1. 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

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

3. The Laplacian Paradigmb c
Graph Laplacians occur in
Spectral graph theory
Optimization
Markov chains b 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

4. 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]

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

6. 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 a 1 2 b a c 1 c
Alternate definition: 0 column sums, LT1 = 0, non-negative off-diagonals

7. 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

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

9. 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

10. 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

11. Stationary Distributionc
Null space of L: Lx = 0 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

12. 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

13. 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

14. Previous Works ON Eulerian Graphs1
[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?

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

16. 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

17. 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

18. 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’

19. 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

20. 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

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

22. 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

23. 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

24. 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

25. 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) 

26. 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

27. 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?