embeddings into trees Iterative Methods Before: only `counterexamples’ for general directed graphs Works on matrices

Directed Laplacian L = D – AT A: adjacency matrix D: diag(out-degrees) Diagonal: out degree Off-diagonal:ij entry: wji a b c 1 b 2 -1 0 -1 1 -2 -1 0 2 a 1 2 b a c 1 c Alternate definition: 0 column sums, LT1 = 0, non-negative off-diagonals

Our Results Input: n × n directed Laplacian L with m non-zeros with poly(n) condition number, vector b Output: vector x s.t. ║x - L+b ║2 < ε Runtime: O(m2O(√logn)log(1/ε)). Implications: similar times for stationary distributions, hitting times, escape times, commute time oracles for irreversible Markov chains Key ideas: Sparsification of directed graphs Preconditioned iterative methods for solving Lx = b

OUtline Undirected vs. Directed Graphs Eulerian graphs Sparsification for iterative methods Full algorithm for Lx = b

Graph Sparsification Undirected: Cut: G and H have similar cuts Spectral: for all vectors x, xTLGx ≈ xTLHx `counterexample’ for cut sparsification: complete bipartite graph Each edge can be isolated by a cut

Algebraic Issue If LG ≈ LH, they must have the same null space Undirected case: all 1s vector xb=1 1 1 xa=1 Null space of directed L: Perron-Frobenius theorem Can differ on graphs D-1 × stationary distribution 1 xc=1 xb=4 1 1 2 xa=2 1 xc=1

Stationary Distribution c Null space of L: Lx = 0 2 -1 0 -1 1 -2 -1 0 2 a b L = D - AT 1 b sb=0.4 c 1 Dx = ATx 2 a sa=0.4 1 c s=Dx: stationary distribution of random walk in A, ATD-1s = s sc=0.2

Eulerian LapaclaiNs In-degree = out-degree 1 AT1 = din = dout = A1 2 1 Null space: all 1s vector 1/2(L+LT): undirected Laplacian, UL Cuts: same amount in each direction 1 Goal: reduce strongly connected Laplacians to Eulerian Laplacians, and only work with Eulerian graphs

Eulerian Rescaling s: stationary, ATD-1s = s Set x = D-1s, and rescale L to Ldiag(x) 1 xb=0.4 0.2 0.4 1 0.2 2 0.2 xa=0.2 1 xc=0.1 Ldiag(x)1 = Lx = 0, Eulerian! [CKPPSV`16]: rescale via solving polylog Eulerian Laplacians, will only work with Eulerian Laplacians from this point on

Previous Works ON Eulerian Graphs 1 [Chung `05]: spectral gap of (L+LT) related to its expansion 2 1 [EMPS`16]: cut sparsifiers and approximate maximum flow on balanced graphs 1 Can sparsify Eulerian case Eulerian case = general case Mincut   Lx = b Solve more than Eulerian graphs? Incorporate into spectral algorithms?

OUtline Undirected vs. Directed Graphs Eulerian graphs Sparsification for iterative methods Full algorithm for Lx = b

Preconditioned Iterative Methods Need: algebraic definition of ≈ that interacts well with iterative methods Solve linear systems in LGx=b by solving a sequence of problems in LH ≈ LG: x’  x + LH-1(b – LGx) Can check: If G = H, done in 1 step fixed point: LG-1b `driver’ for nearly-linear time algorithms for: Solving linear systems Approx maximum flow

Candidate: 2-norm of difference Candidate: ║(L – L’ )x║2 ≤ ε ║ L x║2 Behaves like ≈: Symmetric, triangle inequality, invertible Undirected L: degenerates into L2 ≈ L’2, unfriendly to perturbations 1 1 1 1 1 1 1 ≈ 1 2 1 2 1 2 1

Fix: Divide by Undirected Laplacian L ε-approximates L’, if ║UL+1/2(L – L’) UL+1/2║2+ ≤ ε UL=1/2(L+LT): undirected Laplacian Symmetric Composable / triangle inequality For undirected L, same as spectral approx. Generalizes directed expander from [Chung `05] If L O(1)-approximates L’, then can solve systems in L to error ε using O(log(1/ ε)) solves in L’

Generating Directed Approximations Hard case: forward cycle and undirected cycle are off by factor of n2 ≈ [Tropp `12]: Expanders in U still `easy’: random graph works

Expander Parititioning All but O(nlogn) edges of UL are contained in some expander ║UL+1/2(L – L’) UL+1/2║2+ ≤ ε decomposable onto pieces Algorithm (similar to [ST`04]): Partition undirected UL For each expander Sample to error ε/O(logn) Fix degree Result: O(nlogO(1)nε-2) sized ε-approximations in O(mlogO(1)n) time, and parallelizable

OUtline Undirected vs. Directed Graphs Eulerian graphs Sparsification for iterative methods Full algorithm for Lx = b

Solving Linear Systems Simplification: L Eulerian, L = I - A for a random walk matrix A: ║A║2 < 1 Iterative methods in 1 line: L-1 = (I – A)-1 = I + A + A2 + A3 +… If ║A║2 ≤ ρ, (1 - ρ)-1 terms well approximates (I – A)-1 b Ab A2b Adiameterb Graph interpretation: each term  1 step walk Need Ω(diameter) steps

Fewer Terms via. Squaring (I – A)-1 = (I - A2)-1(I + A) (I – A)-1 = (I + A)(I + A2)…(I + A2i)… A: one step transition of random walk A2i: 2i step transition of random walk A2i mixes for i = Ω(log(κ)), κ: condition number, can make poly(n) I 23

Sparsified Squaring I - A1 ≈ε I – A2 I – A2 ≈ε I – A12 … I – Ai ≈ε I – Ai-12 I - Alogn ≈ I I - A0 (I – Ai)-1 = (I - Ai2)-1(I + Ai) ≈(I - Ai+1)-1(I + Ai) ≈ : approximations via sparsifiers Algorithms involving repeated squaring NC algorithm for shortest path [Reingold `05][Rozenman-Vadhan `05] Logspace connectivity Multiscale methods [P-Spielman `14] Solving Lx = b I - Ad≈ I

Solver I - A0 (I – Ai)-1 ≈(I - Ai+1)-1(I + Ai) Can turn Zi, a preconditioner for Lj, into a preconditioner for Li via: Zi Zj (I + Aj-1) (I + Aj-2)… (I + A1) Error: if Zj is ε-approx. pseudoinverse of Lj, then Zi is exp(O(j – i)) ε-approx. pseudoinverse of Li I - Ad≈ I Directly convert Zd into Z0: need ε < 1/poly(n) 

Fix: Recursive Iterative Method Only work up Δ steps at a time Reduce error to 2-O(Δ) via iterative refinement: O(Δ) branching factor Δ Need ε = 2-O(Δ) in sparsifier O(Δ) branching factor, every Δ layers Overhead: 2(O(Δ))δd/Δ ≤2O(Δ+dlogd/Δ) Optimized at Δ = d1/2 = log1/2n Total: O(m 2O(√logn)log(1/ε)) Δ

Ongoing / Future work Nearly-linear time? Sparse LU decomposition? Applications of directed Laplacian solvers? Formalize `sparsify w.r.t. a problem’? Extensions / generalizations / applications of sparsification of directed graphs?